3.32.0 ∫ (d+cx2)(ax+√ _______ −b+a2x2)5∕4 _________________________________________

Integrand size = 49, antiderivative size = 876 \[ \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=\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 ) \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 \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 \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 ) \text {arctanh}\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 ) \text {arctanh}\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 ) \text {arctanh}\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 \text {arctanh}\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 {arctanh}\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}} \]

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] (veriﬁed)

Time = 2.03 (sec) , antiderivative size = 1407, normalized size of antiderivative = 1.61, number of steps used = 52, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.408, Rules used = {6874, 2145, 477, 481, 592, 541, 536, 220, 218, 212, 209, 217, 1179, 642, 1176, 631, 210, 468, 294, 335} \[ \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=\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 \arctan \left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}-\frac {15 c \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 \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 \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 \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 \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 \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 \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 \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 \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{-b}}\right )}{(-b)^{15/8}}-\frac {15 c \text {arctanh}\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 {arctanh}\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}} \]

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

(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))

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])

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])

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])

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

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

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},

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]

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]

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]

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

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]

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

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]

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]

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]

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]

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]

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]

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])

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

Rubi steps \begin{align*} \text {integral}& = \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 \arctan \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 {arctanh}\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}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(12230\) vs. \(2(876)=1752\).

Time = 51.84 (sec) , antiderivative size = 12230, normalized size of antiderivative = 13.96 \[ \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{x \left (-b+a^2 x^2\right )^{5/2}} \, dx=\text {Result too large to show} \]

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

Maple [F]

\[\int \frac {\left (c \,x^{2}+d \right ) \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {5}{4}}}{x \left (a^{2} x^{2}-b \right )^{\frac {5}{2}}}d x\]

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)

Fricas [C] (veriﬁcation not implemented)

Time = 13.02 (sec) , antiderivative size = 3342, normalized size of antiderivative = 3.82 \[ \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{x \left (-b+a^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

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")

-1/768*(15*sqrt(2)*(-(I + 1)*a^6*b^2*x^4 + (2*I + 2)*a^4*b^3*x^2 - (I + 1)*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)*log((5/2*I + 5/2)*sqrt(2)*a^2*b^2*((500246412961*a^16*d^8 - 413997031416*a^14*b*c*d^7 + 14989547689

2*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 + 1

7166492*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) + 5*(29*a^2*d - 3*b*c)*(a*x

+ sqrt(a^2*x^2 - b))^(1/4)) + 15*sqrt(2)*((I - 1)*a^6*b^2*x^4 - (2*I - 2)*a^4*b^3*x^2 + (I - 1)*a^2*b^4)*((500

246412961*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 + 6

561*b^8*c^8)/(a^16*b^15))^(1/8)*log(-(5/2*I - 5/2)*sqrt(2)*a^2*b^2*((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 - 3318855

12*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) + 5*(2

9*a^2*d - 3*b*c)*(a*x + sqrt(a^2*x^2 - b))^(1/4)) + 15*sqrt(2)*(-(I - 1)*a^6*b^2*x^4 + (2*I - 2)*a^4*b^3*x^2 -

(I - 1)*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 - 310128

57288*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 - 5

07384*a^2*b^7*c^7*d + 6561*b^8*c^8)/(a^16*b^15))^(1/8)*log((5/2*I - 5/2)*sqrt(2)*a^2*b^2*((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) + 5*(29*a^2*d - 3*b*c)*(a*x + sqrt(a^2*x^2 - b))^(1/4)) + 15*sqrt(2)*((I + 1)*a^6*b^2*x^4 - (2

*I + 2)*a^4*b^3*x^2 + (I + 1)*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 + 1716649

2*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)*log(-(5/2*I + 5/2)*sqrt(2)*a^2*b^2

*((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) + 5*(29*a^2*d - 3*b*c)*(a*x + sqrt(a^2*x^2 - b))^(1/4)) + 384*sqrt(2)*((

I + 1)*a^6*b^2*x^4 - (2*I + 2)*a^4*b^3*x^2 + (I + 1)*a^2*b^4)*(-d^8/b^15)^(1/8)*log((I + 1)*sqrt(2)*b^2*(-d^8/

b^15)^(1/8) + 2*(a*x + sqrt(a^2*x^2 - b))^(1/4)*d) + 384*sqrt(2)*(-(I - 1)*a^6*b^2*x^4 + (2*I - 2)*a^4*b^3*x^2

