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

Optimal. Leaf size=400 \[ \frac{e^{3/2} \sqrt{c+d x^2} \left (5 a d (3 d e-c f)-b \left (6 c d e-4 c^2 f\right )\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 c d^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{c+d x^2} \left (10 a d f (2 d e-c f)+b \left (8 c^2 f^2-13 c d e f+3 d^2 e^2\right )\right )}{15 d^3 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left (10 a d f (2 d e-c f)+b \left (8 c^2 f^2-13 c d e f+3 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 d^3 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} (5 a d f-4 b c f+3 b d e)}{15 d^2}+\frac{b x \sqrt{c+d x^2} \left (e+f x^2\right )^{3/2}}{5 d} \]

[Out]

((10*a*d*f*(2*d*e - c*f) + b*(3*d^2*e^2 - 13*c*d*e*f + 8*c^2*f^2))*x*Sqrt[c + d*
x^2])/(15*d^3*Sqrt[e + f*x^2]) + ((3*b*d*e - 4*b*c*f + 5*a*d*f)*x*Sqrt[c + d*x^2
]*Sqrt[e + f*x^2])/(15*d^2) + (b*x*Sqrt[c + d*x^2]*(e + f*x^2)^(3/2))/(5*d) - (S
qrt[e]*(10*a*d*f*(2*d*e - c*f) + b*(3*d^2*e^2 - 13*c*d*e*f + 8*c^2*f^2))*Sqrt[c
+ d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*d^3*Sqrt[f
]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (e^(3/2)*(5*a*d*(3*d*
e - c*f) - b*(6*c*d*e - 4*c^2*f))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/S
qrt[e]], 1 - (d*e)/(c*f)])/(15*c*d^2*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 = 1.21625, antiderivative size = 400, 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 \[ \frac{x \sqrt{c+d x^2} \left (10 a d f (2 d e-c f)+b \left (8 c^2 f^2-13 c d e f+3 d^2 e^2\right )\right )}{15 d^3 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left (10 a d f (2 d e-c f)+b \left (8 c^2 f^2-13 c d e f+3 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 d^3 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{e^{3/2} \sqrt{c+d x^2} (2 b c (3 d e-2 c f)-5 a d (3 d e-c f)) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 c d^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} (5 a d f-4 b c f+3 b d e)}{15 d^2}+\frac{b x \sqrt{c+d x^2} \left (e+f x^2\right )^{3/2}}{5 d} \]

Antiderivative was successfully verified.

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

[Out]

((10*a*d*f*(2*d*e - c*f) + b*(3*d^2*e^2 - 13*c*d*e*f + 8*c^2*f^2))*x*Sqrt[c + d*
x^2])/(15*d^3*Sqrt[e + f*x^2]) + ((3*b*d*e - 4*b*c*f + 5*a*d*f)*x*Sqrt[c + d*x^2
]*Sqrt[e + f*x^2])/(15*d^2) + (b*x*Sqrt[c + d*x^2]*(e + f*x^2)^(3/2))/(5*d) - (S
qrt[e]*(10*a*d*f*(2*d*e - c*f) + b*(3*d^2*e^2 - 13*c*d*e*f + 8*c^2*f^2))*Sqrt[c
+ d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*d^3*Sqrt[f
]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (e^(3/2)*(2*b*c*(3*d*
e - 2*c*f) - 5*a*d*(3*d*e - c*f))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/S
qrt[e]], 1 - (d*e)/(c*f)])/(15*c*d^2*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 [A]  time = 114.798, size = 389, normalized size = 0.97 \[ \frac{b x \sqrt{c + d x^{2}} \left (e + f x^{2}\right )^{\frac{3}{2}}}{5 d} + \frac{\sqrt{c} \sqrt{e + f x^{2}} \left (- 5 a c d f + 15 a d^{2} e + 4 b c^{2} f - 6 b c d e\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{15 d^{\frac{5}{2}} \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} + \frac{x \sqrt{c + d x^{2}} \sqrt{e + f x^{2}} \left (5 a d f - 4 b c f + 3 b d e\right )}{15 d^{2}} - \frac{\sqrt{e} \sqrt{c + d x^{2}} \left (- 10 a c d f^{2} + 20 a d^{2} e f + 8 b c^{2} f^{2} - 13 b c d e f + 3 b d^{2} e^{2}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}\middle | 1 - \frac{d e}{c f}\right )}{15 d^{3} \sqrt{f} \sqrt{\frac{e \left (c + d x^{2}\right )}{c \left (e + f x^{2}\right )}} \sqrt{e + f x^{2}}} + \frac{x \sqrt{c + d x^{2}} \left (- 10 a c d f^{2} + 20 a d^{2} e f + 8 b c^{2} f^{2} - 13 b c d e f + 3 b d^{2} e^{2}\right )}{15 d^{3} \sqrt{e + f x^{2}}} \]

