3.31.36 \(\int \frac {(d+c x^2) (a x+\sqrt {-b+a^2 x^2})^{3/4}}{(-b+a^2 x^2)^{5/2}} \, dx\) [3036]

Optimal. Leaf size=448 \[ \frac {\left (-53 b^2 c+43 a^2 b d+41 a^2 b c x^2-55 a^4 d x^2\right ) \left (\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}\right )^{3/4}}{96 a^3 b^{13/8} \left (-\sqrt {b}+a x\right ) \left (\sqrt {b}+a x\right )}+\frac {\sqrt {-b+a^2 x^2} \left (9 b^2 c x-87 a^2 b d x-41 a^2 b c x^3+55 a^4 d x^3\right ) \left (\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}\right )^{3/4}}{96 a^2 b^{13/8} \left (-\sqrt {b}+a x\right )^2 \left (\sqrt {b}+a x\right )^2}+\frac {\left (41 b c-55 a^2 d\right ) \text {ArcTan}\left (\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{64 a^3 b^{13/8}}+\frac {\left (-41 b c+55 a^2 d\right ) \tanh ^{-1}\left (\sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{64 a^3 b^{13/8}}-\frac {\sqrt [4]{-1} \left (-41 b c+55 a^2 d\right ) \tanh ^{-1}\left (\sqrt [4]{-1} \sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{64 a^3 b^{13/8}}-\frac {(-1)^{3/4} \left (-41 b c+55 a^2 d\right ) \tanh ^{-1}\left ((-1)^{3/4} \sqrt [4]{\frac {a x+\sqrt {-b+a^2 x^2}}{\sqrt {b}}}\right )}{64 a^3 b^{13/8}} \]

[Out]

1/96*(-55*a^4*d*x^2+41*a^2*b*c*x^2+43*a^2*b*d-53*b^2*c)*((a*x+(a^2*x^2-b)^(1/2))/b^(1/2))^(3/4)/a^3/b^(13/8)/(
-b^(1/2)+a*x)/(b^(1/2)+a*x)+1/96*(a^2*x^2-b)^(1/2)*(55*a^4*d*x^3-41*a^2*b*c*x^3-87*a^2*b*d*x+9*b^2*c*x)*((a*x+
(a^2*x^2-b)^(1/2))/b^(1/2))^(3/4)/a^2/b^(13/8)/(-b^(1/2)+a*x)^2/(b^(1/2)+a*x)^2+1/64*(-55*a^2*d+41*b*c)*arctan
(((a*x+(a^2*x^2-b)^(1/2))/b^(1/2))^(1/4))/a^3/b^(13/8)+1/64*(55*a^2*d-41*b*c)*arctanh(((a*x+(a^2*x^2-b)^(1/2))
/b^(1/2))^(1/4))/a^3/b^(13/8)-1/64*(-1)^(1/4)*(55*a^2*d-41*b*c)*arctanh((-1)^(1/4)*((a*x+(a^2*x^2-b)^(1/2))/b^
(1/2))^(1/4))/a^3/b^(13/8)-1/64*(-1)^(3/4)*(55*a^2*d-41*b*c)*arctanh((-1)^(3/4)*((a*x+(a^2*x^2-b)^(1/2))/b^(1/
2))^(1/4))/a^3/b^(13/8)

