3.32.36 \(\int \frac {(d+c x^2) (a x+\sqrt {-b+a^2 x^2})^{5/4}}{x (-b+a^2 x^2)^{5/2}} \, dx\) [3136]

Optimal. Leaf size=876 \[ \frac {\left (-97 b^2 c-a^2 b d+45 a^2 b c x^2-51 a^4 d x^2\right ) \sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}{96 a^2 b^{15/8} \left (-\sqrt {b}+a x\right ) \left (\sqrt {b}+a x\right )}+\frac {\sqrt {-b+a^2 x^2} \left (13 b^2 c x-83 a^2 b d x-45 a^2 b c x^3+51 a^4 d x^3\right ) \sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}{96 a b^{15/8} \left (-\sqrt {b}+a x\right )^2 \left (\sqrt {b}+a x\right )^2}+\frac {5 \left (-3 b c+29 a^2 d\right ) \text {ArcTan}\left (\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{64 a^2 b^{15/8}}-\frac {\sqrt {2-\sqrt {2}} d \text {ArcTan}\left (\frac {-\frac {1}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}{\sqrt {2-\sqrt {2}}}}{\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}\right )}{b^{15/8}}-\frac {\sqrt {2+\sqrt {2}} d \text {ArcTan}\left (\frac {-\frac {1}{\sqrt {2+\sqrt {2}}}+\frac {\sqrt {\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}{\sqrt {2+\sqrt {2}}}}{\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}\right )}{b^{15/8}}+\frac {5 \left (-3 b c+29 a^2 d\right ) \tanh ^{-1}\left (\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{64 a^2 b^{15/8}}-\frac {5 (-1)^{3/4} \left (-3 b c+29 a^2 d\right ) \tanh ^{-1}\left (\sqrt [4]{-1} \sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{64 a^2 b^{15/8}}-\frac {5 \sqrt [4]{-1} \left (-3 b c+29 a^2 d\right ) \tanh ^{-1}\left ((-1)^{3/4} \sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{64 a^2 b^{15/8}}-\frac {\sqrt {2-\sqrt {2}} d \tanh ^{-1}\left (\frac {\frac {1}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}{\sqrt {2-\sqrt {2}}}}{\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}\right )}{b^{15/8}}-\frac {\sqrt {2+\sqrt {2}} d \tanh ^{-1}\left (\frac {\frac {1}{\sqrt {2+\sqrt {2}}}+\frac {\sqrt {\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}{\sqrt {2+\sqrt {2}}}}{\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}}\right )}{b^{15/8}} \]

[Out]

1/96*(-51*a^4*d*x^2+45*a^2*b*c*x^2-a^2*b*d-97*b^2*c)*((a*x+(a^2*x^2-b)^(1/2))/b^(1/2))^(1/4)/a^2/b^(15/8)/(-b^
(1/2)+a*x)/(b^(1/2)+a*x)+1/96*(a^2*x^2-b)^(1/2)*(51*a^4*d*x^3-45*a^2*b*c*x^3-83*a^2*b*d*x+13*b^2*c*x)*((a*x+(a
^2*x^2-b)^(1/2))/b^(1/2))^(1/4)/a/b^(15/8)/(-b^(1/2)+a*x)^2/(b^(1/2)+a*x)^2+5/64*(29*a^2*d-3*b*c)*arctan(((a*x
+(a^2*x^2-b)^(1/2))/b^(1/2))^(1/4))/a^2/b^(15/8)-(2-2^(1/2))^(1/2)*d*arctan((-1/(2-2^(1/2))^(1/2)+((a*x+(a^2*x
^2-b)^(1/2))/b^(1/2))^(1/2)/(2-2^(1/2))^(1/2))/((a*x+(a^2*x^2-b)^(1/2))/b^(1/2))^(1/4))/b^(15/8)-(2+2^(1/2))^(
1/2)*d*arctan((-1/(2+2^(1/2))^(1/2)+((a*x+(a^2*x^2-b)^(1/2))/b^(1/2))^(1/2)/(2+2^(1/2))^(1/2))/((a*x+(a^2*x^2-
b)^(1/2))/b^(1/2))^(1/4))/b^(15/8)+5/64*(29*a^2*d-3*b*c)*arctanh(((a*x+(a^2*x^2-b)^(1/2))/b^(1/2))^(1/4))/a^2/
b^(15/8)-5/64*(-1)^(3/4)*(29*a^2*d-3*b*c)*arctanh((-1)^(1/4)*((a*x+(a^2*x^2-b)^(1/2))/b^(1/2))^(1/4))/a^2/b^(1
5/8)-5/64*(-1)^(1/4)*(29*a^2*d-3*b*c)*arctanh((-1)^(3/4)*((a*x+(a^2*x^2-b)^(1/2))/b^(1/2))^(1/4))/a^2/b^(15/8)
-(2-2^(1/2))^(1/2)*d*arctanh((1/(2-2^(1/2))^(1/2)+((a*x+(a^2*x^2-b)^(1/2))/b^(1/2))^(1/2)/(2-2^(1/2))^(1/2))/(
(a*x+(a^2*x^2-b)^(1/2))/b^(1/2))^(1/4))/b^(15/8)-(2+2^(1/2))^(1/2)*d*arctanh((1/(2+2^(1/2))^(1/2)+((a*x+(a^2*x
^2-b)^(1/2))/b^(1/2))^(1/2)/(2+2^(1/2))^(1/2))/((a*x+(a^2*x^2-b)^(1/2))/b^(1/2))^(1/4))/b^(15/8)

