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

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

[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)])/(1
5*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))]*Sq
rt[e + f*x^2])

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Rubi [A]  time = 2.18842, 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 \[ \frac{c^{3/2} \sqrt{e+f x^2} (b c-a d) (b e-a f)^2 \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a b^3 \sqrt{d} e \sqrt{c+d x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac{f x \sqrt{c+d x^2} (b c-a d) (-3 a d f+b c f+4 b d e)}{3 b^3 d \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (b c-a d) (5 b e-3 a f) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b^3 \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (b c-a d) (-3 a d f+b c f+4 b d e) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 b^3 d \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{f x \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d)}{3 b^2}+\frac{x \sqrt{c+d x^2} \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right )}{15 b d \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left (-2 c^2 f^2+7 c d e f+3 d^2 e^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 b d \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{e^{3/2} \sqrt{c+d x^2} (9 d e-c f) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 b \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{f x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2}}{5 b}+\frac{2 x \sqrt{c+d x^2} \sqrt{e+f x^2} (3 d e-c f)}{15 b} \]

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)*Sqr
t[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)*S
qrt[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))])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [C]  time = 4.50967, size = 445, normalized size = 0.68 \[ \frac{-i a b e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (15 a^2 d^2 f^2-20 a b d f (c f+d e)+3 b^2 \left (c^2 f^2+9 c d e f+d^2 e^2\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )-i a \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (-15 a^3 d^2 f^3+15 a^2 b d f^2 (2 c f+d e)+5 a b^2 f \left (-3 c^2 f^2-7 c d e f+d^2 e^2\right )-3 b^3 e \left (-7 c^2 f^2+c d e f+d^2 e^2\right )\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+f \left (a b^2 x \sqrt{\frac{d}{c}} \left (c+d x^2\right ) \left (e+f x^2\right ) \left (3 b \left (2 c f+2 d e+d f x^2\right )-5 a d f\right )-15 i \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b c-a d)^2 (b e-a f)^2 \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )}{15 a b^4 f \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

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]*Sq
rt[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 = 0.032, size = 1939, normalized size = 2.9 \[ \text{result too large to display} \]

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)*(3*(-d/c)^(1/2)*x^7*a*b^3*d^2*f^3+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+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)*Ellipti
cPi(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+15*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticP
i(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*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Ellipti
cF(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)*El
lipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2*b^2*d^2*e^2*f-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+15*(-d/c)^(1/2)*x^3*a*b^3*c*d*e*f^
2-5*(-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^3-15*((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+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+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-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^3-3
*((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)*El
lipticPi(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+6*(-d/c)^(1/2)*x*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*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*c*d*e^2*f-20*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Elliptic
E(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2*b^2*c*d*e*f^2-35*((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)/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. \[ \int \frac{{\left (d x^{2} + c\right )}^{\frac{3}{2}}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}{b x^{2} + a}\,{d x} \]

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. \[ \text{Timed out} \]

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(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )}^{\frac{3}{2}}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}{b x^{2} + a}\,{d x} \]

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)