3.34 \(\int \frac{\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{9/2}} \, dx\)

Optimal. Leaf size=531 \[ \frac{x \sqrt{e+f x^2} (d e (6 a d+b c)-c f (3 a d+4 b c))}{35 c^2 d^2 \left (c+d x^2\right )^{5/2}}-\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} \left (3 a d \left (c^2 f^2-11 c d e f+8 d^2 e^2\right )+2 b c \left (2 c^2 f^2-c d e f+2 d^2 e^2\right )\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 c^4 d^2 \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{e+f x^2} \left (3 a d \left (-2 c^2 f^2-5 c d e f+8 d^2 e^2\right )+b c \left (-8 c^2 f^2+c d e f+4 d^2 e^2\right )\right )}{105 c^3 d^2 \left (c+d x^2\right )^{3/2} (d e-c f)}+\frac{\sqrt{e+f x^2} \left (6 a d \left (c^3 f^3+2 c^2 d e f^2-12 c d^2 e^2 f+8 d^3 e^3\right )+b c \left (8 c^3 f^3-5 c^2 d e f^2-5 c d^2 e^2 f+8 d^3 e^3\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{105 c^{7/2} d^{5/2} \sqrt{c+d x^2} (d e-c f)^2 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{x \left (e+f x^2\right )^{3/2} (b c-a d)}{7 c d \left (c+d x^2\right )^{7/2}} \]

[Out]

((d*(b*c + 6*a*d)*e - c*(4*b*c + 3*a*d)*f)*x*Sqrt[e + f*x^2])/(35*c^2*d^2*(c + d
*x^2)^(5/2)) + ((b*c*(4*d^2*e^2 + c*d*e*f - 8*c^2*f^2) + 3*a*d*(8*d^2*e^2 - 5*c*
d*e*f - 2*c^2*f^2))*x*Sqrt[e + f*x^2])/(105*c^3*d^2*(d*e - c*f)*(c + d*x^2)^(3/2
)) - ((b*c - a*d)*x*(e + f*x^2)^(3/2))/(7*c*d*(c + d*x^2)^(7/2)) + ((6*a*d*(8*d^
3*e^3 - 12*c*d^2*e^2*f + 2*c^2*d*e*f^2 + c^3*f^3) + b*c*(8*d^3*e^3 - 5*c*d^2*e^2
*f - 5*c^2*d*e*f^2 + 8*c^3*f^3))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sq
rt[c]], 1 - (c*f)/(d*e)])/(105*c^(7/2)*d^(5/2)*(d*e - c*f)^2*Sqrt[c + d*x^2]*Sqr
t[(c*(e + f*x^2))/(e*(c + d*x^2))]) - (e^(3/2)*Sqrt[f]*(3*a*d*(8*d^2*e^2 - 11*c*
d*e*f + c^2*f^2) + 2*b*c*(2*d^2*e^2 - c*d*e*f + 2*c^2*f^2))*Sqrt[c + d*x^2]*Elli
pticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*c^4*d^2*(d*e - c*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 = 1.72548, antiderivative size = 531, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{x \sqrt{e+f x^2} (d e (6 a d+b c)-c f (3 a d+4 b c))}{35 c^2 d^2 \left (c+d x^2\right )^{5/2}}-\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} \left (3 a d \left (c^2 f^2-11 c d e f+8 d^2 e^2\right )+2 b c \left (2 c^2 f^2-c d e f+2 d^2 e^2\right )\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{105 c^4 d^2 \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{e+f x^2} \left (3 a d \left (-2 c^2 f^2-5 c d e f+8 d^2 e^2\right )+b c \left (-8 c^2 f^2+c d e f+4 d^2 e^2\right )\right )}{105 c^3 d^2 \left (c+d x^2\right )^{3/2} (d e-c f)}+\frac{\sqrt{e+f x^2} \left (6 a d \left (c^3 f^3+2 c^2 d e f^2-12 c d^2 e^2 f+8 d^3 e^3\right )+b c \left (8 c^3 f^3-5 c^2 d e f^2-5 c d^2 e^2 f+8 d^3 e^3\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{105 c^{7/2} d^{5/2} \sqrt{c+d x^2} (d e-c f)^2 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{x \left (e+f x^2\right )^{3/2} (b c-a d)}{7 c d \left (c+d x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(9/2),x]

[Out]

((d*(b*c + 6*a*d)*e - c*(4*b*c + 3*a*d)*f)*x*Sqrt[e + f*x^2])/(35*c^2*d^2*(c + d
*x^2)^(5/2)) + ((b*c*(4*d^2*e^2 + c*d*e*f - 8*c^2*f^2) + 3*a*d*(8*d^2*e^2 - 5*c*
d*e*f - 2*c^2*f^2))*x*Sqrt[e + f*x^2])/(105*c^3*d^2*(d*e - c*f)*(c + d*x^2)^(3/2
)) - ((b*c - a*d)*x*(e + f*x^2)^(3/2))/(7*c*d*(c + d*x^2)^(7/2)) + ((6*a*d*(8*d^
3*e^3 - 12*c*d^2*e^2*f + 2*c^2*d*e*f^2 + c^3*f^3) + b*c*(8*d^3*e^3 - 5*c*d^2*e^2
*f - 5*c^2*d*e*f^2 + 8*c^3*f^3))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sq
rt[c]], 1 - (c*f)/(d*e)])/(105*c^(7/2)*d^(5/2)*(d*e - c*f)^2*Sqrt[c + d*x^2]*Sqr
t[(c*(e + f*x^2))/(e*(c + d*x^2))]) - (e^(3/2)*Sqrt[f]*(3*a*d*(8*d^2*e^2 - 11*c*
d*e*f + c^2*f^2) + 2*b*c*(2*d^2*e^2 - c*d*e*f + 2*c^2*f^2))*Sqrt[c + d*x^2]*Elli
pticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(105*c^4*d^2*(d*e - c*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((b*x**2+a)*(f*x**2+e)**(3/2)/(d*x**2+c)**(9/2),x)