________________________________________________________________________________________

Rubi [A]
time = 2.29, antiderivative size = 1407, normalized size of antiderivative = 1.61, number of steps used = 52, number of rules used = 20, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.408, Rules used = {6874, 2145, 477, 481, 592, 541, 536, 220, 218, 212, 209, 217, 1179, 642, 1176, 631, 210, 468, 294, 335} \begin {gather*} \frac {8 c \left (a x+\sqrt {a^2 x^2-b}\right )^{17/4}}{3 a^2 \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^3}-\frac {5 c \left (a x+\sqrt {a^2 x^2-b}\right )^{9/4}}{6 a^2 \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^2}+\frac {8 d \left (a x+\sqrt {a^2 x^2-b}\right )^{9/4}}{3 \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^3}+\frac {15 c \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{16 a^2 \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )}+\frac {39 d \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{16 b \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )}-\frac {7 d \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{2 \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^2}-\frac {2 d \text {ArcTan}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}-\frac {15 c \text {ArcTan}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 a^2 b^{7/8}}+\frac {145 d \text {ArcTan}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 b^{15/8}}+\frac {\sqrt {2} d \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}-\frac {\sqrt {2} d \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}+1\right )}{(-b)^{15/8}}+\frac {15 c \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 \sqrt {2} a^2 b^{7/8}}-\frac {145 d \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 \sqrt {2} b^{15/8}}-\frac {15 c \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}+1\right )}{64 \sqrt {2} a^2 b^{7/8}}+\frac {145 d \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}+1\right )}{64 \sqrt {2} b^{15/8}}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}-\frac {15 c \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 a^2 b^{7/8}}+\frac {145 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 b^{15/8}}+\frac {d \log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}-\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{-b}\right )}{\sqrt {2} (-b)^{15/8}}-\frac {d \log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{-b}\right )}{\sqrt {2} (-b)^{15/8}}+\frac {15 c \log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}\right )}{128 \sqrt {2} a^2 b^{7/8}}-\frac {145 d \log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}\right )}{128 \sqrt {2} b^{15/8}}-\frac {15 c \log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}\right )}{128 \sqrt {2} a^2 b^{7/8}}+\frac {145 d \log \left (\sqrt {a x+\sqrt {a^2 x^2-b}}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {a^2 x^2-b}}+\sqrt [4]{b}\right )}{128 \sqrt {2} b^{15/8}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + c*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(5/4))/(x*(-b + a^2*x^2)^(5/2)),x]

[Out]

