3.274 \(\int \frac{x^2 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^2} \, dx\)

Optimal. Leaf size=122 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (5 c d^2-e (3 b d-a e)\right )}{2 \sqrt{d} e^{7/2}}-\frac{x \left (5 c d^2-e (3 b d-a e)\right )}{2 d e^3}+\frac{x^3 \left (a+\frac{d (c d-b e)}{e^2}\right )}{2 d \left (d+e x^2\right )}+\frac{c x^3}{3 e^2} \]

[Out]

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

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Rubi [A]  time = 0.333695, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (5 c d^2-e (3 b d-a e)\right )}{2 \sqrt{d} e^{7/2}}-\frac{x \left (5 c d^2-e (3 b d-a e)\right )}{2 d e^3}+\frac{x^3 \left (a+\frac{d (c d-b e)}{e^2}\right )}{2 d \left (d+e x^2\right )}+\frac{c x^3}{3 e^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^2*(a + b*x^2 + c*x^4))/(d + e*x^2)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 51.7901, size = 95, normalized size = 0.78 \[ \frac{c x^{3}}{3 e^{2}} + \frac{x \left (b e - 2 c d\right )}{e^{3}} - \frac{x \left (a e^{2} - b d e + c d^{2}\right )}{2 e^{3} \left (d + e x^{2}\right )} + \frac{\left (a e^{2} - 3 b d e + 5 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{2 \sqrt{d} e^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(c*x**4+b*x**2+a)/(e*x**2+d)**2,x)

[Out]

c*x**3/(3*e**2) + x*(b*e - 2*c*d)/e**3 - x*(a*e**2 - b*d*e + c*d**2)/(2*e**3*(d
+ e*x**2)) + (a*e**2 - 3*b*d*e + 5*c*d**2)*atan(sqrt(e)*x/sqrt(d))/(2*sqrt(d)*e*
*(7/2))

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Mathematica [A]  time = 0.114797, size = 102, normalized size = 0.84 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a e^2-3 b d e+5 c d^2\right )}{2 \sqrt{d} e^{7/2}}-\frac{x \left (a e^2-b d e+c d^2\right )}{2 e^3 \left (d+e x^2\right )}+\frac{x (b e-2 c d)}{e^3}+\frac{c x^3}{3 e^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^2*(a + b*x^2 + c*x^4))/(d + e*x^2)^2,x]

[Out]

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

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Maple [A]  time = 0.013, size = 141, normalized size = 1.2 \[{\frac{c{x}^{3}}{3\,{e}^{2}}}+{\frac{bx}{{e}^{2}}}-2\,{\frac{cdx}{{e}^{3}}}-{\frac{ax}{2\,e \left ( e{x}^{2}+d \right ) }}+{\frac{bxd}{2\,{e}^{2} \left ( e{x}^{2}+d \right ) }}-{\frac{cx{d}^{2}}{2\,{e}^{3} \left ( e{x}^{2}+d \right ) }}+{\frac{a}{2\,e}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{3\,bd}{2\,{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{5\,c{d}^{2}}{2\,{e}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(c*x^4+b*x^2+a)/(e*x^2+d)^2,x)

[Out]

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

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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((c*x^4 + b*x^2 + a)*x^2/(e*x^2 + d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.271434, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (5 \, c d^{3} - 3 \, b d^{2} e + a d e^{2} +{\left (5 \, c d^{2} e - 3 \, b d e^{2} + a e^{3}\right )} x^{2}\right )} \log \left (\frac{2 \, d e x +{\left (e x^{2} - d\right )} \sqrt{-d e}}{e x^{2} + d}\right ) + 2 \,{\left (2 \, c e^{2} x^{5} - 2 \,{\left (5 \, c d e - 3 \, b e^{2}\right )} x^{3} - 3 \,{\left (5 \, c d^{2} - 3 \, b d e + a e^{2}\right )} x\right )} \sqrt{-d e}}{12 \,{\left (e^{4} x^{2} + d e^{3}\right )} \sqrt{-d e}}, \frac{3 \,{\left (5 \, c d^{3} - 3 \, b d^{2} e + a d e^{2} +{\left (5 \, c d^{2} e - 3 \, b d e^{2} + a e^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) +{\left (2 \, c e^{2} x^{5} - 2 \,{\left (5 \, c d e - 3 \, b e^{2}\right )} x^{3} - 3 \,{\left (5 \, c d^{2} - 3 \, b d e + a e^{2}\right )} x\right )} \sqrt{d e}}{6 \,{\left (e^{4} x^{2} + d e^{3}\right )} \sqrt{d e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)*x^2/(e*x^2 + d)^2,x, algorithm="fricas")

