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

Optimal. Leaf size=130 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) (a f (3 c f+d e)+b e (c f+3 d e))}{8 e^{5/2} f^{5/2}}-\frac{x (b e (c f+3 d e)-a f (3 c f+d e))}{8 e^2 f^2 \left (e+f x^2\right )}-\frac{x \left (a+b x^2\right ) (d e-c f)}{4 e f \left (e+f x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.313099, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) (a f (3 c f+d e)+b e (c f+3 d e))}{8 e^{5/2} f^{5/2}}-\frac{x (b e (c f+3 d e)-a f (3 c f+d e))}{8 e^2 f^2 \left (e+f x^2\right )}-\frac{x \left (a+b x^2\right ) (d e-c f)}{4 e f \left (e+f x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 29.8022, size = 117, normalized size = 0.9 \[ \frac{x \left (a + b x^{2}\right ) \left (c f - d e\right )}{4 e f \left (e + f x^{2}\right )^{2}} + \frac{x \left (a f \left (3 c f + d e\right ) - b e \left (c f + 3 d e\right )\right )}{8 e^{2} f^{2} \left (e + f x^{2}\right )} + \frac{\left (a f \left (3 c f + d e\right ) + b e \left (c f + 3 d e\right )\right ) \operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}}{8 e^{\frac{5}{2}} f^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a + b*x**2)*(c*f - d*e)/(4*e*f*(e + f*x**2)**2) + x*(a*f*(3*c*f + d*e) - b*e*
(c*f + 3*d*e))/(8*e**2*f**2*(e + f*x**2)) + (a*f*(3*c*f + d*e) + b*e*(c*f + 3*d*
e))*atan(sqrt(f)*x/sqrt(e))/(8*e**(5/2)*f**(5/2))

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Mathematica [A]  time = 0.153391, size = 130, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) (a f (3 c f+d e)+b e (c f+3 d e))}{8 e^{5/2} f^{5/2}}+\frac{x (a f (3 c f+d e)+b e (c f-5 d e))}{8 e^2 f^2 \left (e+f x^2\right )}+\frac{x (b e-a f) (d e-c f)}{4 e f^2 \left (e+f x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 175, normalized size = 1.4 \[{\frac{1}{ \left ( f{x}^{2}+e \right ) ^{2}} \left ({\frac{ \left ( 3\,ac{f}^{2}+aedf+bcef-5\,bd{e}^{2} \right ){x}^{3}}{8\,{e}^{2}f}}+{\frac{ \left ( 5\,ac{f}^{2}-aedf-bcef-3\,bd{e}^{2} \right ) x}{8\,{f}^{2}e}} \right ) }+{\frac{3\,ac}{8\,{e}^{2}}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}+{\frac{ad}{8\,ef}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}+{\frac{bc}{8\,ef}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}+{\frac{3\,bd}{8\,{f}^{2}}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(1/8*(3*a*c*f^2+a*d*e*f+b*c*e*f-5*b*d*e^2)/e^2/f*x^3+1/8*(5*a*c*f^2-a*d*e*f-b*c*
e*f-3*b*d*e^2)/f^2/e*x)/(f*x^2+e)^2+3/8/e^2/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*
a*c+1/8/e/f/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*a*d+1/8/e/f/(e*f)^(1/2)*arctan(x
*f/(e*f)^(1/2))*b*c+3/8/f^2/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.217924, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (3 \, b d e^{4} + 3 \, a c e^{2} f^{2} +{\left (b c + a d\right )} e^{3} f +{\left (3 \, b d e^{2} f^{2} + 3 \, a c f^{4} +{\left (b c + a d\right )} e f^{3}\right )} x^{4} + 2 \,{\left (3 \, b d e^{3} f + 3 \, a c e f^{3} +{\left (b c + a d\right )} e^{2} f^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, e f x +{\left (f x^{2} - e\right )} \sqrt{-e f}}{f x^{2} + e}\right ) - 2 \,{\left ({\left (5 \, b d e^{2} f - 3 \, a c f^{3} -{\left (b c + a d\right )} e f^{2}\right )} x^{3} +{\left (3 \, b d e^{3} - 5 \, a c e f^{2} +{\left (b c + a d\right )} e^{2} f\right )} x\right )} \sqrt{-e f}}{16 \,{\left (e^{2} f^{4} x^{4} + 2 \, e^{3} f^{3} x^{2} + e^{4} f^{2}\right )} \sqrt{-e f}}, \frac{{\left (3 \, b d e^{4} + 3 \, a c e^{2} f^{2} +{\left (b c + a d\right )} e^{3} f +{\left (3 \, b d e^{2} f^{2} + 3 \, a c f^{4} +{\left (b c + a d\right )} e f^{3}\right )} x^{4} + 2 \,{\left (3 \, b d e^{3} f + 3 \, a c e f^{3} +{\left (b c + a d\right )} e^{2} f^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{e f} x}{e}\right ) -{\left ({\left (5 \, b d e^{2} f - 3 \, a c f^{3} -{\left (b c + a d\right )} e f^{2}\right )} x^{3} +{\left (3 \, b d e^{3} - 5 \, a c e f^{2} +{\left (b c + a d\right )} e^{2} f\right )} x\right )} \sqrt{e f}}{8 \,{\left (e^{2} f^{4} x^{4} + 2 \, e^{3} f^{3} x^{2} + e^{4} f^{2}\right )} \sqrt{e f}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*((3*b*d*e^4 + 3*a*c*e^2*f^2 + (b*c + a*d)*e^3*f + (3*b*d*e^2*f^2 + 3*a*c*f
^4 + (b*c + a*d)*e*f^3)*x^4 + 2*(3*b*d*e^3*f + 3*a*c*e*f^3 + (b*c + a*d)*e^2*f^2
)*x^2)*log((2*e*f*x + (f*x^2 - e)*sqrt(-e*f))/(f*x^2 + e)) - 2*((5*b*d*e^2*f - 3
*a*c*f^3 - (b*c + a*d)*e*f^2)*x^3 + (3*b*d*e^3 - 5*a*c*e*f^2 + (b*c + a*d)*e^2*f
)*x)*sqrt(-e*f))/((e^2*f^4*x^4 + 2*e^3*f^3*x^2 + e^4*f^2)*sqrt(-e*f)), 1/8*((3*b
*d*e^4 + 3*a*c*e^2*f^2 + (b*c + a*d)*e^3*f + (3*b*d*e^2*f^2 + 3*a*c*f^4 + (b*c +
a*d)*e*f^3)*x^4 + 2*(3*b*d*e^3*f + 3*a*c*e*f^3 + (b*c + a*d)*e^2*f^2)*x^2)*arct
an(sqrt(e*f)*x/e) - ((5*b*d*e^2*f - 3*a*c*f^3 - (b*c + a*d)*e*f^2)*x^3 + (3*b*d*
e^3 - 5*a*c*e*f^2 + (b*c + a*d)*e^2*f)*x)*sqrt(e*f))/((e^2*f^4*x^4 + 2*e^3*f^3*x
^2 + e^4*f^2)*sqrt(e*f))]

