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

Optimal. Leaf size=142 \[ -\frac{x \left (5 a d f (3 d e-5 c f)-b \left (8 c^2 f^2-25 c d e f+15 d^2 e^2\right )\right )}{15 f^3}-\frac{(b e-a f) (d e-c f)^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} f^{7/2}}-\frac{x \left (c+d x^2\right ) (-5 a d f-4 b c f+5 b d e)}{15 f^2}+\frac{b x \left (c+d x^2\right )^2}{5 f} \]

[Out]

-((5*a*d*f*(3*d*e - 5*c*f) - b*(15*d^2*e^2 - 25*c*d*e*f + 8*c^2*f^2))*x)/(15*f^3
) - ((5*b*d*e - 4*b*c*f - 5*a*d*f)*x*(c + d*x^2))/(15*f^2) + (b*x*(c + d*x^2)^2)
/(5*f) - ((b*e - a*f)*(d*e - c*f)^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(7/2
))

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Rubi [A]  time = 0.534499, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{x \left (5 a d f (3 d e-5 c f)-b \left (8 c^2 f^2-25 c d e f+15 d^2 e^2\right )\right )}{15 f^3}-\frac{(b e-a f) (d e-c f)^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} f^{7/2}}-\frac{x \left (c+d x^2\right ) (-5 a d f-4 b c f+5 b d e)}{15 f^2}+\frac{b x \left (c+d x^2\right )^2}{5 f} \]

Antiderivative was successfully verified.

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

[Out]

-((5*a*d*f*(3*d*e - 5*c*f) - b*(15*d^2*e^2 - 25*c*d*e*f + 8*c^2*f^2))*x)/(15*f^3
) - ((5*b*d*e - 4*b*c*f - 5*a*d*f)*x*(c + d*x^2))/(15*f^2) + (b*x*(c + d*x^2)^2)
/(5*f) - ((b*e - a*f)*(d*e - c*f)^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(7/2
))

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Rubi in Sympy [A]  time = 60.7377, size = 153, normalized size = 1.08 \[ \frac{b x \left (c + d x^{2}\right )^{2}}{5 f} + \frac{d x \left (c \left (5 a f - b e\right ) + x^{2} \left (5 a d f + 4 b c f - 5 b d e\right )\right )}{15 f^{2}} + \frac{x \left (25 a c d f^{2} - 15 a d^{2} e f + 12 b c^{2} f^{2} - 29 b c d e f + 15 b d^{2} e^{2}\right )}{15 f^{3}} + \frac{\left (a f - b e\right ) \left (c f - d e\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}}{\sqrt{e} f^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x*(c + d*x**2)**2/(5*f) + d*x*(c*(5*a*f - b*e) + x**2*(5*a*d*f + 4*b*c*f - 5*b
*d*e))/(15*f**2) + x*(25*a*c*d*f**2 - 15*a*d**2*e*f + 12*b*c**2*f**2 - 29*b*c*d*
e*f + 15*b*d**2*e**2)/(15*f**3) + (a*f - b*e)*(c*f - d*e)**2*atan(sqrt(f)*x/sqrt
(e))/(sqrt(e)*f**(7/2))

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Mathematica [A]  time = 0.12025, size = 115, normalized size = 0.81 \[ -\frac{(b e-a f) (d e-c f)^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} f^{7/2}}+\frac{x \left (a d f (2 c f-d e)+b (d e-c f)^2\right )}{f^3}+\frac{d x^3 (a d f+2 b c f-b d e)}{3 f^2}+\frac{b d^2 x^5}{5 f} \]

Antiderivative was successfully verified.

