3.83 \(\int \frac{\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=980 \[ \frac{e^{3/2} \sqrt{d x^2+c} \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right ) (b c-a d)^3}{a b c \sqrt{f} (b e-a f)^2 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}-\frac{\sqrt{e} (b d e+4 b c f-3 a d f) \sqrt{d x^2+c} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right ) (b c-a d)}{3 b \sqrt{f} (b e-a f)^2 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}+\frac{d (5 b c-3 a d) e^{3/2} \sqrt{d x^2+c} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right ) (b c-a d)}{3 b c \sqrt{f} (b e-a f)^2 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}+\frac{d x \sqrt{d x^2+c} \sqrt{f x^2+e} (b c-a d)}{3 (b e-a f)^2}+\frac{(b d e+4 b c f-3 a d f) x \sqrt{d x^2+c} (b c-a d)}{3 b (b e-a f)^2 \sqrt{f x^2+e}}-\frac{\left (b e \left (6 d^2 e^2-7 c d f e-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d f e+3 c^2 f^2\right )\right ) \sqrt{d x^2+c} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 \sqrt{e} f^{3/2} (b e-a f)^2 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}-\frac{\sqrt{e} \left (2 a d f (2 d e-3 c f)-b \left (3 d^2 e^2-2 c d f e-3 c^2 f^2\right )\right ) \sqrt{d x^2+c} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 f^{3/2} (b e-a f)^2 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}+\frac{d (a f (4 d e-3 c f)-b e (3 d e-2 c f)) x \sqrt{d x^2+c} \sqrt{f x^2+e}}{3 e f (b e-a f)^2}+\frac{(d e-c f) x \left (d x^2+c\right )^{3/2}}{e (b e-a f) \sqrt{f x^2+e}}+\frac{\left (b e \left (6 d^2 e^2-7 c d f e-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d f e+3 c^2 f^2\right )\right ) x \sqrt{d x^2+c}}{3 e f (b e-a f)^2 \sqrt{f x^2+e}} \]

[Out]

((b*c - a*d)*(b*d*e + 4*b*c*f - 3*a*d*f)*x*Sqrt[c + d*x^2])/(3*b*(b*e - a*f)^2*S
qrt[e + f*x^2]) + ((b*e*(6*d^2*e^2 - 7*c*d*e*f - c^2*f^2) - a*f*(8*d^2*e^2 - 13*
c*d*e*f + 3*c^2*f^2))*x*Sqrt[c + d*x^2])/(3*e*f*(b*e - a*f)^2*Sqrt[e + f*x^2]) +
((d*e - c*f)*x*(c + d*x^2)^(3/2))/(e*(b*e - a*f)*Sqrt[e + f*x^2]) + (d*(b*c - a
*d)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*(b*e - a*f)^2) + (d*(a*f*(4*d*e - 3*c*
f) - b*e*(3*d*e - 2*c*f))*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*e*f*(b*e - a*f)^
2) - ((b*c - a*d)*Sqrt[e]*(b*d*e + 4*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*Sqrt[f]*(b*e - a*f)^2*Sqrt[(
e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - ((b*e*(6*d^2*e^2 - 7*c*d*e*f
- c^2*f^2) - a*f*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2))*Sqrt[c + d*x^2]*EllipticE
[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*Sqrt[e]*f^(3/2)*(b*e - a*f)^2
*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (d*(5*b*c - 3*a*d)*(b*
c - a*d)*e^(3/2)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e
)/(c*f)])/(3*b*c*Sqrt[f]*(b*e - a*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqr
t[e + f*x^2]) - (Sqrt[e]*(2*a*d*f*(2*d*e - 3*c*f) - b*(3*d^2*e^2 - 2*c*d*e*f - 3
*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f
)])/(3*f^(3/2)*(b*e - a*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^
2]) + ((b*c - a*d)^3*e^(3/2)*Sqrt[c + d*x^2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[
(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a*b*c*Sqrt[f]*(b*e - a*f)^2*Sqrt[(e*(c
+ d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])

