3.71 \(\int \frac{(c+d x^2)^{3/2} (e+f x^2)^{3/2}}{a+b x^2} \, dx\)

Optimal. Leaf size=659 \[ \frac{e^{3/2} \sqrt{c+d x^2} \left (15 a^2 d^2 f-5 a b d (5 c f+3 d e)+3 b^2 c (3 c f+8 d e)\right ) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right )}{15 b^3 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left (15 a^2 d^2 f^2-20 a b d f (c f+d e)+3 b^2 \left (c^2 f^2+9 c d e f+d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 b^3 d \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{e^{3/2} \sqrt{c+d x^2} (b c-a d)^2 (b e-a f) \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a b^3 c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{f^2 x \sqrt{c+d x^2} (b c-a d)^2}{b^3 d \sqrt{e+f x^2}}+\frac{f x \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d)}{3 b^2}+\frac{2 f x \sqrt{c+d x^2} (b c-a d) (2 d e-c f)}{3 b^2 d \sqrt{e+f x^2}}+\frac{x \sqrt{c+d x^2} \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )}{15 b d \sqrt{e+f x^2}}+\frac{f x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 b}+\frac{2 x \sqrt{c+d x^2} \sqrt{e+f x^2} (3 d e-c f)}{15 b} \]

[Out]

((b*c - a*d)^2*f^2*x*Sqrt[c + d*x^2])/(b^3*d*Sqrt[e + f*x^2]) + (2*(b*c - a*d)*f*(2*d*e - c*f)*x*Sqrt[c + d*x^
2])/(3*b^2*d*Sqrt[e + f*x^2]) + ((3*d^2*e^2 + 7*c*d*e*f - 2*c^2*f^2)*x*Sqrt[c + d*x^2])/(15*b*d*Sqrt[e + f*x^2
]) + ((b*c - a*d)*f*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*b^2) + (2*(3*d*e - c*f)*x*Sqrt[c + d*x^2]*Sqrt[e + f
*x^2])/(15*b) + (f*x*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])/(5*b) - (Sqrt[e]*(15*a^2*d^2*f^2 - 20*a*b*d*f*(d*e + c
*f) + 3*b^2*(d^2*e^2 + 9*c*d*e*f + c^2*f^2))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/
(c*f)])/(15*b^3*d*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (e^(3/2)*(15*a^2*d^2*f + 3*
b^2*c*(8*d*e + 3*c*f) - 5*a*b*d*(3*d*e + 5*c*f))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d
*e)/(c*f)])/(15*b^3*c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + ((b*c - a*d)^2*e^(3/2)*
(b*e - a*f)*Sqrt[c + d*x^2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a*b^3*
c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.752705, antiderivative size = 784, normalized size of antiderivative = 1.19, number of steps used = 14, number of rules used = 9, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.281, Rules used = {545, 416, 528, 531, 418, 492, 411, 543, 539} \[ \frac{c^{3/2} \sqrt{e+f x^2} (b c-a d) (b e-a f)^2 \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a b^3 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{f x \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d)}{3 b^2}+\frac{f x \sqrt{c+d x^2} (b c-a d) (-3 a d f+b c f+4 b d e)}{3 b^3 d \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (b c-a d) (5 b e-3 a f) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b^3 \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (b c-a d) (-3 a d f+b c f+4 b d e) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b^3 d \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{c+d x^2} \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )}{15 b d \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 b d \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{e^{3/2} \sqrt{c+d x^2} (9 d e-c f) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 b \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{f x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 b}+\frac{2 x \sqrt{c+d x^2} \sqrt{e+f x^2} (3 d e-c f)}{15 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c - a*d)*f*(4*b*d*e + b*c*f - 3*a*d*f)*x*Sqrt[c + d*x^2])/(3*b^3*d*Sqrt[e + f*x^2]) + ((3*d^2*e^2 + 7*c*d*
e*f - 2*c^2*f^2)*x*Sqrt[c + d*x^2])/(15*b*d*Sqrt[e + f*x^2]) + ((b*c - a*d)*f*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2
])/(3*b^2) + (2*(3*d*e - c*f)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(15*b) + (f*x*(c + d*x^2)^(3/2)*Sqrt[e + f*x^
2])/(5*b) - ((b*c - a*d)*Sqrt[e]*Sqrt[f]*(4*b*d*e + b*c*f - 3*a*d*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]
*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b^3*d*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (Sqrt[e]*(3*
d^2*e^2 + 7*c*d*e*f - 2*c^2*f^2)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*
b*d*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + ((b*c - a*d)*Sqrt[e]*Sqrt[f]*(5*b*e - 3*a
*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b^3*Sqrt[(e*(c + d*x^2))/(c*(e
+ f*x^2))]*Sqrt[e + f*x^2]) + (e^(3/2)*(9*d*e - c*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1
- (d*e)/(c*f)])/(15*b*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (c^(3/2)*(b*c - a*d)*(
b*e - a*f)^2*Sqrt[e + f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(a*b^3
*Sqrt[d]*e*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])