________________________________________________________________________________________

Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(1077\) vs. \(2(448)=896\).
time = 1.23, antiderivative size = 1077, normalized size of antiderivative = 2.40, number of steps used = 38, number of rules used = 18, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6874, 2147, 294, 296, 335, 306, 303, 1176, 631, 210, 1179, 642, 304, 209, 212, 2145, 474, 468} \begin {gather*} -\frac {11 c \left (a x+\sqrt {a^2 x^2-b}\right )^{11/4}}{6 a^3 \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^2}+\frac {8 b c \left (a x+\sqrt {a^2 x^2-b}\right )^{11/4}}{3 a^3 \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^3}+\frac {8 d \left (a x+\sqrt {a^2 x^2-b}\right )^{11/4}}{3 a \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^3}+\frac {41 c \left (a x+\sqrt {a^2 x^2-b}\right )^{3/4}}{48 a^3 \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )}+\frac {11 d \left (a x+\sqrt {a^2 x^2-b}\right )^{3/4}}{16 a b \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )}-\frac {11 d \left (a x+\sqrt {a^2 x^2-b}\right )^{3/4}}{6 a \left (b-\left (a x+\sqrt {a^2 x^2-b}\right )^2\right )^2}+\frac {41 c \text {ArcTan}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 a^3 b^{5/8}}-\frac {55 d \text {ArcTan}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 a b^{13/8}}+\frac {41 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^3 b^{5/8}}-\frac {55 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} a b^{13/8}}-\frac {41 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^3 b^{5/8}}+\frac {55 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} a b^{13/8}}-\frac {41 c \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 a^3 b^{5/8}}+\frac {55 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{\sqrt [8]{b}}\right )}{64 a b^{13/8}}-\frac {41 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^3 b^{5/8}}+\frac {55 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} a b^{13/8}}+\frac {41 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^3 b^{5/8}}-\frac {55 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} a b^{13/8}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + c*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(3/4))/(-b + a^2*x^2)^(5/2),x]

[Out]

(8*b*c*(a*x + Sqrt[-b + a^2*x^2])^(11/4))/(3*a^3*(b - (a*x + Sqrt[-b + a^2*x^2])^2)^3) + (8*d*(a*x + Sqrt[-b +
a^2*x^2])^(11/4))/(3*a*(b - (a*x + Sqrt[-b + a^2*x^2])^2)^3) - (11*d*(a*x + Sqrt[-b + a^2*x^2])^(3/4))/(6*a*(
b - (a*x + Sqrt[-b + a^2*x^2])^2)^2) - (11*c*(a*x + Sqrt[-b + a^2*x^2])^(11/4))/(6*a^3*(b - (a*x + Sqrt[-b + a
^2*x^2])^2)^2) + (41*c*(a*x + Sqrt[-b + a^2*x^2])^(3/4))/(48*a^3*(b - (a*x + Sqrt[-b + a^2*x^2])^2)) + (11*d*(
a*x + Sqrt[-b + a^2*x^2])^(3/4))/(16*a*b*(b - (a*x + Sqrt[-b + a^2*x^2])^2)) + (41*c*ArcTan[(a*x + Sqrt[-b + a
^2*x^2])^(1/4)/b^(1/8)])/(64*a^3*b^(5/8)) - (55*d*ArcTan[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)])/(64*a*b^(1
3/8)) + (41*c*ArcTan[1 - (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/b^(1/8)])/(64*Sqrt[2]*a^3*b^(5/8)) - (55*d
*ArcTan[1 - (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/b^(1/8)])/(64*Sqrt[2]*a*b^(13/8)) - (41*c*ArcTan[1 + (S
qrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/b^(1/8)])/(64*Sqrt[2]*a^3*b^(5/8)) + (55*d*ArcTan[1 + (Sqrt[2]*(a*x +
Sqrt[-b + a^2*x^2])^(1/4))/b^(1/8)])/(64*Sqrt[2]*a*b^(13/8)) - (41*c*ArcTanh[(a*x + Sqrt[-b + a^2*x^2])^(1/4)
/b^(1/8)])/(64*a^3*b^(5/8)) + (55*d*ArcTanh[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/b^(1/8)])/(64*a*b^(13/8)) - (41*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^3*b^(5/8)) + (55*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]*a*b^(13/8)) + (41*c*Log[b^(1/4) + Sqrt[2]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4) + S
qrt[a*x + Sqrt[-b + a^2*x^2]]])/(128*Sqrt[2]*a^3*b^(5/8)) - (55*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]*a*b^(13/8))

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 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 296