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

[Out]

((b*(d*e - c*f)^2 + a*d*f*(-(d*e) + 2*c*f))*x)/f^3 + (d*(-(b*d*e) + 2*b*c*f + a*
d*f)*x^3)/(3*f^2) + (b*d^2*x^5)/(5*f) - ((b*e - a*f)*(d*e - c*f)^2*ArcTan[(Sqrt[
f]*x)/Sqrt[e]])/(Sqrt[e]*f^(7/2))

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Maple [A]  time = 0.007, size = 243, normalized size = 1.7 \[{\frac{b{d}^{2}{x}^{5}}{5\,f}}+{\frac{{x}^{3}a{d}^{2}}{3\,f}}+{\frac{2\,{x}^{3}bcd}{3\,f}}-{\frac{{x}^{3}b{d}^{2}e}{3\,{f}^{2}}}+2\,{\frac{acdx}{f}}-{\frac{a{d}^{2}ex}{{f}^{2}}}+{\frac{b{c}^{2}x}{f}}-2\,{\frac{bcdex}{{f}^{2}}}+{\frac{b{d}^{2}{e}^{2}x}{{f}^{3}}}+{a{c}^{2}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}-2\,{\frac{acde}{f\sqrt{ef}}\arctan \left ({\frac{fx}{\sqrt{ef}}} \right ) }+{\frac{a{d}^{2}{e}^{2}}{{f}^{2}}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}-{\frac{b{c}^{2}e}{f}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}+2\,{\frac{bcd{e}^{2}}{{f}^{2}\sqrt{ef}}\arctan \left ({\frac{fx}{\sqrt{ef}}} \right ) }-{\frac{b{d}^{2}{e}^{3}}{{f}^{3}}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)^2/(f*x^2 + e),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.215693, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (b d^{2} e^{3} - a c^{2} f^{3} -{\left (2 \, b c d + a d^{2}\right )} e^{2} f +{\left (b c^{2} + 2 \, a c d\right )} e f^{2}\right )} \log \left (\frac{2 \, e f x +{\left (f x^{2} - e\right )} \sqrt{-e f}}{f x^{2} + e}\right ) - 2 \,{\left (3 \, b d^{2} f^{2} x^{5} - 5 \,{\left (b d^{2} e f -{\left (2 \, b c d + a d^{2}\right )} f^{2}\right )} x^{3} + 15 \,{\left (b d^{2} e^{2} -{\left (2 \, b c d + a d^{2}\right )} e f +{\left (b c^{2} + 2 \, a c d\right )} f^{2}\right )} x\right )} \sqrt{-e f}}{30 \, \sqrt{-e f} f^{3}}, -\frac{15 \,{\left (b d^{2} e^{3} - a c^{2} f^{3} -{\left (2 \, b c d + a d^{2}\right )} e^{2} f +{\left (b c^{2} + 2 \, a c d\right )} e f^{2}\right )} \arctan \left (\frac{\sqrt{e f} x}{e}\right ) -{\left (3 \, b d^{2} f^{2} x^{5} - 5 \,{\left (b d^{2} e f -{\left (2 \, b c d + a d^{2}\right )} f^{2}\right )} x^{3} + 15 \,{\left (b d^{2} e^{2} -{\left (2 \, b c d + a d^{2}\right )} e f +{\left (b c^{2} + 2 \, a c d\right )} f^{2}\right )} x\right )} \sqrt{e f}}{15 \, \sqrt{e f} f^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/30*(15*(b*d^2*e^3 - a*c^2*f^3 - (2*b*c*d + a*d^2)*e^2*f + (b*c^2 + 2*a*c*d)*
e*f^2)*log((2*e*f*x + (f*x^2 - e)*sqrt(-e*f))/(f*x^2 + e)) - 2*(3*b*d^2*f^2*x^5
- 5*(b*d^2*e*f - (2*b*c*d + a*d^2)*f^2)*x^3 + 15*(b*d^2*e^2 - (2*b*c*d + a*d^2)*
e*f + (b*c^2 + 2*a*c*d)*f^2)*x)*sqrt(-e*f))/(sqrt(-e*f)*f^3), -1/15*(15*(b*d^2*e
^3 - a*c^2*f^3 - (2*b*c*d + a*d^2)*e^2*f + (b*c^2 + 2*a*c*d)*e*f^2)*arctan(sqrt(
e*f)*x/e) - (3*b*d^2*f^2*x^5 - 5*(b*d^2*e*f - (2*b*c*d + a*d^2)*f^2)*x^3 + 15*(b
*d^2*e^2 - (2*b*c*d + a*d^2)*e*f + (b*c^2 + 2*a*c*d)*f^2)*x)*sqrt(e*f))/(sqrt(e*
f)*f^3)]