[Out]

Timed out

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Mathematica [C]  time = 3.27776, size = 545, normalized size = 1.03 \[ \frac{\sqrt{\frac{d}{c}} \left (-x \sqrt{\frac{d}{c}} \left (e+f x^2\right ) \left (15 c^3 (b c-a d) (d e-c f)^3-c \left (c+d x^2\right )^2 (d e-c f) \left (3 a d \left (-2 c^2 f^2-5 c d e f+8 d^2 e^2\right )+b c \left (-8 c^2 f^2+c d e f+4 d^2 e^2\right )\right )-3 c^2 \left (c+d x^2\right ) (d e-c f)^2 (2 a d (c f+3 d e)+b c (d e-9 c f))-\left (c+d x^2\right )^3 \left (6 a d \left (c^3 f^3+2 c^2 d e f^2-12 c d^2 e^2 f+8 d^3 e^3\right )+b c \left (8 c^3 f^3-5 c^2 d e f^2-5 c d^2 e^2 f+8 d^3 e^3\right )\right )\right )+i e \left (c+d x^2\right )^3 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (\left (6 a d \left (c^3 f^3+2 c^2 d e f^2-12 c d^2 e^2 f+8 d^3 e^3\right )+b c \left (8 c^3 f^3-5 c^2 d e f^2-5 c d^2 e^2 f+8 d^3 e^3\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )-(c f-d e) \left (3 a d \left (c^2 f^2+16 c d e f-16 d^2 e^2\right )+b c \left (4 c^2 f^2+c d e f-8 d^2 e^2\right )\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )\right )}{105 c^3 d^3 \left (c+d x^2\right )^{7/2} \sqrt{e+f x^2} (d e-c f)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(9/2),x]

[Out]

(Sqrt[d/c]*(-(Sqrt[d/c]*x*(e + f*x^2)*(15*c^3*(b*c - a*d)*(d*e - c*f)^3 - 3*c^2*
(d*e - c*f)^2*(b*c*(d*e - 9*c*f) + 2*a*d*(3*d*e + c*f))*(c + d*x^2) - c*(d*e - c
*f)*(b*c*(4*d^2*e^2 + c*d*e*f - 8*c^2*f^2) + 3*a*d*(8*d^2*e^2 - 5*c*d*e*f - 2*c^
2*f^2))*(c + d*x^2)^2 - (6*a*d*(8*d^3*e^3 - 12*c*d^2*e^2*f + 2*c^2*d*e*f^2 + c^3
*f^3) + b*c*(8*d^3*e^3 - 5*c*d^2*e^2*f - 5*c^2*d*e*f^2 + 8*c^3*f^3))*(c + d*x^2)
^3)) + I*e*(c + d*x^2)^3*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*((6*a*d*(8*d^3*
e^3 - 12*c*d^2*e^2*f + 2*c^2*d*e*f^2 + c^3*f^3) + b*c*(8*d^3*e^3 - 5*c*d^2*e^2*f
- 5*c^2*d*e*f^2 + 8*c^3*f^3))*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] -
(-(d*e) + c*f)*(3*a*d*(-16*d^2*e^2 + 16*c*d*e*f + c^2*f^2) + b*c*(-8*d^2*e^2 + c
*d*e*f + 4*c^2*f^2))*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])))/(105*c^3*
d^3*(d*e - c*f)^2*(c + d*x^2)^(7/2)*Sqrt[e + f*x^2])

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Maple [B]  time = 0.1, size = 5113, normalized size = 9.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)*(f*x^2+e)^(3/2)/(d*x^2+c)^(9/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}{{\left (d x^{2} + c\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*(f*x^2 + e)^(3/2)/(d*x^2 + c)^(9/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)*(f*x^2 + e)^(3/2)/(d*x^2 + c)^(9/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b f x^{4} +{\left (b e + a f\right )} x^{2} + a e\right )} \sqrt{f x^{2} + e}}{{\left (d^{4} x^{8} + 4 \, c d^{3} x^{6} + 6 \, c^{2} d^{2} x^{4} + 4 \, c^{3} d x^{2} + c^{4}\right )} \sqrt{d x^{2} + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*(f*x^2 + e)^(3/2)/(d*x^2 + c)^(9/2),x, algorithm="fricas")

[Out]

integral((b*f*x^4 + (b*e + a*f)*x^2 + a*e)*sqrt(f*x^2 + e)/((d^4*x^8 + 4*c*d^3*x
^6 + 6*c^2*d^2*x^4 + 4*c^3*d*x^2 + c^4)*sqrt(d*x^2 + c)), x)

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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((b*x**2+a)*(f*x**2+e)**(3/2)/(d*x**2+c)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}{{\left (d x^{2} + c\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*(f*x^2 + e)^(3/2)/(d*x^2 + c)^(9/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)*(f*x^2 + e)^(3/2)/(d*x^2 + c)^(9/2), x)