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)**(1/2),x)

[Out]

b*x*sqrt(c + d*x**2)*(e + f*x**2)**(3/2)/(5*d) + sqrt(c)*sqrt(e + f*x**2)*(-5*a*
c*d*f + 15*a*d**2*e + 4*b*c**2*f - 6*b*c*d*e)*elliptic_f(atan(sqrt(d)*x/sqrt(c))
, -c*f/(d*e) + 1)/(15*d**(5/2)*sqrt(c*(e + f*x**2)/(e*(c + d*x**2)))*sqrt(c + d*
x**2)) + x*sqrt(c + d*x**2)*sqrt(e + f*x**2)*(5*a*d*f - 4*b*c*f + 3*b*d*e)/(15*d
**2) - sqrt(e)*sqrt(c + d*x**2)*(-10*a*c*d*f**2 + 20*a*d**2*e*f + 8*b*c**2*f**2
- 13*b*c*d*e*f + 3*b*d**2*e**2)*elliptic_e(atan(sqrt(f)*x/sqrt(e)), 1 - d*e/(c*f
))/(15*d**3*sqrt(f)*sqrt(e*(c + d*x**2)/(c*(e + f*x**2)))*sqrt(e + f*x**2)) + x*
sqrt(c + d*x**2)*(-10*a*c*d*f**2 + 20*a*d**2*e*f + 8*b*c**2*f**2 - 13*b*c*d*e*f
+ 3*b*d**2*e**2)/(15*d**3*sqrt(e + f*x**2))

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Mathematica [C]  time = 1.51848, size = 275, normalized size = 0.69 \[ \frac{-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (10 a d f (2 d e-c f)+b \left (8 c^2 f^2-13 c d e f+3 d^2 e^2\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+f x \left (-\sqrt{\frac{d}{c}}\right ) \left (c+d x^2\right ) \left (e+f x^2\right ) \left (-5 a d f+4 b c f-3 b d \left (2 e+f x^2\right )\right )+i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) (-5 a d f+4 b c f-3 b d e) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{15 c^2 f \left (\frac{d}{c}\right )^{5/2} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-(Sqrt[d/c]*f*x*(c + d*x^2)*(e + f*x^2)*(4*b*c*f - 5*a*d*f - 3*b*d*(2*e + f*x^2
))) - I*e*(10*a*d*f*(2*d*e - c*f) + b*(3*d^2*e^2 - 13*c*d*e*f + 8*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*e*(-(d*e) + c*f)*(-3*b*d*e + 4*b*c*f - 5*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)])/(15*c^2*(d/c)^(5/
2)*f*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

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Maple [B]  time = 0.027, size = 870, normalized size = 2.2 \[ \text{result too large to display} \]

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)^(1/2),x)

[Out]

1/15*(f*x^2+e)^(1/2)*(d*x^2+c)^(1/2)*(3*(-d/c)^(1/2)*x^7*b*d^2*f^3+5*(-d/c)^(1/2
)*x^5*a*d^2*f^3-(-d/c)^(1/2)*x^5*b*c*d*f^3+9*(-d/c)^(1/2)*x^5*b*d^2*e*f^2+5*(-d/
c)^(1/2)*x^3*a*c*d*f^3+5*(-d/c)^(1/2)*x^3*a*d^2*e*f^2-4*(-d/c)^(1/2)*x^3*b*c^2*f
^3+5*(-d/c)^(1/2)*x^3*b*c*d*e*f^2+6*(-d/c)^(1/2)*x^3*b*d^2*e^2*f+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*c*d*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*d^2*e^2*f-4*((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))*b*c^2*e*f^2+7*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Ellip
ticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c*d*e^2*f-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))*b*d^2*e^3-10*((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*c*d*e*f^2+
20*((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*d^2*e^2*f+8*((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))*b*c^2*e*f^2-13*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Elli
pticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c*d*e^2*f+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))*b*d^2*e^3+5*(-d/c)^(1/2)*x
*a*c*d*e*f^2-4*(-d/c)^(1/2)*x*b*c^2*e*f^2+6*(-d/c)^(1/2)*x*b*c*d*e^2*f)/f/d^2/(d
*f*x^4+c*f*x^2+d*e*x^2+c*e)/(-d/c)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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