(8*d*(a*x + Sqrt[-b + a^2*x^2])^(9/4))/(3*(b - (a*x + Sqrt[-b + a^2*x^2])^2)^3) + (8*c*(a*x + Sqrt[-b + a^2*x^
2])^(17/4))/(3*a^2*(b - (a*x + Sqrt[-b + a^2*x^2])^2)^3) - (7*d*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(2*(b - (a*x
+ Sqrt[-b + a^2*x^2])^2)^2) - (5*c*(a*x + Sqrt[-b + a^2*x^2])^(9/4))/(6*a^2*(b - (a*x + Sqrt[-b + a^2*x^2])^2
)^2) + (15*c*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(16*a^2*(b - (a*x + Sqrt[-b + a^2*x^2])^2)) + (39*d*(a*x + Sqrt
[-b + a^2*x^2])^(1/4))/(16*b*(b - (a*x + Sqrt[-b + a^2*x^2])^2)) - (2*d*ArcTan[(a*x + Sqrt[-b + a^2*x^2])^(1/4
)/(-b)^(1/8)])/(-b)^(15/8) - (15*c*ArcTan[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)])/(64*a^2*b^(7/8)) + (145*d
*ArcTan[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)])/(64*b^(15/8)) + (Sqrt[2]*d*ArcTan[1 - (Sqrt[2]*(a*x + Sqrt[
-b + a^2*x^2])^(1/4))/(-b)^(1/8)])/(-b)^(15/8) - (Sqrt[2]*d*ArcTan[1 + (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/
4))/(-b)^(1/8)])/(-b)^(15/8) + (15*c*ArcTan[1 - (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/b^(1/8)])/(64*Sqrt[
2]*a^2*b^(7/8)) - (145*d*ArcTan[1 - (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/b^(1/8)])/(64*Sqrt[2]*b^(15/8))
- (15*c*ArcTan[1 + (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/b^(1/8)])/(64*Sqrt[2]*a^2*b^(7/8)) + (145*d*Arc
Tan[1 + (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/b^(1/8)])/(64*Sqrt[2]*b^(15/8)) - (2*d*ArcTanh[(a*x + Sqrt[
-b + a^2*x^2])^(1/4)/(-b)^(1/8)])/(-b)^(15/8) - (15*c*ArcTanh[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)])/(64*a
^2*b^(7/8)) + (145*d*ArcTanh[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)])/(64*b^(15/8)) + (d*Log[(-b)^(1/4) - Sq
rt[2]*(-b)^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/(Sqrt[2]*(-b)^(15/8)) - (
d*Log[(-b)^(1/4) + Sqrt[2]*(-b)^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/(Sqr
t[2]*(-b)^(15/8)) + (15*c*Log[b^(1/4) - Sqrt[2]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4) + Sqrt[a*x + Sqrt[-b
+ a^2*x^2]]])/(128*Sqrt[2]*a^2*b^(7/8)) - (145*d*Log[b^(1/4) - Sqrt[2]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4
) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/(128*Sqrt[2]*b^(15/8)) - (15*c*Log[b^(1/4) + Sqrt[2]*b^(1/8)*(a*x + Sqrt[
-b + a^2*x^2])^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/(128*Sqrt[2]*a^2*b^(7/8)) + (145*d*Log[b^(1/4) + Sqrt[
2]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/(128*Sqrt[2]*b^(15/8))

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 220

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]
}, Dist[r/(2*a), Int[1/(r - s*x^(n/2)), x], x] + Dist[r/(2*a), Int[1/(r + s*x^(n/2)), x], x]] /; FreeQ[{a, b},
x] && IGtQ[n/4, 1] &&  !GtQ[a/b, 0]

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 468

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d
))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b*e*n*(p + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a
*b*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0]
&& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0]
&& LeQ[-1, m, (-n)*(p + 1)]))

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 481

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-a)*e^(
2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Dist[e^
(2*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1)
+ (a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 592

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Simp[g^(n - 1)*(b*e - a*f)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c -
a*d)*(p + 1))), x] - Dist[g^n/(b*n*(b*c - a*d)*(p + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*S
imp[c*(b*e - a*f)*(m - n + 1) + (d*(b*e - a*f)*(m + n*q + 1) - b*n*(c*f - d*e)*(p + 1))*x^n, x], x], x] /; Fre
eQ[{a, b, c, d, e, f, g, q}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, 0]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 2145

Int[(x_)^(p_.)*((g_) + (i_.)*(x_)^2)^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :>
Dist[(1/(2^(2*m + p + 1)*e^(p + 1)*f^(2*m)))*(i/c)^m, Subst[Int[x^(n - 2*m - p - 2)*((-a)*f^2 + x^2)^p*(a*f^2
+ x^2)^(2*m + 1), x], x, e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0
] && EqQ[c*g - a*i, 0] && IntegersQ[p, 2*m] && (IntegerQ[m] || GtQ[i/c, 0])