Rule 545

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

Rule 416

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^(q - 1))/(b*(n*(p + q) + 1)), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 528

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

Rule 531

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

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticF[ArcT
an[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[c + d*x^2]), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rule 411

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticE[ArcTan
[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rule 543

Int[(((c_) + (d_.)*(x_)^2)^(3/2)*Sqrt[(e_) + (f_.)*(x_)^2])/((a_) + (b_.)*(x_)^2), x_Symbol] :> Dist[(b*c - a*
d)^2/b^2, Int[Sqrt[e + f*x^2]/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] + Dist[d/b^2, Int[((2*b*c - a*d + b*d*x^2)
*Sqrt[e + f*x^2])/Sqrt[c + d*x^2], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c] && PosQ[f/e]

Rule 539

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(c*Sqrt[e +
f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[Rt[d/c, 2]*x], 1 - (c*f)/(d*e)])/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sq
rt[(c*(e + f*x^2))/(e*(c + d*x^2))]), x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c]

Rubi steps

\begin{align*} \int \frac{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{a+b x^2} \, dx &=\frac{d \int \sqrt{c+d x^2} \left (e+f x^2\right )^{3/2} \, dx}{b}+\frac{(b c-a d) \int \frac{\sqrt{c+d x^2} \left (e+f x^2\right )^{3/2}}{a+b x^2} \, dx}{b}\\ &=\frac{f x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 b}+\frac{\int \frac{\sqrt{c+d x^2} \left (e (5 d e-c f)+2 f (3 d e-c f) x^2\right )}{\sqrt{e+f x^2}} \, dx}{5 b}+\frac{((b c-a d) f) \int \frac{\sqrt{c+d x^2} \left (2 b e-a f+b f x^2\right )}{\sqrt{e+f x^2}} \, dx}{b^3}+\frac{\left ((b c-a d) (b e-a f)^2\right ) \int \frac{\sqrt{c+d x^2}}{\left (a+b x^2\right ) \sqrt{e+f x^2}} \, dx}{b^3}\\ &=\frac{(b c-a d) f x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b^2}+\frac{2 (3 d e-c f) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{15 b}+\frac{f x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 b}+\frac{c^{3/2} (b c-a d) (b e-a f)^2 \sqrt{e+f x^2} \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a b^3 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{(b c-a d) \int \frac{c f (5 b e-3 a f)+f (4 b d e+b c f-3 a d f) x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{3 b^3}+\frac{\int \frac{c e f (9 d e-c f)+f \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right ) x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{15 b f}\\ &=\frac{(b c-a d) f x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b^2}+\frac{2 (3 d e-c f) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{15 b}+\frac{f x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 b}+\frac{c^{3/2} (b c-a d) (b e-a f)^2 \sqrt{e+f x^2} \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a b^3 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{(c (b c-a d) f (5 b e-3 a f)) \int \frac{1}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{3 b^3}+\frac{(c e (9 d e-c f)) \int \frac{1}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{15 b}+\frac{((b c-a d) f (4 b d e+b c f-3 a d f)) \int \frac{x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{3 b^3}+\frac{\left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right ) \int \frac{x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{15 b}\\ &=\frac{(b c-a d) f (4 b d e+b c f-3 a d f) x \sqrt{c+d x^2}}{3 b^3 d \sqrt{e+f x^2}}+\frac{\left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right ) x \sqrt{c+d x^2}}{15 b d \sqrt{e+f x^2}}+\frac{(b c-a