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

Rule 303

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

Rule 304

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

Rule 306

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

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 474

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

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 2147

Int[((g_) + (i_.)*(x_)^2)^(m_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Dis
t[(1/(2^(2*m + 1)*e*f^(2*m)))*(i/c)^m, Subst[Int[x^n*((d^2 + a*f^2 - 2*d*x + x^2)^(2*m + 1)/(-d + x)^(2*(m + 1
))), x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, d, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0] && E
qQ[c*g - a*i, 0] && IntegerQ[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 )^{3/4}}{\left (-b+a^2 x^2\right )^{5/2}} \, dx &=\int \left (\frac {d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{\left (-b+a^2 x^2\right )^{5/2}}+\frac {c x^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{\left (-b+a^2 x^2\right )^{5/2}}\right ) \, dx\\ &=c \int \frac {x^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{\left (-b+a^2 x^2\right )^{5/2}} \, dx+d \int \frac {\left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{\left (-b+a^2 x^2\right )^{5/2}} \, dx\\ &=\frac {(4 c) \text {Subst}\left (\int \frac {x^{7/4} \left (b+x^2\right )^2}{\left (-b+x^2\right )^4} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{a^3}+\frac {(16 d) \text {Subst}\left (\int \frac {x^{15/4}}{\left (-b+x^2\right )^4} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{a}\\ &=\frac {8 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {(2 c) \text {Subst}\left (\int \frac {x^{7/4} \left (5 b^2+6 b x^2\right )}{\left (-b+x^2\right )^3} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{3 a^3 b}+\frac {(22 d) \text {Subst}\left (\int \frac {x^{7/4}}{\left (-b+x^2\right )^3} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{3 a}\\ &=\frac {8 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{6 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {11 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{6 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {(41 c) \text {Subst}\left (\int \frac {x^{7/4}}{\left (-b+x^2\right )^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{24 a^3}+\frac {(11 d) \text {Subst}\left (\int \frac {1}{\sqrt [4]{x} \left (-b+x^2\right )^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{8 a}\\ &=\frac {8 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{6 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {11 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{6 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {41 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{48 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{16 a b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {(41 c) \text {Subst}\left (\int \frac {1}{\sqrt [4]{x} \left (-b+x^2\right )} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{64 a^3}-\frac {(55 d) \text {Subst}\left (\int \frac {1}{\sqrt [4]{x} \left (-b+x^2\right )} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{64 a b}\\ &=\frac {8 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{6 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {11 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{6 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {41 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{48 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{16 a b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {(41 c) \text {Subst}\left (\int \frac {x^2}{-b+x^8} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{16 a^3}-\frac {(55 d) \text {Subst}\left (\int \frac {x^2}{-b+x^8} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{16 a b}\\ &=\frac {8 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{6 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {11 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{6 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {41 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{48 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{16 a b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {(41 c) \text {Subst}\left (\int \frac {x^2}{\sqrt {b}-x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{32 a^3 \sqrt {b}}-\frac {(41 c) \text {Subst}\left (\int \frac {x^2}{\sqrt {b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{32 a^3 \sqrt {b}}+\frac {(55 d) \text {Subst}\left (\int \frac {x^2}{\sqrt {b}-x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{32 a b^{3/2}}+\frac {(55 d) \text {Subst}\left (\int \frac {x^2}{\sqrt {b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{32 a b^{3/2}}\\ &=\frac {8 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{6 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {11 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{6 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {41 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{48 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{16 a b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}-\frac {(41 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^3 \sqrt {b}}+\frac {(41 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^3 \sqrt {b}}+\frac {(41 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^3 \sqrt {b}}-\frac {(41 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^3 \sqrt {b}}+\frac {(55 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 a b^{3/2}}-\frac {(55 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 a b^{3/2}}-\frac {(55 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 a b^{3/2}}+\frac {(55 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 a b^{3/2}}\\ &=\frac {8 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{6 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {11 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{6 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {41 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{48 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{16 a b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {41 c \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^3 b^{5/8}}-\frac {55 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a b^{13/8}}-\frac {41 c \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^3 b^{5/8}}+\frac {55 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a b^{13/8}}-\frac {(41 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^3 b^{5/8}}-\frac {(41 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^3 b^{5/8}}-\frac {(41 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^3 \sqrt {b}}-\frac {(41 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^3 \sqrt {b}}+\frac {(55 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} a b^{13/8}}+\frac {(55 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} a b^{13/8}}+\frac {(55 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 a b^{3/2}}+\frac {(55 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 a b^{3/2}}\\ &=\frac {8 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{6 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {11 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{6 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {41 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{48 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{16 a b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {41 c \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^3 b^{5/8}}-\frac {55 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a b^{13/8}}-\frac {41 c \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^3 b^{5/8}}+\frac {55 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a b^{13/8}}-\frac {41 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^3 b^{5/8}}+\frac {55 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} a b^{13/8}}+\frac {41 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^3 b^{5/8}}-\frac {55 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} a b^{13/8}}-\frac {(41 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^3 b^{5/8}}+\frac {(41 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^3 b^{5/8}}+\frac {(55 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} a b^{13/8}}-\frac {(55 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} a b^{13/8}}\\ &=\frac {8 b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}+\frac {8 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{3 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^3}-\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{6 a \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}-\frac {11 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{6 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )^2}+\frac {41 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{48 a^3 \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {11 d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{16 a b \left (b-\left (a x+\sqrt {-b+a^2 x^2}\right )^2\right )}+\frac {41 c \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^3 b^{5/8}}-\frac {55 d \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a b^{13/8}}+\frac {41 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^3 b^{5/8}}-\frac {55 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} a b^{13/8}}-\frac {41 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^3 b^{5/8}}+\frac {55 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} a b^{13/8}}-\frac {41 c \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a^3 b^{5/8}}+\frac {55 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}\right )}{64 a b^{13/8}}-\frac {41 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^3 b^{5/8}}+\frac {55 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} a b^{13/8}}+\frac {41 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^3 b^{5/8}}-\frac {55 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} a b^{13/8}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]
time = 10.19, 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 )^{3/4}}{\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])^(3/4))/(-b + a^2*x^2)^(5/2),x]