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{x \left (-b+a^2 x^2\right )^{5/2}} \, dx &=\int \left (\frac {d \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{x \left (-b+a^2 x^2\right )^{5/2}}+\frac {c x \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{5/2}}\right ) \, dx\\ &=c \int \frac {x \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{\left (-b+a^2 x^2\right )^{5/2}} \, dx+d \int \frac {\left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{x \left (-b+a^2 x^2\right )^{5/2}} \, dx\\ &=\frac {(8 c) \text {Subst}\left (\int \frac {x^{13/4} \left (b+x^2\right )}{\left (-b+x^2\right )^4} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{a^2}+(32 d) \text {Subst}\left (\int \frac {x^{21/4}}{\left (-b+x^2\right )^4 \left (b+x^2\right )} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )\\ &=\frac {8 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{17/4}}{3 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {(10 c) \text {Subst}\left (\int \frac {x^{13/4}}{\left (-b+x^2\right )^3} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{3 a^2}+(128 d) \text {Subst}\left (\int \frac {x^{24}}{\left (-b+x^8\right )^4 \left (b+x^8\right )} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )\\ &=\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{17/4}}{3 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {5 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{6 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {(15 c) \text {Subst}\left (\int \frac {x^{5/4}}{\left (-b+x^2\right )^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{8 a^2}-\frac {(8 d) \text {Subst}\left (\int \frac {x^8 \left (-9 b^2-33 b x^8\right )}{\left (-b+x^8\right )^3 \left (b+x^8\right )} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{3 b}\\ &=\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{17/4}}{3 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {7 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {5 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{6 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {15 c \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {(15 c) \text {Subst}\left (\int \frac {1}{x^{3/4} \left (-b+x^2\right )} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{64 a^2}-\frac {d \text {Subst}\left (\int \frac {-42 b^3-426 b^2 x^8}{\left (-b+x^8\right )^2 \left (b+x^8\right )} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{12 b^2}\\ &=\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{17/4}}{3 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {7 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {5 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{6 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {15 c \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {39 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {(15 c) \text {Subst}\left (\int \frac {1}{-b+x^8} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{16 a^2}-\frac {d \text {Subst}\left (\int \frac {204 b^4+3276 b^3 x^8}{\left (-b+x^8\right ) \left (b+x^8\right )} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{192 b^4}\\ &=\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{17/4}}{3 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {7 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {5 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{6 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {15 c \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {39 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {(15 c) \text {Subst}\left (\int \frac {1}{\sqrt {b}-x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{32 a^2 \sqrt {b}}-\frac {(15 c) \text {Subst}\left (\int \frac {1}{\sqrt {b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{32 a^2 \sqrt {b}}-\frac {(8 d) \text {Subst}\left (\int \frac {1}{b+x^8} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{b}-\frac {(145 d) \text {Subst}\left (\int \frac {1}{-b+x^8} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{16 b}\\ &=\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{17/4}}{3 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {7 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {5 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{6 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {15 c \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {39 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {(15 c) \text {Subst}\left (\int \frac {1}{\sqrt [4]{b}-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{64 a^2 b^{3/4}}-\frac {(15 c) \text {Subst}\left (\int \frac {1}{\sqrt [4]{b}+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{64 a^2 b^{3/4}}-\frac {(15 c) \text {Subst}\left (\int \frac {\sqrt [4]{b}-x^2}{\sqrt {b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{64 a^2 b^{3/4}}-\frac {(15 c) \text {Subst}\left (\int \frac {\sqrt [4]{b}+x^2}{\sqrt {b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{64 a^2 b^{3/4}}-\frac {(4 d) \text {Subst}\left (\int \frac {1}{\sqrt {-b}-x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{(-b)^{3/2}}-\frac {(4 d) \text {Subst}\left (\int \frac {1}{\sqrt {-b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{(-b)^{3/2}}+\frac {(145 d) \text {Subst}\left (\int \frac {1}{\sqrt {b}-x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{32 b^{3/2}}+\frac {(145 d) \text {Subst}\left (\int \frac {1}{\sqrt {b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{32 b^{3/2}}\\ &=\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{17/4}}{3 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {7 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {5 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{6 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {15 c \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {39 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {15 c \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^2 b^{7/8}}-\frac {15 c \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^2 b^{7/8}}+\frac {(15 c) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [8]{b}+2 x}{-\sqrt [4]{b}-\sqrt {2} \sqrt [8]{b} x-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{128 \sqrt {2} a^2 b^{7/8}}+\frac {(15 c) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [8]{b}-2 x}{-\sqrt [4]{b}+\sqrt {2} \sqrt [8]{b} x-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{128 \sqrt {2} a^2 b^{7/8}}-\frac {(15 c) \text {Subst}\left (\int \frac {1}{\sqrt [4]{b}-\sqrt {2} \sqrt [8]{b} x+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{128 a^2 b^{3/4}}-\frac {(15 c) \text {Subst}\left (\int \frac {1}{\sqrt [4]{b}+\sqrt {2} \sqrt [8]{b} x+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{128 a^2 b^{3/4}}-\frac {(2 d) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b}-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{(-b)^{7/4}}-\frac {(2 d) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b}+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{(-b)^{7/4}}-\frac {(2 d) \text {Subst}\left (\int \frac {\sqrt [4]{-b}-x^2}{\sqrt {-b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{(-b)^{7/4}}-\frac {(2 d) \text {Subst}\left (\int \frac {\sqrt [4]{-b}+x^2}{\sqrt {-b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{(-b)^{7/4}}+\frac {(145 d) \text {Subst}\left (\int \frac {1}{\sqrt [4]{b}-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{64 b^{7/4}}+\frac {(145 d) \text {Subst}\left (\int \frac {1}{\sqrt [4]{b}+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{64 b^{7/4}}+\frac {(145 d) \text {Subst}\left (\int \frac {\sqrt [4]{b}-x^2}{\sqrt {b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{64 b^{7/4}}+\frac {(145 d) \text {Subst}\left (\int \frac {\sqrt [4]{b}+x^2}{\sqrt {b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{64 b^{7/4}}\\ &=\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{17/4}}{3 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {7 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {5 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{6 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {15 c \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {39 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {2 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}-\frac {15 c \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^2 b^{7/8}}+\frac {145 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 b^{15/8}}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}-\frac {15 c \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^2 b^{7/8}}+\frac {145 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 b^{15/8}}+\frac {15 c \log \left (\sqrt [4]{b}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{128 \sqrt {2} a^2 b^{7/8}}-\frac {15 c \log \left (\sqrt [4]{b}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{128 \sqrt {2} a^2 b^{7/8}}-\frac {(15 c) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 \sqrt {2} a^2 b^{7/8}}+\frac {(15 c) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 \sqrt {2} a^2 b^{7/8}}+\frac {d \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [8]{-b}+2 x}{-\sqrt [4]{-b}-\sqrt {2} \sqrt [8]{-b} x-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2} (-b)^{15/8}}+\frac {d \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [8]{-b}-2 x}{-\sqrt [4]{-b}+\sqrt {2} \sqrt [8]{-b} x-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2} (-b)^{15/8}}-\frac {d \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b}-\sqrt {2} \sqrt [8]{-b} x+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{(-b)^{7/4}}-\frac {d \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b}+\sqrt {2} \sqrt [8]{-b} x+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{(-b)^{7/4}}-\frac {(145 d) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [8]{b}+2 x}{-\sqrt [4]{b}-\sqrt {2} \sqrt [8]{b} x-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{128 \sqrt {2} b^{15/8}}-\frac {(145 d) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [8]{b}-2 x}{-\sqrt [4]{b}+\sqrt {2} \sqrt [8]{b} x-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{128 \sqrt {2} b^{15/8}}+\frac {(145 d) \text {Subst}\left (\int \frac {1}{\sqrt [4]{b}-\sqrt {2} \sqrt [8]{b} x+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{128 b^{7/4}}+\frac {(145 d) \text {Subst}\left (\int \frac {1}{\sqrt [4]{b}+\sqrt {2} \sqrt [8]{b} x+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{128 b^{7/4}}\\ &=\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{17/4}}{3 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {7 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {5 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{6 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {15 c \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {39 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {2 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}-\frac {15 c \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^2 b^{7/8}}+\frac {145 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 b^{15/8}}+\frac {15 c \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 \sqrt {2} a^2 b^{7/8}}-\frac {15 c \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 \sqrt {2} a^2 b^{7/8}}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}-\frac {15 c \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^2 b^{7/8}}+\frac {145 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 b^{15/8}}+\frac {d \log \left (\sqrt [4]{-b}-\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2} (-b)^{15/8}}-\frac {d \log \left (\sqrt [4]{-b}+\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2} (-b)^{15/8}}+\frac {15 c \log \left (\sqrt [4]{b}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{128 \sqrt {2} a^2 b^{7/8}}-\frac {145 d \log \left (\sqrt [4]{b}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{128 \sqrt {2} b^{15/8}}-\frac {15 c \log \left (\sqrt [4]{b}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{128 \sqrt {2} a^2 b^{7/8}}+\frac {145 d \log \left (\sqrt [4]{b}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{128 \sqrt {2} b^{15/8}}-\frac {\left (\sqrt {2} d\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}+\frac {\left (\sqrt {2} d\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}+\frac {(145 d) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 \sqrt {2} b^{15/8}}-\frac {(145 d) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 \sqrt {2} b^{15/8}}\\ &=\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{17/4}}{3 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {7 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {5 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}{6 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {15 c \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 a^2 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {39 d \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{16 b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {2 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}-\frac {15 c \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^2 b^{7/8}}+\frac {145 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 b^{15/8}}+\frac {\sqrt {2} d \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}-\frac {\sqrt {2} d \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}+\frac {15 c \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 \sqrt {2} a^2 b^{7/8}}-\frac {145 d \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 \sqrt {2} b^{15/8}}-\frac {15 c \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 \sqrt {2} a^2 b^{7/8}}+\frac {145 d \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 \sqrt {2} b^{15/8}}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}-\frac {15 c \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^2 b^{7/8}}+\frac {145 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 b^{15/8}}+\frac {d \log \left (\sqrt [4]{-b}-\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2} (-b)^{15/8}}-\frac {d \log \left (\sqrt [4]{-b}+\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2} (-b)^{15/8}}+\frac {15 c \log \left (\sqrt [4]{b}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{128 \sqrt {2} a^2 b^{7/8}}-\frac {145 d \log \left (\sqrt [4]{b}-\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{128 \sqrt {2} b^{15/8}}-\frac {15 c \log \left (\sqrt [4]{b}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{128 \sqrt {2} a^2 b^{7/8}}+\frac {145 d \log \left (\sqrt [4]{b}+\sqrt {2} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{128 \sqrt {2} b^{15/8}}\\ \end {align*}