d) f x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b^2}+\frac{2 (3 d e-c f) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{15 b}+\frac{f x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 b}+\frac{(b c-a d) \sqrt{e} \sqrt{f} (5 b e-3 a f) \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b^3 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{e^{3/2} (9 d e-c f) \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 b \sqrt{f} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{c^{3/2} (b c-a d) (b e-a f)^2 \sqrt{e+f x^2} \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a b^3 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{((b c-a d) e f (4 b d e+b c f-3 a d f)) \int \frac{\sqrt{c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{3 b^3 d}-\frac{\left (e \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )\right ) \int \frac{\sqrt{c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{15 b d}\\ &=\frac{(b c-a d) f (4 b d e+b c f-3 a d f) x \sqrt{c+d x^2}}{3 b^3 d \sqrt{e+f x^2}}+\frac{\left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right ) x \sqrt{c+d x^2}}{15 b d \sqrt{e+f x^2}}+\frac{(b c-a d) f x \sqrt{c+d x^2} \sqrt{e+f x^2}}{3 b^2}+\frac{2 (3 d e-c f) x \sqrt{c+d x^2} \sqrt{e+f x^2}}{15 b}+\frac{f x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 b}-\frac{(b c-a d) \sqrt{e} \sqrt{f} (4 b d e+b c f-3 a d f) \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b^3 d \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}-\frac{\sqrt{e} \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right ) \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 b d \sqrt{f} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{(b c-a d) \sqrt{e} \sqrt{f} (5 b e-3 a f) \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b^3 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{e^{3/2} (9 d e-c f) \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 b \sqrt{f} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{c^{3/2} (b c-a d) (b e-a f)^2 \sqrt{e+f x^2} \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a b^3 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}\\ \end{align*}

Mathematica [C]  time = 2.49514, size = 445, normalized size = 0.68 \[ \frac{-i a \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (15 a^2 b d f^2 (2 c f+d e)-15 a^3 d^2 f^3+5 a b^2 f \left (-3 c^2 f^2-7 c d e f+d^2 e^2\right )-3 b^3 e \left (-7 c^2 f^2+c d e f+d^2 e^2\right )\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{d}{c}}\right ),\frac{c f}{d e}\right )-i a b e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (15 a^2 d^2 f^2-20 a b d f (c f+d e)+3 b^2 \left (c^2 f^2+9 c d e f+d^2 e^2\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+f \left (a b^2 x \sqrt{\frac{d}{c}} \left (c+d x^2\right ) \left (e+f x^2\right ) \left (3 b \left (2 c f+2 d e+d f x^2\right )-5 a d f\right )-15 i \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b c-a d)^2 (b e-a f)^2 \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )}{15 a b^4 f \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*a*b*e*(15*a^2*d^2*f^2 - 20*a*b*d*f*(d*e + c*f) + 3*b^2*(d^2*e^2 + 9*c*d*e*f + c^2*f^2))*Sqrt[1 + (d*x^2)
/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*a*(-15*a^3*d^2*f^3 + 15*a^2*b*d*f^2
*(d*e + 2*c*f) - 3*b^3*e*(d^2*e^2 + c*d*e*f - 7*c^2*f^2) + 5*a*b^2*f*(d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2))*Sqrt[1
+ (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + f*(a*b^2*Sqrt[d/c]*x*(c + d
*x^2)*(e + f*x^2)*(-5*a*d*f + 3*b*(2*d*e + 2*c*f + d*f*x^2)) - (15*I)*(b*c - a*d)^2*(b*e - a*f)^2*Sqrt[1 + (d*
x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(15*a*b^4*Sqrt[d/c]
*f*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

