3.82 \(\int \frac{1}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2}} \, dx\)

Optimal. Leaf size=435 \[ \frac{b^2 \sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{-c}}\right )|\frac{c f}{d e}\right )}{a \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d)^2}-\frac{d^{3/2} \sqrt{e+f x^2} (b c (5 d e-7 c f)-2 a d (d e-2 c f)) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{3 c^{3/2} \sqrt{c+d x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{d \sqrt{e} \sqrt{f} \sqrt{c+d x^2} (a d (d e-3 c f)-2 b c (2 d e-3 c f)) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c^2 \sqrt{e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{d^2 x \sqrt{e+f x^2}}{3 c \left (c+d x^2\right )^{3/2} (b c-a d) (d e-c f)} \]

[Out]

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

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Rubi [A]  time = 1.69787, antiderivative size = 435, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219 \[ \frac{b^2 \sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{-c}}\right )|\frac{c f}{d e}\right )}{a \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d)^2}-\frac{d^{3/2} \sqrt{e+f x^2} (b c (5 d e-7 c f)-2 a d (d e-2 c f)) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{3 c^{3/2} \sqrt{c+d x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{d \sqrt{e} \sqrt{f} \sqrt{c+d x^2} (a d (d e-3 c f)-2 b c (2 d e-3 c f)) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c^2 \sqrt{e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{d^2 x \sqrt{e+f x^2}}{3 c \left (c+d x^2\right )^{3/2} (b c-a d) (d e-c f)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^2)*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2]),x]

[Out]

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

[Out]

Timed out

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Mathematica [C]  time = 6.34525, size = 433, normalized size = 1. \[ \frac{-3 i b^2 c^2 \left (c+d x^2\right ) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (d e-c f)^2 \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+a c d x \left (\frac{d}{c}\right )^{3/2} \left (e+f x^2\right ) \left (a d \left (-5 c^2 f+c d \left (3 e-4 f x^2\right )+2 d^2 e x^2\right )+b c \left (8 c^2 f-6 c d e+7 c d f x^2-5 d^2 e x^2\right )\right )+i a d^2 e \left (c+d x^2\right ) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (2 a d (d e-2 c f)+b c (7 c f-5 d e)) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+i a d \left (c+d x^2\right ) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) (a d (2 d e-3 c f)+b c (6 c f-5 d e)) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{3 a c^2 \sqrt{\frac{d}{c}} \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (b c-a d)^2 (d e-c f)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x^2)*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2]),x]

[Out]

(a*c*d*(d/c)^(3/2)*x*(e + f*x^2)*(b*c*(-6*c*d*e + 8*c^2*f - 5*d^2*e*x^2 + 7*c*d*
f*x^2) + a*d*(-5*c^2*f + 2*d^2*e*x^2 + c*d*(3*e - 4*f*x^2))) + I*a*d^2*e*(2*a*d*
(d*e - 2*c*f) + b*c*(-5*d*e + 7*c*f))*(c + d*x^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*(-(d*e) + c*f)*
(a*d*(2*d*e - 3*c*f) + b*c*(-5*d*e + 6*c*f))*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqr
t[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (3*I)*b^2*c^2*
(d*e - c*f)^2*(c + d*x^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)])/(3*a*c^2*Sqrt[d/c]*(b*c - a*d)^2
*(d*e - c*f)^2*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])

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Maple [B]  time = 0.068, size = 2062, normalized size = 4.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2+a)/(d*x^2+c)^(5/2)/(f*x^2+e)^(1/2),x)

[Out]

1/3*(-7*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b*c^3*d^2*e*f*((d*x^2+c)/c)^
(1/2)*((f*x^2+e)/e)^(1/2)-5*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a^2*c*
d^4*e*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-6*EllipticF(x*(-d/c)^(1/2),(c*f/
d/e)^(1/2))*x^2*a*b*c^3*d^2*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-5*Ellipt
icF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*b*c*d^4*e^2*((d*x^2+c)/c)^(1/2)*((f*x^
2+e)/e)^(1/2)+4*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a^2*c*d^4*e*f*((d*
x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+5*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*
x^2*a*b*c*d^4*e^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-4*x^5*a^2*c*d^4*f^2*(-
d/c)^(1/2)+11*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*b*c^2*d^3*e*f*((d*
x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-7*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*
x^2*a*b*c^2*d^3*e*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+2*x^3*a^2*d^5*e^2*(-
d/c)^(1/2)-6*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*x^2*b^
2*c^3*d^2*e*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+11*EllipticF(x*(-d/c)^(1/2
),(c*f/d/e)^(1/2))*a*b*c^3*d^2*e*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+x^3*a
*b*c^2*d^3*e*f*(-d/c)^(1/2)+8*x*a*b*c^3*d^2*e*f*(-d/c)^(1/2)+3*EllipticF(x*(-d/c
)^(1/2),(c*f/d/e)^(1/2))*x^2*a^2*c^2*d^3*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))*x^2*b^2*c^4*
d*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))*x^2*b^2*c^2*d^3*e^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/
e)^(1/2)-5*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2*c^2*d^3*e*f*((d*x^2+c)/
c)^(1/2)*((f*x^2+e)/e)^(1/2)-6*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b*c^4
*d*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-5*EllipticF(x*(-d/c)^(1/2),(c*f/d
/e)^(1/2))*a*b*c^2*d^3*e^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+4*EllipticE(x
*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2*c^2*d^3*e*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)
^(1/2)+5*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b*c^2*d^3*e^2*((d*x^2+c)/c)
^(1/2)*((f*x^2+e)/e)^(1/2)-6*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/
c)^(1/2))*b^2*c^4*d*e*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-5*x^5*a*b*c*d^4*
e*f*(-d/c)^(1/2)+2*x^5*a^2*d^5*e*f*(-d/c)^(1/2)-5*x^3*a^2*c^2*d^3*f^2*(-d/c)^(1/
2)+3*x*a^2*c*d^4*e^2*(-d/c)^(1/2)+3*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/
2)/(-d/c)^(1/2))*b^2*c^5*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-5*x^3*a*b*c
*d^4*e^2*(-d/c)^(1/2)-5*x*a^2*c^2*d^3*e*f*(-d/c)^(1/2)-6*x*a*b*c^2*d^3*e^2*(-d/c
)^(1/2)+2*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a^2*d^5*e^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))*x^2*a^2*
d^5*e^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*c^3*d^2*f^2*((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^2*c*d^4*e^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^2*c*d^4*e^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))*b^2*c^3*d^2*e^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+7*x^5*a*b*c^2*d^3
*f^2*(-d/c)^(1/2)-x^3*a^2*c*d^4*e*f*(-d/c)^(1/2)+8*x^3*a*b*c^3*d^2*f^2*(-d/c)^(1
/2))/(f*x^2+e)^(1/2)/(a*d-b*c)^2/(-d/c)^(1/2)/a/(c*f-d*e)^2/c^2/(d*x^2+c)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{5}{2}} \sqrt{f x^{2} + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*sqrt(f*x^2 + e)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*sqrt(f*x^2 + e)), 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(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*sqrt(f*x^2 + e)),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(1/(b*x**2+a)/(d*x**2+c)**(5/2)/(f*x**2+e)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{5}{2}} \sqrt{f x^{2} + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*sqrt(f*x^2 + e)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*sqrt(f*x^2 + e)), x)