3.832 \(\int (e x)^{5/2} (a+b x^2)^2 (c+d x^2)^{3/2} \, dx\)

Optimal. Leaf size=530 \[ -\frac{4 c^{13/4} e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (51 a^2 d^2+b c (11 b c-42 a d)\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),\frac{1}{2}\right )}{3315 d^{15/4} \sqrt{c+d x^2}}-\frac{8 c^3 e^2 \sqrt{e x} \sqrt{c+d x^2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{3315 d^{7/2} \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{8 c^{13/4} e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (51 a^2 d^2+b c (11 b c-42 a d)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{3315 d^{15/4} \sqrt{c+d x^2}}+\frac{8 c^2 e (e x)^{3/2} \sqrt{c+d x^2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{9945 d^3}+\frac{2 (e x)^{7/2} \left (c+d x^2\right )^{3/2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{663 d^2 e}+\frac{4 c (e x)^{7/2} \sqrt{c+d x^2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{1989 d^2 e}-\frac{2 b (e x)^{7/2} \left (c+d x^2\right )^{5/2} (11 b c-42 a d)}{357 d^2 e}+\frac{2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3} \]

[Out]

(8*c^2*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*e*(e*x)^(3/2)*Sqrt[c + d*x^2])/(9945*d^3) + (4*c*(51*a^2*d^2 + b*c
*(11*b*c - 42*a*d))*(e*x)^(7/2)*Sqrt[c + d*x^2])/(1989*d^2*e) - (8*c^3*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*e^
2*Sqrt[e*x]*Sqrt[c + d*x^2])/(3315*d^(7/2)*(Sqrt[c] + Sqrt[d]*x)) + (2*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*(e
*x)^(7/2)*(c + d*x^2)^(3/2))/(663*d^2*e) - (2*b*(11*b*c - 42*a*d)*(e*x)^(7/2)*(c + d*x^2)^(5/2))/(357*d^2*e) +
(2*b^2*(e*x)^(11/2)*(c + d*x^2)^(5/2))/(21*d*e^3) + (8*c^(13/4)*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*e^(5/2)*
(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4
)*Sqrt[e])], 1/2])/(3315*d^(15/4)*Sqrt[c + d*x^2]) - (4*c^(13/4)*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*e^(5/2)*
(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4
)*Sqrt[e])], 1/2])/(3315*d^(15/4)*Sqrt[c + d*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.562325, antiderivative size = 530, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {464, 459, 279, 321, 329, 305, 220, 1196} \[ -\frac{8 c^3 e^2 \sqrt{e x} \sqrt{c+d x^2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{3315 d^{7/2} \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{4 c^{13/4} e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (51 a^2 d^2+b c (11 b c-42 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{3315 d^{15/4} \sqrt{c+d x^2}}+\frac{8 c^{13/4} e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (51 a^2 d^2+b c (11 b c-42 a d)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{3315 d^{15/4} \sqrt{c+d x^2}}+\frac{8 c^2 e (e x)^{3/2} \sqrt{c+d x^2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{9945 d^3}+\frac{2 (e x)^{7/2} \left (c+d x^2\right )^{3/2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{663 d^2 e}+\frac{4 c (e x)^{7/2} \sqrt{c+d x^2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{1989 d^2 e}-\frac{2 b (e x)^{7/2} \left (c+d x^2\right )^{5/2} (11 b c-42 a d)}{357 d^2 e}+\frac{2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*c^2*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*e*(e*x)^(3/2)*Sqrt[c + d*x^2])/(9945*d^3) + (4*c*(51*a^2*d^2 + b*c
*(11*b*c - 42*a*d))*(e*x)^(7/2)*Sqrt[c + d*x^2])/(1989*d^2*e) - (8*c^3*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*e^
2*Sqrt[e*x]*Sqrt[c + d*x^2])/(3315*d^(7/2)*(Sqrt[c] + Sqrt[d]*x)) + (2*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*(e
*x)^(7/2)*(c + d*x^2)^(3/2))/(663*d^2*e) - (2*b*(11*b*c - 42*a*d)*(e*x)^(7/2)*(c + d*x^2)^(5/2))/(357*d^2*e) +
(2*b^2*(e*x)^(11/2)*(c + d*x^2)^(5/2))/(21*d*e^3) + (8*c^(13/4)*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*e^(5/2)*
(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4
)*Sqrt[e])], 1/2])/(3315*d^(15/4)*Sqrt[c + d*x^2]) - (4*c^(13/4)*(51*a^2*d^2 + b*c*(11*b*c - 42*a*d))*e^(5/2)*
(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4
)*Sqrt[e])], 1/2])/(3315*d^(15/4)*Sqrt[c + d*x^2])