- (I - 1)*a^2*b^4)*(-d^8/b^15)^(1/8)*log(-(I - 1)*sqrt(2)*b^2*(-d^8/b^15)^(1/8) + 2*(a*x + sqrt(a^2*x^2 - b))

^(1/4)*d) + 384*sqrt(2)*((I - 1)*a^6*b^2*x^4 - (2*I - 2)*a^4*b^3*x^2 + (I - 1)*a^2*b^4)*(-d^8/b^15)^(1/8)*log(

(I - 1)*sqrt(2)*b^2*(-d^8/b^15)^(1/8) + 2*(a*x + sqrt(a^2*x^2 - b))^(1/4)*d) + 384*sqrt(2)*(-(I + 1)*a^6*b^2*x

^4 + (2*I + 2)*a^4*b^3*x^2 - (I + 1)*a^2*b^4)*(-d^8/b^15)^(1/8)*log(-(I + 1)*sqrt(2)*b^2*(-d^8/b^15)^(1/8) + 2

*(a*x + sqrt(a^2*x^2 - b))^(1/4)*d) - 30*(a^6*b^2*x^4 - 2*a^4*b^3*x^2 + a^2*b^4)*((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)/(a^16*b^15)

)^(1/8)*log(5*a^2*b^2*((500246412961*a^16*d^8 - 413997031416*a^14*b*c*d^7 + 149895476892*a^12*b^2*c^2*d^6 - 31

012857288*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) + 5*(29*a^2*d - 3*b*c)*(a*x + sqrt(a^2*x^2 - b))^(1

/4)) + 30*(-I*a^6*b^2*x^4 + 2*I*a^4*b^3*x^2 - I*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)*log(5*I*a^2*b^2*

((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) + 5*(29*a^2*d - 3*b*c)*(a*x + sqrt(a^2*x^2 - b))^(1/4)) + 30*(I*a^6*b^2*x

^4 - 2*I*a^4*b^3*x^2 + I*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)*log(-5*I*a^2*b^2*((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^1

6*b^15))^(1/8) + 5*(29*a^2*d - 3*b*c)*(a*x + sqrt(a^2*x^2 - b))^(1/4)) + 30*(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)*log(-5*a^2*b^2*((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) + 5*(29*a^2*d -

3*b*c)*(a*x + sqrt(a^2*x^2 - b))^(1/4)) + 768*(a^6*b^2*x^4 - 2*a^4*b^3*x^2 + a^2*b^4)*(-d^8/b^15)^(1/8)*log(b

^2*(-d^8/b^15)^(1/8) + (a*x + sqrt(a^2*x^2 - b))^(1/4)*d) + 768*(I*a^6*b^2*x^4 - 2*I*a^4*b^3*x^2 + I*a^2*b^4)*

(-d^8/b^15)^(1/8)*log(I*b^2*(-d^8/b^15)^(1/8) + (a*x + sqrt(a^2*x^2 - b))^(1/4)*d) + 768*(-I*a^6*b^2*x^4 + 2*I

*a^4*b^3*x^2 - I*a^2*b^4)*(-d^8/b^15)^(1/8)*log(-I*b^2*(-d^8/b^15)^(1/8) + (a*x + sqrt(a^2*x^2 - b))^(1/4)*d)

- 768*(a^6*b^2*x^4 - 2*a^4*b^3*x^2 + a^2*b^4)*(-d^8/b^15)^(1/8)*log(-b^2*(-d^8/b^15)^(1/8) + (a*x + sqrt(a^2*x

^2 - b))^(1/4)*d) - 8*(a^2*b^2*d - 3*(17*a^6*d - 15*a^4*b*c)*x^4 + 97*b^3*c + 2*(25*a^4*b*d - 71*a^2*b^2*c)*x^

2 + sqrt(a^2*x^2 - b)*(3*(17*a^5*d - 15*a^3*b*c)*x^3 - (83*a^3*b*d - 13*a*b^2*c)*x))*(a*x + sqrt(a^2*x^2 - b))

^(1/4))/(a^6*b^2*x^4 - 2*a^4*b^3*x^2 + a^2*b^4)

Sympy [F]

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

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)

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

Maxima [F]

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

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")

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

Giac [F(-1)]

Timed out. \[ \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{x \left (-b+a^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]

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")

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+c x^2\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}{x \left (-b+a^2 x^2\right )^{5/2}} \, dx=\int \frac {{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{5/4}\,\left (c\,x^2+d\right )}{x\,{\left (a^2\,x^2-b\right )}^{5/2}} \,d x \]

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)

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)

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

Optimal result