3.76 \(\int \frac{\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=639 \[ \frac{b^2 e^{3/2} \sqrt{c+d x^2} (b e-a f) \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^3 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (3 b c (3 d e-2 c f)-a d (4 d e-c f)) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 c^3 \sqrt{e+f x^2} (b c-a d)^2 (d e-c f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{e+f x^2} (3 b c (3 d e-c f)-2 a d (c f+2 d e))}{15 c^2 \left (c+d x^2\right )^{3/2} (b c-a d)^2}+\frac{\sqrt{e+f x^2} \left (a d \left (-2 c^2 f^2-3 c d e f+8 d^2 e^2\right )-3 b c \left (c^2 f^2-6 c d e f+6 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{15 c^{5/2} \sqrt{d} \sqrt{c+d x^2} (b c-a d)^2 (d e-c f) \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{x \sqrt{e+f x^2} (d e-c f)}{5 c \left (c+d x^2\right )^{5/2} (b c-a d)}-\frac{b \sqrt{d} \sqrt{e+f x^2} (b e-a f) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{\sqrt{c} \sqrt{c+d x^2} (b c-a d)^3 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}} \]

[Out]

-((d*e - c*f)*x*Sqrt[e + f*x^2])/(5*c*(b*c - a*d)*(c + d*x^2)^(5/2)) - ((3*b*c*(
3*d*e - c*f) - 2*a*d*(2*d*e + c*f))*x*Sqrt[e + f*x^2])/(15*c^2*(b*c - a*d)^2*(c
+ d*x^2)^(3/2)) - (b*Sqrt[d]*(b*e - a*f)*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[
d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(Sqrt[c]*(b*c - a*d)^3*Sqrt[c + d*x^2]*Sqrt[(c
*(e + f*x^2))/(e*(c + d*x^2))]) + ((a*d*(8*d^2*e^2 - 3*c*d*e*f - 2*c^2*f^2) - 3*
b*c*(6*d^2*e^2 - 6*c*d*e*f + c^2*f^2))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]
*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(15*c^(5/2)*Sqrt[d]*(b*c - a*d)^2*(d*e - c*f)*Sq
rt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) + (e^(3/2)*Sqrt[f]*(3*b*c*(
3*d*e - 2*c*f) - a*d*(4*d*e - c*f))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)
/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*c^3*(b*c - a*d)^2*(d*e - c*f)*Sqrt[(e*(c + d*x^
2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b^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*c
*(b*c - a*d)^3*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])

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Rubi [A]  time = 2.15564, antiderivative size = 639, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{b^2 e^{3/2} \sqrt{c+d x^2} (b e-a f) \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^3 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (3 b c (3 d e-2 c f)-a d (4 d e-c f)) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 c^3 \sqrt{e+f x^2} (b c-a d)^2 (d e-c f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{e+f x^2} (3 b c (3 d e-c f)-2 a d (c f+2 d e))}{15 c^2 \left (c+d x^2\right )^{3/2} (b c-a d)^2}+\frac{\sqrt{e+f x^2} \left (a d \left (-2 c^2 f^2-3 c d e f+8 d^2 e^2\right )-3 b c \left (c^2 f^2-6 c d e f+6 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{15 c^{5/2} \sqrt{d} \sqrt{c+d x^2} (b c-a d)^2 (d e-c f) \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{x \sqrt{e+f x^2} (d e-c f)}{5 c \left (c+d x^2\right )^{5/2} (b c-a d)}-\frac{b \sqrt{d} \sqrt{e+f x^2} (b e-a f) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{\sqrt{c} \sqrt{c+d x^2} (b c-a d)^3 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}} \]

Antiderivative was successfully verified.

[In]  Int[(e + f*x^2)^(3/2)/((a + b*x^2)*(c + d*x^2)^(7/2)),x]

[Out]

-((d*e - c*f)*x*Sqrt[e + f*x^2])/(5*c*(b*c - a*d)*(c + d*x^2)^(5/2)) - ((3*b*c*(
3*d*e - c*f) - 2*a*d*(2*d*e + c*f))*x*Sqrt[e + f*x^2])/(15*c^2*(b*c - a*d)^2*(c
+ d*x^2)^(3/2)) - (b*Sqrt[d]*(b*e - a*f)*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[
d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(Sqrt[c]*(b*c - a*d)^3*Sqrt[c + d*x^2]*Sqrt[(c
*(e + f*x^2))/(e*(c + d*x^2))]) + ((a*d*(8*d^2*e^2 - 3*c*d*e*f - 2*c^2*f^2) - 3*
b*c*(6*d^2*e^2 - 6*c*d*e*f + c^2*f^2))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]
*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(15*c^(5/2)*Sqrt[d]*(b*c - a*d)^2*(d*e - c*f)*Sq
rt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) + (e^(3/2)*Sqrt[f]*(3*b*c*(
3*d*e - 2*c*f) - a*d*(4*d*e - c*f))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)
/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*c^3*(b*c - a*d)^2*(d*e - c*f)*Sqrt[(e*(c + d*x^
2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b^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*c
*(b*c - a*d)^3*Sqrt[f]*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((f*x**2+e)**(3/2)/(b*x**2+a)/(d*x**2+c)**(7/2),x)