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Sympy [A]  time = 2.51235, size = 343, normalized size = 2.42 \[ \frac{b d^{2} x^{5}}{5 f} - \frac{\sqrt{- \frac{1}{e f^{7}}} \left (a f - b e\right ) \left (c f - d e\right )^{2} \log{\left (- \frac{e f^{3} \sqrt{- \frac{1}{e f^{7}}} \left (a f - b e\right ) \left (c f - d e\right )^{2}}{a c^{2} f^{3} - 2 a c d e f^{2} + a d^{2} e^{2} f - b c^{2} e f^{2} + 2 b c d e^{2} f - b d^{2} e^{3}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{e f^{7}}} \left (a f - b e\right ) \left (c f - d e\right )^{2} \log{\left (\frac{e f^{3} \sqrt{- \frac{1}{e f^{7}}} \left (a f - b e\right ) \left (c f - d e\right )^{2}}{a c^{2} f^{3} - 2 a c d e f^{2} + a d^{2} e^{2} f - b c^{2} e f^{2} + 2 b c d e^{2} f - b d^{2} e^{3}} + x \right )}}{2} + \frac{x^{3} \left (a d^{2} f + 2 b c d f - b d^{2} e\right )}{3 f^{2}} + \frac{x \left (2 a c d f^{2} - a d^{2} e f + b c^{2} f^{2} - 2 b c d e f + b d^{2} e^{2}\right )}{f^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*d**2*x**5/(5*f) - sqrt(-1/(e*f**7))*(a*f - b*e)*(c*f - d*e)**2*log(-e*f**3*sqr
t(-1/(e*f**7))*(a*f - b*e)*(c*f - d*e)**2/(a*c**2*f**3 - 2*a*c*d*e*f**2 + a*d**2
*e**2*f - b*c**2*e*f**2 + 2*b*c*d*e**2*f - b*d**2*e**3) + x)/2 + sqrt(-1/(e*f**7
))*(a*f - b*e)*(c*f - d*e)**2*log(e*f**3*sqrt(-1/(e*f**7))*(a*f - b*e)*(c*f - d*
e)**2/(a*c**2*f**3 - 2*a*c*d*e*f**2 + a*d**2*e**2*f - b*c**2*e*f**2 + 2*b*c*d*e*
*2*f - b*d**2*e**3) + x)/2 + x**3*(a*d**2*f + 2*b*c*d*f - b*d**2*e)/(3*f**2) + x
*(2*a*c*d*f**2 - a*d**2*e*f + b*c**2*f**2 - 2*b*c*d*e*f + b*d**2*e**2)/f**3

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GIAC/XCAS [A]  time = 0.228411, size = 240, normalized size = 1.69 \[ \frac{{\left (a c^{2} f^{3} - b c^{2} f^{2} e - 2 \, a c d f^{2} e + 2 \, b c d f e^{2} + a d^{2} f e^{2} - b d^{2} e^{3}\right )} \arctan \left (\sqrt{f} x e^{\left (-\frac{1}{2}\right )}\right ) e^{\left (-\frac{1}{2}\right )}}{f^{\frac{7}{2}}} + \frac{3 \, b d^{2} f^{4} x^{5} + 10 \, b c d f^{4} x^{3} + 5 \, a d^{2} f^{4} x^{3} - 5 \, b d^{2} f^{3} x^{3} e + 15 \, b c^{2} f^{4} x + 30 \, a c d f^{4} x - 30 \, b c d f^{3} x e - 15 \, a d^{2} f^{3} x e + 15 \, b d^{2} f^{2} x e^{2}}{15 \, f^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a*c^2*f^3 - b*c^2*f^2*e - 2*a*c*d*f^2*e + 2*b*c*d*f*e^2 + a*d^2*f*e^2 - b*d^2*e
^3)*arctan(sqrt(f)*x*e^(-1/2))*e^(-1/2)/f^(7/2) + 1/15*(3*b*d^2*f^4*x^5 + 10*b*c
*d*f^4*x^3 + 5*a*d^2*f^4*x^3 - 5*b*d^2*f^3*x^3*e + 15*b*c^2*f^4*x + 30*a*c*d*f^4
*x - 30*b*c*d*f^3*x*e - 15*a*d^2*f^3*x*e + 15*b*d^2*f^2*x*e^2)/f^5