Rule 464

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

Rule 459

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

Rule 279

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

Rule 321

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*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 329

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 305

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x],
x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 1196

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

Rubi steps

\begin{align*} \int (e x)^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx &=\frac{2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3}+\frac{2 \int (e x)^{5/2} \left (c+d x^2\right )^{3/2} \left (\frac{21 a^2 d}{2}-\frac{1}{2} b (11 b c-42 a d) x^2\right ) \, dx}{21 d}\\ &=-\frac{2 b (11 b c-42 a d) (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{357 d^2 e}+\frac{2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3}+\frac{1}{51} \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right ) \int (e x)^{5/2} \left (c+d x^2\right )^{3/2} \, dx\\ &=\frac{2 \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right ) (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{663 e}-\frac{2 b (11 b c-42 a d) (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{357 d^2 e}+\frac{2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3}+\frac{1}{221} \left (2 c \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right )\right ) \int (e x)^{5/2} \sqrt{c+d x^2} \, dx\\ &=\frac{4 c \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right ) (e x)^{7/2} \sqrt{c+d x^2}}{1989 e}+\frac{2 \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right ) (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{663 e}-\frac{2 b (11 b c-42 a d) (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{357 d^2 e}+\frac{2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3}+\frac{\left (4 c^2 \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right )\right ) \int \frac{(e x)^{5/2}}{\sqrt{c+d x^2}} \, dx}{1989}\\ &=\frac{8 c^2 \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right ) e (e x)^{3/2} \sqrt{c+d x^2}}{9945 d}+\frac{4 c \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right ) (e x)^{7/2} \sqrt{c+d x^2}}{1989 e}+\frac{2 \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right ) (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{663 e}-\frac{2 b (11 b c-42 a d) (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{357 d^2 e}+\frac{2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3}-\frac{\left (4 c^3 \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right ) e^2\right ) \int \frac{\sqrt{e x}}{\sqrt{c+d x^2}} \, dx}{3315 d}\\ &=\frac{8 c^2 \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right ) e (e x)^{3/2} \sqrt{c+d x^2}}{9945 d}+\frac{4 c \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right ) (e x)^{7/2} \sqrt{c+d x^2}}{1989 e}+\frac{2 \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right ) (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{663 e}-\frac{2 b (11 b c-42 a d) (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{357 d^2 e}+\frac{2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3}-\frac{\left (8 c^3 \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right ) e\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{3315 d}\\ &=\frac{8 c^2 \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right ) e (e x)^{3/2} \sqrt{c+d x^2}}{9945 d}+\frac{4 c \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right ) (e x)^{7/2} \sqrt{c+d x^2}}{1989 e}+\frac{2 \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right ) (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{663 e}-\frac{2 b (11 b c-42 a d) (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{357 d^2 e}+\frac{2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3}-\frac{\left (8 c^{7/2} \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right ) e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{3315 d^{3/2}}+\frac{\left (8 c^{7/2} \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right ) e^2\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{d} x^2}{\sqrt{c} e}}{\sqrt{c+\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{3315 d^{3/2}}\\ &=\frac{8 c^2 \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right ) e (e x)^{3/2} \sqrt{c+d x^2}}{9945 d}+\frac{4 c \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right ) (e x)^{7/2} \sqrt{c+d x^2}}{1989 e}-\frac{8 c^3 \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right ) e^2 \sqrt{e x} \sqrt{c+d x^2}}{3315 d^{3/2} \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{2 \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right ) (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{663 e}-\frac{2 b (11 b c-42 a d) (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{357 d^2 e}+\frac{2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3}+\frac{8 c^{13/4} \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right ) e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{3315 d^{7/4} \sqrt{c+d x^2}}-\frac{4 c^{13/4} \left (51 a^2+\frac{b c (11 b c-42 a d)}{d^2}\right ) e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{3315 d^{7/4} \sqrt{c+d x^2}}\\ \end{align*}