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Sympy [A]  time = 7.20104, size = 246, normalized size = 1.89 \[ - \frac{\sqrt{- \frac{1}{e^{5} f^{5}}} \left (3 a c f^{2} + a d e f + b c e f + 3 b d e^{2}\right ) \log{\left (- e^{3} f^{2} \sqrt{- \frac{1}{e^{5} f^{5}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{e^{5} f^{5}}} \left (3 a c f^{2} + a d e f + b c e f + 3 b d e^{2}\right ) \log{\left (e^{3} f^{2} \sqrt{- \frac{1}{e^{5} f^{5}}} + x \right )}}{16} + \frac{x^{3} \left (3 a c f^{3} + a d e f^{2} + b c e f^{2} - 5 b d e^{2} f\right ) + x \left (5 a c e f^{2} - a d e^{2} f - b c e^{2} f - 3 b d e^{3}\right )}{8 e^{4} f^{2} + 16 e^{3} f^{3} x^{2} + 8 e^{2} f^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-1/(e**5*f**5))*(3*a*c*f**2 + a*d*e*f + b*c*e*f + 3*b*d*e**2)*log(-e**3*f*
*2*sqrt(-1/(e**5*f**5)) + x)/16 + sqrt(-1/(e**5*f**5))*(3*a*c*f**2 + a*d*e*f + b
*c*e*f + 3*b*d*e**2)*log(e**3*f**2*sqrt(-1/(e**5*f**5)) + x)/16 + (x**3*(3*a*c*f
**3 + a*d*e*f**2 + b*c*e*f**2 - 5*b*d*e**2*f) + x*(5*a*c*e*f**2 - a*d*e**2*f - b
*c*e**2*f - 3*b*d*e**3))/(8*e**4*f**2 + 16*e**3*f**3*x**2 + 8*e**2*f**4*x**4)

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GIAC/XCAS [A]  time = 0.227891, size = 182, normalized size = 1.4 \[ \frac{{\left (3 \, a c f^{2} + b c f e + a d f e + 3 \, b d e^{2}\right )} \arctan \left (\sqrt{f} x e^{\left (-\frac{1}{2}\right )}\right ) e^{\left (-\frac{5}{2}\right )}}{8 \, f^{\frac{5}{2}}} + \frac{{\left (3 \, a c f^{3} x^{3} + b c f^{2} x^{3} e + a d f^{2} x^{3} e - 5 \, b d f x^{3} e^{2} + 5 \, a c f^{2} x e - b c f x e^{2} - a d f x e^{2} - 3 \, b d x e^{3}\right )} e^{\left (-2\right )}}{8 \,{\left (f x^{2} + e\right )}^{2} f^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(3*a*c*f^2 + b*c*f*e + a*d*f*e + 3*b*d*e^2)*arctan(sqrt(f)*x*e^(-1/2))*e^(-5
/2)/f^(5/2) + 1/8*(3*a*c*f^3*x^3 + b*c*f^2*x^3*e + a*d*f^2*x^3*e - 5*b*d*f*x^3*e
^2 + 5*a*c*f^2*x*e - b*c*f*x*e^2 - a*d*f*x*e^2 - 3*b*d*x*e^3)*e^(-2)/((f*x^2 + e
)^2*f^2)