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Mathematica [F]
time = 10.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{x \left (-b+a^2 x^2\right )^{5/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((d + c*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(5/4))/(x*(-b + a^2*x^2)^(5/2)),x]

[Out]

Integrate[((d + c*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(5/4))/(x*(-b + a^2*x^2)^(5/2)), x]

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (c \,x^{2}+d \right ) \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {5}{4}}}{x \left (a^{2} x^{2}-b \right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+d)*(a*x+(a^2*x^2-b)^(1/2))^(5/4)/x/(a^2*x^2-b)^(5/2),x)

[Out]

int((c*x^2+d)*(a*x+(a^2*x^2-b)^(1/2))^(5/4)/x/(a^2*x^2-b)^(5/2),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+d)*(a*x+(a^2*x^2-b)^(1/2))^(5/4)/x/(a^2*x^2-b)^(5/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + d)*(a*x + sqrt(a^2*x^2 - b))^(5/4)/((a^2*x^2 - b)^(5/2)*x), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5382 vs. \(2 (693) = 1386\).
time = 13.76, size = 5382, normalized size = 6.14 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+d)*(a*x+(a^2*x^2-b)^(1/2))^(5/4)/x/(a^2*x^2-b)^(5/2),x, algorithm="fricas")

[Out]

-1/768*(60*sqrt(2)*(a^6*b^2*x^4 - 2*a^4*b^3*x^2 + a^2*b^4)*((500246412961*a^16*d^8 - 413997031416*a^14*b*c*d^7
+ 149895476892*a^12*b^2*c^2*d^6 - 31012857288*a^10*b^3*c^3*d^5 + 4010283270*a^8*b^4*c^4*d^4 - 331885512*a^6*b
^5*c^5*d^3 + 17166492*a^4*b^6*c^6*d^2 - 507384*a^2*b^7*c^7*d + 6561*b^8*c^8)/(a^16*b^15))^(1/8)*arctan((sqrt(2
)*sqrt(a^4*b^4*((500246412961*a^16*d^8 - 413997031416*a^14*b*c*d^7 + 149895476892*a^12*b^2*c^2*d^6 - 310128572
88*a^10*b^3*c^3*d^5 + 4010283270*a^8*b^4*c^4*d^4 - 331885512*a^6*b^5*c^5*d^3 + 17166492*a^4*b^6*c^6*d^2 - 5073
84*a^2*b^7*c^7*d + 6561*b^8*c^8)/(a^16*b^15))^(1/4) + sqrt(2)*(29*a^4*b^2*d - 3*a^2*b^3*c)*(a*x + sqrt(a^2*x^2
- b))^(1/4)*((500246412961*a^16*d^8 - 413997031416*a^14*b*c*d^7 + 149895476892*a^12*b^2*c^2*d^6 - 31012857288
*a^10*b^3*c^3*d^5 + 4010283270*a^8*b^4*c^4*d^4 - 331885512*a^6*b^5*c^5*d^3 + 17166492*a^4*b^6*c^6*d^2 - 507384
*a^2*b^7*c^7*d + 6561*b^8*c^8)/(a^16*b^15))^(1/8) + (841*a^4*d^2 - 174*a^2*b*c*d + 9*b^2*c^2)*sqrt(a*x + sqrt(
a^2*x^2 - b)))*a^14*b^13*((500246412961*a^16*d^8 - 413997031416*a^14*b*c*d^7 + 149895476892*a^12*b^2*c^2*d^6 -