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Maple [B]  time = 0.023, size = 1939, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)*(-15*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(-d/c)^(1/2),b
*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a^4*d^2*f^3+35*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(-d/c)^(1
/2),(c*f/d/e)^(1/2))*a^2*b^2*c*d*e*f^2+30*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(-d/c)^(1/2),b*
c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a*b^3*c*d*e^2*f-27*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE(x*(-d/c)
^(1/2),(c*f/d/e)^(1/2))*a*b^3*c*d*e^2*f-60*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(-d/c)^(1/2),b
*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a^2*b^2*c*d*e*f^2+3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(-d/
c)^(1/2),(c*f/d/e)^(1/2))*a*b^3*d^2*e^3-3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE(x*(-d/c)^(1/2),(c*
f/d/e)^(1/2))*a*b^3*d^2*e^3-15*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e
)^(1/2)/(-d/c)^(1/2))*a^2*b^2*c^2*f^3-15*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(-d/c)^(1/2),b*c
/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*b^4*c^2*e^2*f+15*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(-d/c)^(1
/2),(c*f/d/e)^(1/2))*a^2*b^2*c^2*f^3-9*(-d/c)^(1/2)*x^5*a*b^3*c*d*f^3-9*(-d/c)^(1/2)*x^5*a*b^3*d^2*e*f^2+5*(-d
/c)^(1/2)*x^3*a^2*b^2*c*d*f^3+5*(-d/c)^(1/2)*x^3*a^2*b^2*d^2*e*f^2-6*(-d/c)^(1/2)*x^3*a*b^3*d^2*e^2*f-6*(-d/c)
^(1/2)*x*a*b^3*c^2*e*f^2+15*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*
a^4*d^2*f^3-3*(-d/c)^(1/2)*x^7*a*b^3*d^2*f^3+5*(-d/c)^(1/2)*x^5*a^2*b^2*d^2*f^3-6*(-d/c)^(1/2)*x^3*a*b^3*c^2*f
^3+3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b^3*c*d*e^2*f+20*((d*
x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2*b^2*c*d*e*f^2-15*((d*x^2+c)/
c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a^2*b^2*d^2*e^2*f+30
*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a*b^3*c^
2*e*f^2-15*(-d/c)^(1/2)*x^3*a*b^3*c*d*e*f^2+5*(-d/c)^(1/2)*x*a^2*b^2*c*d*e*f^2-6*(-d/c)^(1/2)*x*a*b^3*c*d*e^2*
f-30*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^3*b*c*d*f^3-15*((d*x^
2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^3*b*d^2*e*f^2-5*((d*x^2+c)/c)^(1
/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2*b^2*d^2*e^2*f-21*((d*x^2+c)/c)^(1/2)*((f
*x^2+e)/e)^(1/2)*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b^3*c^2*e*f^2-15*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e
)^(1/2)*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^3*b*d^2*e*f^2+20*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*E
llipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2*b^2*d^2*e^2*f-3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE
(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b^3*c^2*e*f^2+30*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(-d/c
)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a^3*b*c*d*f^3+30*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi
(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a^3*b*d^2*e*f^2)/f/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)/b^4/(-d/c)
^(1/2)/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [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((d*x^2+c)^(3/2)*(f*x^2+e)^(3/2)/(b*x^2+a),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c + d*x**2)**(3/2)*(e + f*x**2)**(3/2)/(a + b*x**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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