[Out]

[1/12*(3*(5*c*d^3 - 3*b*d^2*e + a*d*e^2 + (5*c*d^2*e - 3*b*d*e^2 + a*e^3)*x^2)*l
og((2*d*e*x + (e*x^2 - d)*sqrt(-d*e))/(e*x^2 + d)) + 2*(2*c*e^2*x^5 - 2*(5*c*d*e
- 3*b*e^2)*x^3 - 3*(5*c*d^2 - 3*b*d*e + a*e^2)*x)*sqrt(-d*e))/((e^4*x^2 + d*e^3
)*sqrt(-d*e)), 1/6*(3*(5*c*d^3 - 3*b*d^2*e + a*d*e^2 + (5*c*d^2*e - 3*b*d*e^2 +
a*e^3)*x^2)*arctan(sqrt(d*e)*x/d) + (2*c*e^2*x^5 - 2*(5*c*d*e - 3*b*e^2)*x^3 - 3
*(5*c*d^2 - 3*b*d*e + a*e^2)*x)*sqrt(d*e))/((e^4*x^2 + d*e^3)*sqrt(d*e))]

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Sympy [A]  time = 4.5117, size = 160, normalized size = 1.31 \[ \frac{c x^{3}}{3 e^{2}} - \frac{x \left (a e^{2} - b d e + c d^{2}\right )}{2 d e^{3} + 2 e^{4} x^{2}} - \frac{\sqrt{- \frac{1}{d e^{7}}} \left (a e^{2} - 3 b d e + 5 c d^{2}\right ) \log{\left (- d e^{3} \sqrt{- \frac{1}{d e^{7}}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{d e^{7}}} \left (a e^{2} - 3 b d e + 5 c d^{2}\right ) \log{\left (d e^{3} \sqrt{- \frac{1}{d e^{7}}} + x \right )}}{4} + \frac{x \left (b e - 2 c d\right )}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(c*x**4+b*x**2+a)/(e*x**2+d)**2,x)

[Out]

c*x**3/(3*e**2) - x*(a*e**2 - b*d*e + c*d**2)/(2*d*e**3 + 2*e**4*x**2) - sqrt(-1
/(d*e**7))*(a*e**2 - 3*b*d*e + 5*c*d**2)*log(-d*e**3*sqrt(-1/(d*e**7)) + x)/4 +
sqrt(-1/(d*e**7))*(a*e**2 - 3*b*d*e + 5*c*d**2)*log(d*e**3*sqrt(-1/(d*e**7)) + x
)/4 + x*(b*e - 2*c*d)/e**3

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GIAC/XCAS [A]  time = 0.270334, size = 123, normalized size = 1.01 \[ \frac{{\left (5 \, c d^{2} - 3 \, b d e + a e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{7}{2}\right )}}{2 \, \sqrt{d}} + \frac{1}{3} \,{\left (c x^{3} e^{4} - 6 \, c d x e^{3} + 3 \, b x e^{4}\right )} e^{\left (-6\right )} - \frac{{\left (c d^{2} x - b d x e + a x e^{2}\right )} e^{\left (-3\right )}}{2 \,{\left (x^{2} e + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)*x^2/(e*x^2 + d)^2,x, algorithm="giac")

[Out]

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