31012857288*a^10*b^3*c^3*d^5 + 4010283270*a^8*b^4*c^4*d^4 - 331885512*a^6*b^5*c^5*d^3 + 17166492*a^4*b^6*c^6*
d^2 - 507384*a^2*b^7*c^7*d + 6561*b^8*c^8)/(a^16*b^15))^(7/8) - 500246412961*a^16*d^8 + 413997031416*a^14*b*c*
d^7 - 149895476892*a^12*b^2*c^2*d^6 + 31012857288*a^10*b^3*c^3*d^5 - 4010283270*a^8*b^4*c^4*d^4 + 331885512*a^
6*b^5*c^5*d^3 - 17166492*a^4*b^6*c^6*d^2 + 507384*a^2*b^7*c^7*d - 6561*b^8*c^8 - sqrt(2)*(29*a^16*b^13*d - 3*a
^14*b^14*c)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*((500246412961*a^16*d^8 - 413997031416*a^14*b*c*d^7 + 149895476892
*a^12*b^2*c^2*d^6 - 31012857288*a^10*b^3*c^3*d^5 + 4010283270*a^8*b^4*c^4*d^4 - 331885512*a^6*b^5*c^5*d^3 + 17
166492*a^4*b^6*c^6*d^2 - 507384*a^2*b^7*c^7*d + 6561*b^8*c^8)/(a^16*b^15))^(7/8))/(500246412961*a^16*d^8 - 413
997031416*a^14*b*c*d^7 + 149895476892*a^12*b^2*c^2*d^6 - 31012857288*a^10*b^3*c^3*d^5 + 4010283270*a^8*b^4*c^4
*d^4 - 331885512*a^6*b^5*c^5*d^3 + 17166492*a^4*b^6*c^6*d^2 - 507384*a^2*b^7*c^7*d + 6561*b^8*c^8)) + 60*sqrt(
2)*(a^6*b^2*x^4 - 2*a^4*b^3*x^2 + a^2*b^4)*((500246412961*a^16*d^8 - 413997031416*a^14*b*c*d^7 + 149895476892*
a^12*b^2*c^2*d^6 - 31012857288*a^10*b^3*c^3*d^5 + 4010283270*a^8*b^4*c^4*d^4 - 331885512*a^6*b^5*c^5*d^3 + 171
66492*a^4*b^6*c^6*d^2 - 507384*a^2*b^7*c^7*d + 6561*b^8*c^8)/(a^16*b^15))^(1/8)*arctan((sqrt(2)*sqrt(a^4*b^4*(
(500246412961*a^16*d^8 - 413997031416*a^14*b*c*d^7 + 149895476892*a^12*b^2*c^2*d^6 - 31012857288*a^10*b^3*c^3*
d^5 + 4010283270*a^8*b^4*c^4*d^4 - 331885512*a^6*b^5*c^5*d^3 + 17166492*a^4*b^6*c^6*d^2 - 507384*a^2*b^7*c^7*d
+ 6561*b^8*c^8)/(a^16*b^15))^(1/4) - sqrt(2)*(29*a^4*b^2*d - 3*a^2*b^3*c)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*((5
00246412961*a^16*d^8 - 413997031416*a^14*b*c*d^7 + 149895476892*a^12*b^2*c^2*d^6 - 31012857288*a^10*b^3*c^3*d^
5 + 4010283270*a^8*b^4*c^4*d^4 - 331885512*a^6*b^5*c^5*d^3 + 17166492*a^4*b^6*c^6*d^2 - 507384*a^2*b^7*c^7*d +
6561*b^8*c^8)/(a^16*b^15))^(1/8) + (841*a^4*d^2 - 174*a^2*b*c*d + 9*b^2*c^2)*sqrt(a*x + sqrt(a^2*x^2 - b)))*a
^14*b^13*((500246412961*a^16*d^8 - 413997031416*a^14*b*c*d^7 + 149895476892*a^12*b^2*c^2*d^6 - 31012857288*a^1
0*b^3*c^3*d^5 + 4010283270*a^8*b^4*c^4*d^4 - 331885512*a^6*b^5*c^5*d^3 + 17166492*a^4*b^6*c^6*d^2 - 507384*a^2
*b^7*c^7*d + 6561*b^8*c^8)/(a^16*b^15))^(7/8) + 500246412961*a^16*d^8 - 413997031416*a^14*b*c*d^7 + 1498954768
92*a^12*b^2*c^2*d^6 - 31012857288*a^10*b^3*c^3*d^5 + 4010283270*a^8*b^4*c^4*d^4 - 331885512*a^6*b^5*c^5*d^3 +