[Out]

Integrate[((d + c*x^2)*(a*x + Sqrt[-b + a^2*x^2])^(3/4))/(-b + a^2*x^2)^(5/2), x]

________________________________________________________________________________________

Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (c \,x^{2}+d \right ) \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {3}{4}}}{\left (a^{2} x^{2}-b \right )^{\frac {5}{2}}}\, dx\]

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))^(3/4)/(a^2*x^2-b)^(5/2),x)

[Out]

int((c*x^2+d)*(a*x+(a^2*x^2-b)^(1/2))^(3/4)/(a^2*x^2-b)^(5/2),x)

________________________________________________________________________________________

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))^(3/4)/(a^2*x^2-b)^(5/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + d)*(a*x + sqrt(a^2*x^2 - b))^(3/4)/(a^2*x^2 - b)^(5/2), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4991 vs. \(2 (370) = 740\).
time = 0.50, size = 4991, normalized size = 11.14 \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))^(3/4)/(a^2*x^2-b)^(5/2),x, algorithm="fricas")

[Out]

-1/768*(12*sqrt(2)*(a^7*b^2*x^4 - 2*a^5*b^3*x^2 + a^3*b^4)*((83733937890625*a^16*d^8 - 499358756875000*a^14*b*
c*d^7 + 1302872392937500*a^12*b^2*c^2*d^6 - 1942464294925000*a^10*b^3*c^3*d^5 + 1810023547543750*a^8*b^4*c^4*d
^4 - 1079432224717000*a^6*b^5*c^5*d^3 + 402333829212700*a^4*b^6*c^6*d^2 - 85691880507640*a^2*b^7*c^7*d + 79849
25229121*b^8*c^8)/(a^24*b^13))^(1/8)*arctan((sqrt(2)*sqrt(a^18*b^10*((83733937890625*a^16*d^8 - 49935875687500
0*a^14*b*c*d^7 + 1302872392937500*a^12*b^2*c^2*d^6 - 1942464294925000*a^10*b^3*c^3*d^5 + 1810023547543750*a^8*
b^4*c^4*d^4 - 1079432224717000*a^6*b^5*c^5*d^3 + 402333829212700*a^4*b^6*c^6*d^2 - 85691880507640*a^2*b^7*c^7*
d + 7984925229121*b^8*c^8)/(a^24*b^13))^(3/4) + sqrt(2)*(166375*a^15*b^5*d^3 - 372075*a^13*b^6*c*d^2 + 277365*
a^11*b^7*c^2*d - 68921*a^9*b^8*c^3)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*((83733937890625*a^16*d^8 - 49935875687500
0*a^14*b*c*d^7 + 1302872392937500*a^12*b^2*c^2*d^6 - 1942464294925000*a^10*b^3*c^3*d^5 + 1810023547543750*a^8*
b^4*c^4*d^4 - 1079432224717000*a^6*b^5*c^5*d^3 + 402333829212700*a^4*b^6*c^6*d^2 - 85691880507640*a^2*b^7*c^7*
d + 7984925229121*b^8*c^8)/(a^24*b^13))^(3/8) + (27680640625*a^12*d^6 - 123807956250*a^10*b*c*d^5 + 2307330093
75*a^8*b^2*c^2*d^4 - 229334627500*a^6*b^3*c^3*d^3 + 128218905375*a^4*b^4*c^4*d^2 - 38232546330*a^2*b^5*c^5*d +
