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

Optimal. Leaf size=358 \[ -\frac{\sqrt{e} \sqrt{c+d x^2} (-3 a d f-3 b c f+4 b d e) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{\sqrt{c+d x^2} (b e (8 d e-7 c f)-3 a f (2 d e-c f)) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 \sqrt{e} f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2} (4 b e-3 a f)}{3 e f^2}-\frac{x \sqrt{c+d x^2} (b e (8 d e-7 c f)-3 a f (2 d e-c f))}{3 e f^2 \sqrt{e+f x^2}}-\frac{x \left (c+d x^2\right )^{3/2} (b e-a f)}{e f \sqrt{e+f x^2}} \]

[Out]

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

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Rubi [A]  time = 1.09576, antiderivative size = 358, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\sqrt{e} \sqrt{c+d x^2} (-3 a d f-3 b c f+4 b d e) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{\sqrt{c+d x^2} (b e (8 d e-7 c f)-3 a f (2 d e-c f)) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 \sqrt{e} f^{5/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2} (4 b e-3 a f)}{3 e f^2}-\frac{x \sqrt{c+d x^2} (b e (8 d e-7 c f)-3 a f (2 d e-c f))}{3 e f^2 \sqrt{e+f x^2}}-\frac{x \left (c+d x^2\right )^{3/2} (b e-a f)}{e f \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-((b*e*(8*d*e - 7*c*f) - 3*a*f*(2*d*e - c*f))*x*Sqrt[c + d*x^2])/(3*e*f^2*Sqrt[e
+ f*x^2]) - ((b*e - a*f)*x*(c + d*x^2)^(3/2))/(e*f*Sqrt[e + f*x^2]) + (d*(4*b*e
- 3*a*f)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*e*f^2) + ((b*e*(8*d*e - 7*c*f) -
3*a*f*(2*d*e - c*f))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 -
(d*e)/(c*f)])/(3*Sqrt[e]*f^(5/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e +
f*x^2]) - (Sqrt[e]*(4*b*d*e - 3*b*c*f - 3*a*d*f)*Sqrt[c + d*x^2]*EllipticF[ArcT
an[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*f^(5/2)*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 = 113.906, size = 345, normalized size = 0.96 \[ \frac{\sqrt{c} \sqrt{d} \sqrt{e + f x^{2}} \left (3 a c f^{2} - 6 a d e f - 7 b c e f + 8 b d e^{2}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{3 e f^{3} \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} - \frac{d x \sqrt{c + d x^{2}} \sqrt{e + f x^{2}} \left (3 a f - 4 b e\right )}{3 e f^{2}} - \frac{d x \sqrt{e + f x^{2}} \left (3 a c f^{2} - 6 a d e f - 7 b c e f + 8 b d e^{2}\right )}{3 e f^{3} \sqrt{c + d x^{2}}} + \frac{\sqrt{e} \sqrt{c + d x^{2}} \left (3 a d f + 3 b c f - 4 b d e\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}\middle | 1 - \frac{d e}{c f}\right )}{3 f^{\frac{5}{2}} \sqrt{\frac{e \left (c + d x^{2}\right )}{c \left (e + f x^{2}\right )}} \sqrt{e + f x^{2}}} + \frac{x \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a f - b e\right )}{e f \sqrt{e + f x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c)*sqrt(d)*sqrt(e + f*x**2)*(3*a*c*f**2 - 6*a*d*e*f - 7*b*c*e*f + 8*b*d*e**
2)*elliptic_e(atan(sqrt(d)*x/sqrt(c)), -c*f/(d*e) + 1)/(3*e*f**3*sqrt(c*(e + f*x
**2)/(e*(c + d*x**2)))*sqrt(c + d*x**2)) - d*x*sqrt(c + d*x**2)*sqrt(e + f*x**2)
*(3*a*f - 4*b*e)/(3*e*f**2) - d*x*sqrt(e + f*x**2)*(3*a*c*f**2 - 6*a*d*e*f - 7*b
*c*e*f + 8*b*d*e**2)/(3*e*f**3*sqrt(c + d*x**2)) + sqrt(e)*sqrt(c + d*x**2)*(3*a
*d*f + 3*b*c*f - 4*b*d*e)*elliptic_f(atan(sqrt(f)*x/sqrt(e)), 1 - d*e/(c*f))/(3*
f**(5/2)*sqrt(e*(c + d*x**2)/(c*(e + f*x**2)))*sqrt(e + f*x**2)) + x*(c + d*x**2
)**(3/2)*(a*f - b*e)/(e*f*sqrt(e + f*x**2))

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Mathematica [C]  time = 1.30167, size = 260, normalized size = 0.73 \[ \frac{f x \sqrt{\frac{d}{c}} \left (c+d x^2\right ) \left (3 a f (c f-d e)+b e \left (-3 c f+4 d e+d f x^2\right )\right )-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) (6 a d f+3 b c f-8 b d e) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )-i d e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b e (7 c f-8 d e)-3 a f (c f-2 d e)) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{3 e f^3 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.038, size = 750, normalized size = 2.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)*((-d/c)^(1/2)*x^5*b*d^2*e*f^2+3*(-d/c)^(1/2)
*x^3*a*c*d*f^3-3*(-d/c)^(1/2)*x^3*a*d^2*e*f^2-2*(-d/c)^(1/2)*x^3*b*c*d*e*f^2+4*(
-d/c)^(1/2)*x^3*b*d^2*e^2*f+6*((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^2-6*((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+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*c^2*e*f^2-11*(
(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*d*e^2*f+8*((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^3-3*((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*c*d*e*f^2+6*((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+7*((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*d*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
*d^2*e^3+3*x*a*c^2*f^3*(-d/c)^(1/2)-3*(-d/c)^(1/2)*x*a*c*d*e*f^2-3*(-d/c)^(1/2)*
x*b*c^2*e*f^2+4*(-d/c)^(1/2)*x*b*c*d*e^2*f)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)/f^3/e/
(-d/c)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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