17166492*a^4*b^6*c^6*d^2 - 507384*a^2*b^7*c^7*d + 6561*b^8*c^8 - sqrt(2)*(29*a^16*b^13*d - 3*a^14*b^14*c)*(a*x
+ sqrt(a^2*x^2 - b))^(1/4)*((500246412961*a^16*d^8 - 413997031416*a^14*b*c*d^7 + 149895476892*a^12*b^2*c^2*d^
6 - 31012857288*a^10*b^3*c^3*d^5 + 4010283270*a^8*b^4*c^4*d^4 - 331885512*a^6*b^5*c^5*d^3 + 17166492*a^4*b^6*c
^6*d^2 - 507384*a^2*b^7*c^7*d + 6561*b^8*c^8)/(a^16*b^15))^(7/8))/(500246412961*a^16*d^8 - 413997031416*a^14*b
*c*d^7 + 149895476892*a^12*b^2*c^2*d^6 - 31012857288*a^10*b^3*c^3*d^5 + 4010283270*a^8*b^4*c^4*d^4 - 331885512
*a^6*b^5*c^5*d^3 + 17166492*a^4*b^6*c^6*d^2 - 507384*a^2*b^7*c^7*d + 6561*b^8*c^8)) + 1536*sqrt(2)*(a^6*b^2*x^
4 - 2*a^4*b^3*x^2 + a^2*b^4)*(-d^8/b^15)^(1/8)*arctan(-(sqrt(2)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*b^13*d*(-d^8/b
^15)^(7/8) - sqrt(2)*sqrt(b^4*(-d^8/b^15)^(1/4) + sqrt(2)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*b^2*d*(-d^8/b^15)^(1
/8) + sqrt(a*x + sqrt(a^2*x^2 - b))*d^2)*b^13*(-d^8/b^15)^(7/8) - d^8)/d^8) + 1536*sqrt(2)*(a^6*b^2*x^4 - 2*a^
4*b^3*x^2 + a^2*b^4)*(-d^8/b^15)^(1/8)*arctan(-(sqrt(2)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*b^13*d*(-d^8/b^15)^(7/
8) - sqrt(2)*sqrt(b^4*(-d^8/b^15)^(1/4) - sqrt(2)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*b^2*d*(-d^8/b^15)^(1/8) + sq
rt(a*x + sqrt(a^2*x^2 - b))*d^2)*b^13*(-d^8/b^15)^(7/8) + d^8)/d^8) - 15*sqrt(2)*(a^6*b^2*x^4 - 2*a^4*b^3*x^2
+ a^2*b^4)*((500246412961*a^16*d^8 - 413997031416*a^14*b*c*d^7 + 149895476892*a^12*b^2*c^2*d^6 - 31012857288*a
^10*b^3*c^3*d^5 + 4010283270*a^8*b^4*c^4*d^4 - ...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x + \sqrt {a^{2} x^{2} - b}\right )^{\frac {5}{4}} \left (c x^{2} + d\right )}{x \left (a^{2} x^{2} - b\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+d)*(a*x+(a**2*x**2-b)**(1/2))**(5/4)/x/(a**2*x**2-b)**(5/2),x)

[Out]

Integral((a*x + sqrt(a**2*x**2 - b))**(5/4)*(c*x**2 + d)/(x*(a**2*x**2 - b)**(5/2)), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+d)*(a*x+(a^2*x^2-b)^(1/2))^(5/4)/x/(a^2*x^2-b)^(5/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{5/4}\,\left (c\,x^2+d\right )}{x\,{\left (a^2\,x^2-b\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x + (a^2*x^2 - b)^(1/2))^(5/4)*(d + c*x^2))/(x*(a^2*x^2 - b)^(5/2)),x)

[Out]

int(((a*x + (a^2*x^2 - b)^(1/2))^(5/4)*(d + c*x^2))/(x*(a^2*x^2 - b)^(5/2)), x)

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