4750104241*b^6*c^6)*sqrt(a*x + sqrt(a^2*x^2 - b)))*a^15*b^8*((83733937890625*a^16*d^8 - 499358756875000*a^14*
b*c*d^7 + 1302872392937500*a^12*b^2*c^2*d^6 - 1942464294925000*a^10*b^3*c^3*d^5 + 1810023547543750*a^8*b^4*c^4
*d^4 - 1079432224717000*a^6*b^5*c^5*d^3 + 402333829212700*a^4*b^6*c^6*d^2 - 85691880507640*a^2*b^7*c^7*d + 798
4925229121*b^8*c^8)/(a^24*b^13))^(5/8) - 83733937890625*a^16*d^8 + 499358756875000*a^14*b*c*d^7 - 130287239293
7500*a^12*b^2*c^2*d^6 + 1942464294925000*a^10*b^3*c^3*d^5 - 1810023547543750*a^8*b^4*c^4*d^4 + 107943222471700
0*a^6*b^5*c^5*d^3 - 402333829212700*a^4*b^6*c^6*d^2 + 85691880507640*a^2*b^7*c^7*d - 7984925229121*b^8*c^8 - s
qrt(2)*(166375*a^21*b^8*d^3 - 372075*a^19*b^9*c*d^2 + 277365*a^17*b^10*c^2*d - 68921*a^15*b^11*c^3)*(a*x + sqr
t(a^2*x^2 - b))^(1/4)*((83733937890625*a^16*d^8 - 499358756875000*a^14*b*c*d^7 + 1302872392937500*a^12*b^2*c^2
*d^6 - 1942464294925000*a^10*b^3*c^3*d^5 + 1810023547543750*a^8*b^4*c^4*d^4 - 1079432224717000*a^6*b^5*c^5*d^3
+ 402333829212700*a^4*b^6*c^6*d^2 - 85691880507640*a^2*b^7*c^7*d + 7984925229121*b^8*c^8)/(a^24*b^13))^(5/8))
/(83733937890625*a^16*d^8 - 499358756875000*a^14*b*c*d^7 + 1302872392937500*a^12*b^2*c^2*d^6 - 194246429492500
0*a^10*b^3*c^3*d^5 + 1810023547543750*a^8*b^4*c^4*d^4 - 1079432224717000*a^6*b^5*c^5*d^3 + 402333829212700*a^4
*b^6*c^6*d^2 - 85691880507640*a^2*b^7*c^7*d + 7984925229121*b^8*c^8)) + 12*sqrt(2)*(a^7*b^2*x^4 - 2*a^5*b^3*x^
2 + a^3*b^4)*((83733937890625*a^16*d^8 - 499358756875000*a^14*b*c*d^7 + 1302872392937500*a^12*b^2*c^2*d^6 - 19
42464294925000*a^10*b^3*c^3*d^5 + 1810023547543750*a^8*b^4*c^4*d^4 - 1079432224717000*a^6*b^5*c^5*d^3 + 402333
829212700*a^4*b^6*c^6*d^2 - 85691880507640*a^2*b^7*c^7*d + 7984925229121*b^8*c^8)/(a^24*b^13))^(1/8)*arctan((s
qrt(2)*sqrt(a^18*b^10*((83733937890625*a^16*d^8 - 499358756875000*a^14*b*c*d^7 + 1302872392937500*a^12*b^2*c^2
*d^6 - 1942464294925000*a^10*b^3*c^3*d^5 + 1810023547543750*a^8*b^4*c^4*d^4 - 1079432224717000*a^6*b^5*c^5*d^3
+ 402333829212700*a^4*b^6*c^6*d^2 - 85691880507640*a^2*b^7*c^7*d + 7984925229121*b^8*c^8)/(a^24*b^13))^(3/4)
- sqrt(2)*(166375*a^15*b^5*d^3 - 372075*a^13*b^6*c*d^2 + 277365*a^11*b^7*c^2*d - 68921*a^9*b^8*c^3)*(a*x + sqr
t(a^2*x^2 - b))^(1/4)*((83733937890625*a^16*d^8 - 499358756875000*a^14*b*c*d^7 + 1302872392937500*a^12*b^2*c^2