Mathematica [C]  time = 0.167632, size = 210, normalized size = 0.4 \[ \frac{2 e (e x)^{3/2} \left (\left (c+d x^2\right ) \left (357 a^2 d^2 \left (4 c^2+25 c d x^2+15 d^2 x^4\right )+42 a b d \left (20 c^2 d x^2-28 c^3+285 c d^2 x^4+195 d^3 x^6\right )+b^2 \left (180 c^2 d^2 x^4-220 c^3 d x^2+308 c^4+4485 c d^3 x^6+3315 d^4 x^8\right )\right )-84 c^3 \sqrt{\frac{c}{d x^2}+1} \left (51 a^2 d^2-42 a b c d+11 b^2 c^2\right ) \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c}{d x^2}\right )\right )}{69615 d^3 \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*e*(e*x)^(3/2)*((c + d*x^2)*(357*a^2*d^2*(4*c^2 + 25*c*d*x^2 + 15*d^2*x^4) + 42*a*b*d*(-28*c^3 + 20*c^2*d*x^
2 + 285*c*d^2*x^4 + 195*d^3*x^6) + b^2*(308*c^4 - 220*c^3*d*x^2 + 180*c^2*d^2*x^4 + 4485*c*d^3*x^6 + 3315*d^4*
x^8)) - 84*c^3*(11*b^2*c^2 - 42*a*b*c*d + 51*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*Hypergeometric2F1[-1/4, 1/2, 3/4, -(
c/(d*x^2))]))/(69615*d^3*Sqrt[c + d*x^2])

________________________________________________________________________________________

Maple [A]  time = 0.034, size = 743, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/69615/x*e^2*(e*x)^(1/2)/(d*x^2+c)^(1/2)/d^4*(-3315*x^12*b^2*d^6-8190*x^10*a*b*d^6-7800*x^10*b^2*c*d^5-5355*
x^8*a^2*d^6-20160*x^8*a*b*c*d^5-4665*x^8*b^2*c^2*d^4-14280*x^6*a^2*c*d^5-12810*x^6*a*b*c^2*d^4+40*x^6*b^2*c^3*
d^3+4284*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(
1/2)*d)^(1/2)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^2*c^4*d^2-3528*((d*x+(-c*d)^(1/
2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticE((
(d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b*c^5*d+924*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1
/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1
/2))^(1/2),1/2*2^(1/2))*b^2*c^6-2142*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*
d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^2*c
^4*d^2+1764*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d
)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b*c^5*d-462*((d*x+(-c*d)^(1/
2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF((
(d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^6-10353*x^4*a^2*c^2*d^4+336*x^4*a*b*c^3*d^3-88*x^4*b
^2*c^4*d^2-1428*x^2*a^2*c^3*d^3+1176*x^2*a*b*c^4*d^2-308*x^2*b^2*c^5*d)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(5/2)*(b*x^2+a)^2*(d*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{2} d e^{2} x^{8} +{\left (b^{2} c + 2 \, a b d\right )} e^{2} x^{6} + a^{2} c e^{2} x^{2} +{\left (2 \, a b c + a^{2} d\right )} e^{2} x^{4}\right )} \sqrt{d x^{2} + c} \sqrt{e x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(5/2)*(b*x^2+a)^2*(d*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

integral((b^2*d*e^2*x^8 + (b^2*c + 2*a*b*d)*e^2*x^6 + a^2*c*e^2*x^2 + (2*a*b*c + a^2*d)*e^2*x^4)*sqrt(d*x^2 +
c)*sqrt(e*x), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**(5/2)*(b*x**2+a)**2*(d*x**2+c)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(5/2)*(b*x^2+a)^2*(d*x^2+c)^(3/2),x, algorithm="giac")

[Out]

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