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

Optimal. Leaf size=67 \[ \frac{d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 \sqrt{a} c^{3/2}}+\frac{e \log \left (a+c x^2\right )}{2 c^2}-\frac{x (d+e x)}{2 c \left (a+c x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0869026, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 \sqrt{a} c^{3/2}}+\frac{e \log \left (a+c x^2\right )}{2 c^2}-\frac{x (d+e x)}{2 c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.273, size = 61, normalized size = 0.91 \[ - \frac{x \left (2 d + 2 e x\right )}{4 c \left (a + c x^{2}\right )} + \frac{e \log{\left (a + c x^{2} \right )}}{2 c^{2}} + \frac{d \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 \sqrt{a} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*(2*d + 2*e*x)/(4*c*(a + c*x**2)) + e*log(a + c*x**2)/(2*c**2) + d*atan(sqrt(c
)*x/sqrt(a))/(2*sqrt(a)*c**(3/2))

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Mathematica [A]  time = 0.0727357, size = 62, normalized size = 0.93 \[ \frac{\frac{a e-c d x}{a+c x^2}+\frac{\sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a}}+e \log \left (a+c x^2\right )}{2 c^2} \]

Antiderivative was successfully verified.

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

[Out]

((a*e - c*d*x)/(a + c*x^2) + (Sqrt[c]*d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[a] + e
*Log[a + c*x^2])/(2*c^2)

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Maple [A]  time = 0.016, size = 63, normalized size = 0.9 \[{\frac{1}{c{x}^{2}+a} \left ( -{\frac{dx}{2\,c}}+{\frac{ae}{2\,{c}^{2}}} \right ) }+{\frac{e\ln \left ( c \left ( c{x}^{2}+a \right ) \right ) }{2\,{c}^{2}}}+{\frac{d}{2\,c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(e*x+d)/(c*x^2+a)^2,x)

[Out]

(-1/2*d*x/c+1/2*a*e/c^2)/(c*x^2+a)+1/2*e/c^2*ln(c*(c*x^2+a))+1/2*d/c/(a*c)^(1/2)
*arctan(c*x/(a*c)^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.285773, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (c^{2} d x^{2} + a c d\right )} \log \left (\frac{2 \, a c x +{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) - 2 \,{\left (c d x - a e -{\left (c e x^{2} + a e\right )} \log \left (c x^{2} + a\right )\right )} \sqrt{-a c}}{4 \,{\left (c^{3} x^{2} + a c^{2}\right )} \sqrt{-a c}}, \frac{{\left (c^{2} d x^{2} + a c d\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (c d x - a e -{\left (c e x^{2} + a e\right )} \log \left (c x^{2} + a\right )\right )} \sqrt{a c}}{2 \,{\left (c^{3} x^{2} + a c^{2}\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*((c^2*d*x^2 + a*c*d)*log((2*a*c*x + (c*x^2 - a)*sqrt(-a*c))/(c*x^2 + a)) -
2*(c*d*x - a*e - (c*e*x^2 + a*e)*log(c*x^2 + a))*sqrt(-a*c))/((c^3*x^2 + a*c^2)*
sqrt(-a*c)), 1/2*((c^2*d*x^2 + a*c*d)*arctan(sqrt(a*c)*x/a) - (c*d*x - a*e - (c*
e*x^2 + a*e)*log(c*x^2 + a))*sqrt(a*c))/((c^3*x^2 + a*c^2)*sqrt(a*c))]

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Sympy [A]  time = 2.56301, size = 162, normalized size = 2.42 \[ \left (\frac{e}{2 c^{2}} - \frac{d \sqrt{- a c^{5}}}{4 a c^{4}}\right ) \log{\left (x + \frac{4 a c^{2} \left (\frac{e}{2 c^{2}} - \frac{d \sqrt{- a c^{5}}}{4 a c^{4}}\right ) - 2 a e}{c d} \right )} + \left (\frac{e}{2 c^{2}} + \frac{d \sqrt{- a c^{5}}}{4 a c^{4}}\right ) \log{\left (x + \frac{4 a c^{2} \left (\frac{e}{2 c^{2}} + \frac{d \sqrt{- a c^{5}}}{4 a c^{4}}\right ) - 2 a e}{c d} \right )} - \frac{- a e + c d x}{2 a c^{2} + 2 c^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.271168, size = 84, normalized size = 1.25 \[ \frac{d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} c} + \frac{e{\rm ln}\left (c x^{2} + a\right )}{2 \, c^{2}} - \frac{d x - \frac{a e}{c}}{2 \,{\left (c x^{2} + a\right )} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*d*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*c) + 1/2*e*ln(c*x^2 + a)/c^2 - 1/2*(d*x -
a*e/c)/((c*x^2 + a)*c)