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Rubi [A]  time = 3.20434, antiderivative size = 980, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.281 \[ \frac{e^{3/2} \sqrt{d x^2+c} \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right ) (b c-a d)^3}{a b c \sqrt{f} (b e-a f)^2 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}-\frac{\sqrt{e} (b d e+4 b c f-3 a d f) \sqrt{d x^2+c} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right ) (b c-a d)}{3 b \sqrt{f} (b e-a f)^2 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}+\frac{d (5 b c-3 a d) e^{3/2} \sqrt{d x^2+c} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right ) (b c-a d)}{3 b c \sqrt{f} (b e-a f)^2 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}+\frac{d x \sqrt{d x^2+c} \sqrt{f x^2+e} (b c-a d)}{3 (b e-a f)^2}+\frac{(b d e+4 b c f-3 a d f) x \sqrt{d x^2+c} (b c-a d)}{3 b (b e-a f)^2 \sqrt{f x^2+e}}-\frac{\left (b e \left (6 d^2 e^2-7 c d f e-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d f e+3 c^2 f^2\right )\right ) \sqrt{d x^2+c} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 \sqrt{e} f^{3/2} (b e-a f)^2 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}-\frac{\sqrt{e} \left (2 a d f (2 d e-3 c f)-b \left (3 d^2 e^2-2 c d f e-3 c^2 f^2\right )\right ) \sqrt{d x^2+c} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 f^{3/2} (b e-a f)^2 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}+\frac{d (a f (4 d e-3 c f)-b e (3 d e-2 c f)) x \sqrt{d x^2+c} \sqrt{f x^2+e}}{3 e f (b e-a f)^2}+\frac{(d e-c f) x \left (d x^2+c\right )^{3/2}}{e (b e-a f) \sqrt{f x^2+e}}+\frac{\left (b e \left (6 d^2 e^2-7 c d f e-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d f e+3 c^2 f^2\right )\right ) x \sqrt{d x^2+c}}{3 e f (b e-a f)^2 \sqrt{f x^2+e}} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)*(b*d*e + 4*b*c*f - 3*a*d*f)*x*Sqrt[c + d*x^2])/(3*b*(b*e - a*f)^2*S
qrt[e + f*x^2]) + ((b*e*(6*d^2*e^2 - 7*c*d*e*f - c^2*f^2) - a*f*(8*d^2*e^2 - 13*
c*d*e*f + 3*c^2*f^2))*x*Sqrt[c + d*x^2])/(3*e*f*(b*e - a*f)^2*Sqrt[e + f*x^2]) +
((d*e - c*f)*x*(c + d*x^2)^(3/2))/(e*(b*e - a*f)*Sqrt[e + f*x^2]) + (d*(b*c - a
*d)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*(b*e - a*f)^2) + (d*(a*f*(4*d*e - 3*c*
f) - b*e*(3*d*e - 2*c*f))*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*e*f*(b*e - a*f)^
2) - ((b*c - a*d)*Sqrt[e]*(b*d*e + 4*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*Sqrt[f]*(b*e - a*f)^2*Sqrt[(
e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - ((b*e*(6*d^2*e^2 - 7*c*d*e*f
- c^2*f^2) - a*f*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2))*Sqrt[c + d*x^2]*EllipticE
[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*Sqrt[e]*f^(3/2)*(b*e - a*f)^2
*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (d*(5*b*c - 3*a*d)*(b*
c - a*d)*e^(3/2)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e
)/(c*f)])/(3*b*c*Sqrt[f]*(b*e - a*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqr
t[e + f*x^2]) - (Sqrt[e]*(2*a*d*f*(2*d*e - 3*c*f) - b*(3*d^2*e^2 - 2*c*d*e*f - 3
*c^2*f^2))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f
)])/(3*f^(3/2)*(b*e - a*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^
2]) + ((b*c - a*d)^3*e^(3/2)*Sqrt[c + d*x^2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[
(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a*b*c*Sqrt[f]*(b*e - a*f)^2*Sqrt[(e*(c
+ d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*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)**(5/2)/(b*x**2+a)/(f*x**2+e)**(3/2),x)

[Out]

Timed out

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Mathematica [C]  time = 2.5149, size = 352, normalized size = 0.36 \[ \frac{-f \left (a b^2 x \sqrt{\frac{d}{c}} \left (c+d x^2\right ) (d e-c f)^2+i e f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b c-a d)^3 \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )-i a b d e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (b \left (c^2 f^2-2 c d e f+2 d^2 e^2\right )-a d^2 e f\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )-i a d^2 e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b e-a f) (-a d f+3 b c f-2 b d e) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{a b^2 e f^2 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2} (b e-a f)} \]

Antiderivative was successfully verified.

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

[Out]

((-I)*a*b*d*e*(-(a*d^2*e*f) + b*(2*d^2*e^2 - 2*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*
d^2*e*(b*e - a*f)*(-2*b*d*e + 3*b*c*f - a*d*f)*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]*(d*e
- c*f)^2*x*(c + d*x^2) + I*(b*c - a*d)^3*e*f*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)]))/(a*b^2*Sqrt[
d/c]*e*f^2*(b*e - a*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

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Maple [A]  time = 0.047, size = 1063, normalized size = 1.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x^3*a*b^2*c^2*d*f^3*(-d/c)^(1/2)-2*x^3*a*b^2*c*d^2*e*f^2*(-d/c)^(1/2)+x^3*a*b^2
*d^3*e^2*f*(-d/c)^(1/2)-EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^3*d^3*e*f^2*
((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+3*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/
2))*a^2*b*c*d^2*e*f^2*((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*d^3*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-
3*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b^2*c*d^2*e^2*f*((d*x^2+c)/c)^(1/2
)*((f*x^2+e)/e)^(1/2)+2*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b^2*d^3*e^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^2*b*d^3*e^2*f*((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^2*c^2*d*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+2*
EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b^2*c*d^2*e^2*f*((d*x^2+c)/c)^(1/2)*
((f*x^2+e)/e)^(1/2)-2*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b^2*d^3*e^3*((
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*d^3*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-3*Ellip
ticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*a^2*b*c*d^2*e*f^2*((d*x^
2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+3*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/
2)/(-d/c)^(1/2))*a*b^2*c^2*d*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-Ellip
ticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*b^3*c^3*e*f^2*((d*x^2+c)
/c)^(1/2)*((f*x^2+e)/e)^(1/2)+x*a*b^2*c^3*f^3*(-d/c)^(1/2)-2*x*a*b^2*c^2*d*e*f^2
*(-d/c)^(1/2)+x*a*b^2*c*d^2*e^2*f*(-d/c)^(1/2))*(f*x^2+e)^(1/2)*(d*x^2+c)^(1/2)/
e/f^2/a/(-d/c)^(1/2)/b^2/(a*f-b*e)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )}^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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