[Out]

Timed out

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Mathematica [C]  time = 5.03297, size = 570, normalized size = 0.89 \[ \frac{-a x \sqrt{\frac{d}{c}} \left (e+f x^2\right ) \left (\left (c+d x^2\right )^2 \left (a^2 d^2 \left (-2 c^2 f^2-3 c d e f+8 d^2 e^2\right )+2 a b c d \left (7 c^2 f^2+3 c d e f-13 d^2 e^2\right )+3 b^2 c^2 \left (c^2 f^2-11 c d e f+11 d^2 e^2\right )\right )+3 c^2 (b c-a d)^2 (d e-c f)^2+c \left (c+d x^2\right ) (b c-a d) (c f-d e) (2 a d (c f+2 d e)+3 b c (c f-3 d e))\right )+i \left (c+d x^2\right )^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (a e \left (a^2 d^2 \left (2 c^2 f^2+3 c d e f-8 d^2 e^2\right )-2 a b c d \left (7 c^2 f^2+3 c d e f-13 d^2 e^2\right )-3 b^2 c^2 \left (c^2 f^2-11 c d e f+11 d^2 e^2\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+(d e-c f) \left (a \left (a^2 d^2 e (c f+8 d e)+a b c \left (15 c^2 f^2-7 c d e f-26 d^2 e^2\right )+3 b^2 c^2 e (11 d e-8 c f)\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )-15 b c^3 (b e-a f)^2 \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )\right )}{15 a c^3 \sqrt{\frac{d}{c}} \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2} (b c-a d)^3 (d e-c f)} \]

Antiderivative was successfully verified.

[In]  Integrate[(e + f*x^2)^(3/2)/((a + b*x^2)*(c + d*x^2)^(7/2)),x]

[Out]

(-(a*Sqrt[d/c]*x*(e + f*x^2)*(3*c^2*(b*c - a*d)^2*(d*e - c*f)^2 + c*(b*c - a*d)*
(-(d*e) + c*f)*(3*b*c*(-3*d*e + c*f) + 2*a*d*(2*d*e + c*f))*(c + d*x^2) + (a^2*d
^2*(8*d^2*e^2 - 3*c*d*e*f - 2*c^2*f^2) + 3*b^2*c^2*(11*d^2*e^2 - 11*c*d*e*f + c^
2*f^2) + 2*a*b*c*d*(-13*d^2*e^2 + 3*c*d*e*f + 7*c^2*f^2))*(c + d*x^2)^2)) + I*(c
+ d*x^2)^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*(a*e*(-3*b^2*c^2*(11*d^2*e^2
- 11*c*d*e*f + c^2*f^2) + a^2*d^2*(-8*d^2*e^2 + 3*c*d*e*f + 2*c^2*f^2) - 2*a*b*
c*d*(-13*d^2*e^2 + 3*c*d*e*f + 7*c^2*f^2))*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*
f)/(d*e)] + (d*e - c*f)*(a*(3*b^2*c^2*e*(11*d*e - 8*c*f) + a^2*d^2*e*(8*d*e + c*
f) + a*b*c*(-26*d^2*e^2 - 7*c*d*e*f + 15*c^2*f^2))*EllipticF[I*ArcSinh[Sqrt[d/c]
*x], (c*f)/(d*e)] - 15*b*c^3*(b*e - a*f)^2*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqr
t[d/c]*x], (c*f)/(d*e)])))/(15*a*c^3*Sqrt[d/c]*(b*c - a*d)^3*(d*e - c*f)*(c + d*
x^2)^(5/2)*Sqrt[e + f*x^2])

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Maple [B]  time = 0.11, size = 6211, normalized size = 9.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^2+e)^(3/2)/(b*x^2+a)/(d*x^2+c)^(7/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (f x^{2} + e\right )}^{\frac{3}{2}}}{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^2 + e)^(3/2)/((b*x^2 + a)*(d*x^2 + c)^(7/2)),x, algorithm="maxima")

[Out]

integrate((f*x^2 + e)^(3/2)/((b*x^2 + a)*(d*x^2 + c)^(7/2)), 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((f*x^2 + e)^(3/2)/((b*x^2 + a)*(d*x^2 + c)^(7/2)),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((f*x**2+e)**(3/2)/(b*x**2+a)/(d*x**2+c)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (f x^{2} + e\right )}^{\frac{3}{2}}}{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^2 + e)^(3/2)/((b*x^2 + a)*(d*x^2 + c)^(7/2)),x, algorithm="giac")

[Out]

integrate((f*x^2 + e)^(3/2)/((b*x^2 + a)*(d*x^2 + c)^(7/2)), x)