*d^6 - 1942464294925000*a^10*b^3*c^3*d^5 + 1810023547543750*a^8*b^4*c^4*d^4 - 1079432224717000*a^6*b^5*c^5*d^3
+ 402333829212700*a^4*b^6*c^6*d^2 - 85691880507640*a^2*b^7*c^7*d + 7984925229121*b^8*c^8)/(a^24*b^13))^(3/8)
+ (27680640625*a^12*d^6 - 123807956250*a^10*b*c*d^5 + 230733009375*a^8*b^2*c^2*d^4 - 229334627500*a^6*b^3*c^3*
d^3 + 128218905375*a^4*b^4*c^4*d^2 - 38232546330*a^2*b^5*c^5*d + 4750104241*b^6*c^6)*sqrt(a*x + sqrt(a^2*x^2 -
b)))*a^15*b^8*((83733937890625*a^16*d^8 - 499358756875000*a^14*b*c*d^7 + 1302872392937500*a^12*b^2*c^2*d^6 -
1942464294925000*a^10*b^3*c^3*d^5 + 1810023547543750*a^8*b^4*c^4*d^4 - 1079432224717000*a^6*b^5*c^5*d^3 + 4023
33829212700*a^4*b^6*c^6*d^2 - 85691880507640*a^2*b^7*c^7*d + 7984925229121*b^8*c^8)/(a^24*b^13))^(5/8) + 83733
937890625*a^16*d^8 - 499358756875000*a^14*b*c*d^7 + 1302872392937500*a^12*b^2*c^2*d^6 - 1942464294925000*a^10*
b^3*c^3*d^5 + 1810023547543750*a^8*b^4*c^4*d^4 - 1079432224717000*a^6*b^5*c^5*d^3 + 402333829212700*a^4*b^6*c^
6*d^2 - 85691880507640*a^2*b^7*c^7*d + 7984925229121*b^8*c^8 - sqrt(2)*(166375*a^21*b^8*d^3 - 372075*a^19*b^9*
c*d^2 + 277365*a^17*b^10*c^2*d - 68921*a^15*b^11*c^3)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*((83733937890625*a^16*d^
8 - 499358756875000*a^14*b*c*d^7 + 1302872392937500*a^12*b^2*c^2*d^6 - 1942464294925000*a^10*b^3*c^3*d^5 + 181
0023547543750*a^8*b^4*c^4*d^4 - 107943222471700...

________________________________________________________________________________________

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 {3}{4}} \left (c x^{2} + d\right )}{\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))**(3/4)/(a**2*x**2-b)**(5/2),x)

[Out]

Integral((a*x + sqrt(a**2*x**2 - b))**(3/4)*(c*x**2 + d)/(a**2*x**2 - b)**(5/2), x)

________________________________________________________________________________________

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))^(3/4)/(a^2*x^2-b)^(5/2),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

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 )}^{3/4}\,\left (c\,x^2+d\right )}{{\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))^(3/4)*(d + c*x^2))/(a^2*x^2 - b)^(5/2),x)

[Out]

int(((a*x + (a^2*x^2 - b)^(1/2))^(3/4)*(d + c*x^2))/(a^2*x^2 - b)^(5/2), x)

________________________________________________________________________________________