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

Optimal. Leaf size=77 \[ \frac{x (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac{(2 a e+c d) \log (a-c x)}{4 a c^4}-\frac{(c d-2 a e) \log (a+c x)}{4 a c^4} \]

[Out]

(x*(d + e*x))/(2*c^2*(a^2 - c^2*x^2)) + ((c*d + 2*a*e)*Log[a - c*x])/(4*a*c^4) -
((c*d - 2*a*e)*Log[a + c*x])/(4*a*c^4)

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Rubi [A]  time = 0.124827, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{x (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac{(2 a e+c d) \log (a-c x)}{4 a c^4}-\frac{(c d-2 a e) \log (a+c x)}{4 a c^4} \]

Antiderivative was successfully verified.

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

[Out]

(x*(d + e*x))/(2*c^2*(a^2 - c^2*x^2)) + ((c*d + 2*a*e)*Log[a - c*x])/(4*a*c^4) -
((c*d - 2*a*e)*Log[a + c*x])/(4*a*c^4)

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Rubi in Sympy [A]  time = 17.8011, size = 68, normalized size = 0.88 \[ \frac{x \left (2 d + 2 e x\right )}{4 c^{2} \left (a^{2} - c^{2} x^{2}\right )} + \frac{\left (2 a e - c d\right ) \log{\left (a + c x \right )}}{4 a c^{4}} + \frac{\left (2 a e + c d\right ) \log{\left (a - c x \right )}}{4 a c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0652929, size = 64, normalized size = 0.83 \[ \frac{\frac{a^2 e+c^2 d x}{a^2-c^2 x^2}+e \log \left (a^2-c^2 x^2\right )-\frac{c d \tanh ^{-1}\left (\frac{c x}{a}\right )}{a}}{2 c^4} \]

Antiderivative was successfully verified.

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

[Out]

((a^2*e + c^2*d*x)/(a^2 - c^2*x^2) - (c*d*ArcTanh[(c*x)/a])/a + e*Log[a^2 - c^2*
x^2])/(2*c^4)

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Maple [A]  time = 0.019, size = 118, normalized size = 1.5 \[{\frac{ae}{4\,{c}^{4} \left ( cx+a \right ) }}-{\frac{d}{4\,{c}^{3} \left ( cx+a \right ) }}+{\frac{\ln \left ( cx+a \right ) e}{2\,{c}^{4}}}-{\frac{\ln \left ( cx+a \right ) d}{4\,a{c}^{3}}}-{\frac{ae}{4\,{c}^{4} \left ( cx-a \right ) }}-{\frac{d}{4\,{c}^{3} \left ( cx-a \right ) }}+{\frac{\ln \left ( cx-a \right ) e}{2\,{c}^{4}}}+{\frac{\ln \left ( cx-a \right ) d}{4\,a{c}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4/c^4/(c*x+a)*a*e-1/4/c^3/(c*x+a)*d+1/2/c^4*ln(c*x+a)*e-1/4/a/c^3*ln(c*x+a)*d-
1/4/c^4/(c*x-a)*a*e-1/4/c^3/(c*x-a)*d+1/2/c^4*ln(c*x-a)*e+1/4/a/c^3*ln(c*x-a)*d

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Maxima [A]  time = 0.686512, size = 107, normalized size = 1.39 \[ -\frac{c^{2} d x + a^{2} e}{2 \,{\left (c^{6} x^{2} - a^{2} c^{4}\right )}} - \frac{{\left (c d - 2 \, a e\right )} \log \left (c x + a\right )}{4 \, a c^{4}} + \frac{{\left (c d + 2 \, a e\right )} \log \left (c x - a\right )}{4 \, a c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(c^2*d*x + a^2*e)/(c^6*x^2 - a^2*c^4) - 1/4*(c*d - 2*a*e)*log(c*x + a)/(a*c
^4) + 1/4*(c*d + 2*a*e)*log(c*x - a)/(a*c^4)

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Fricas [A]  time = 0.283485, size = 155, normalized size = 2.01 \[ -\frac{2 \, a c^{2} d x + 2 \, a^{3} e -{\left (a^{2} c d - 2 \, a^{3} e -{\left (c^{3} d - 2 \, a c^{2} e\right )} x^{2}\right )} \log \left (c x + a\right ) +{\left (a^{2} c d + 2 \, a^{3} e -{\left (c^{3} d + 2 \, a c^{2} e\right )} x^{2}\right )} \log \left (c x - a\right )}{4 \,{\left (a c^{6} x^{2} - a^{3} c^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(2*a*c^2*d*x + 2*a^3*e - (a^2*c*d - 2*a^3*e - (c^3*d - 2*a*c^2*e)*x^2)*log(
c*x + a) + (a^2*c*d + 2*a^3*e - (c^3*d + 2*a*c^2*e)*x^2)*log(c*x - a))/(a*c^6*x^
2 - a^3*c^4)

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Sympy [A]  time = 2.71784, size = 109, normalized size = 1.42 \[ - \frac{a^{2} e + c^{2} d x}{- 2 a^{2} c^{4} + 2 c^{6} x^{2}} + \frac{\left (2 a e - c d\right ) \log{\left (x + \frac{2 a^{2} e - a \left (2 a e - c d\right )}{c^{2} d} \right )}}{4 a c^{4}} + \frac{\left (2 a e + c d\right ) \log{\left (x + \frac{2 a^{2} e - a \left (2 a e + c d\right )}{c^{2} d} \right )}}{4 a c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a**2*e + c**2*d*x)/(-2*a**2*c**4 + 2*c**6*x**2) + (2*a*e - c*d)*log(x + (2*a**
2*e - a*(2*a*e - c*d))/(c**2*d))/(4*a*c**4) + (2*a*e + c*d)*log(x + (2*a**2*e -
a*(2*a*e + c*d))/(c**2*d))/(4*a*c**4)

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GIAC/XCAS [A]  time = 0.281245, size = 115, normalized size = 1.49 \[ -\frac{d x + \frac{a^{2} e}{c^{2}}}{2 \,{\left (c x + a\right )}{\left (c x - a\right )} c^{2}} - \frac{{\left (c d - 2 \, a e\right )}{\rm ln}\left ({\left | c x + a \right |}\right )}{4 \, a c^{4}} + \frac{{\left (c d + 2 \, a e\right )}{\rm ln}\left ({\left | c x - a \right |}\right )}{4 \, a c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(d*x + a^2*e/c^2)/((c*x + a)*(c*x - a)*c^2) - 1/4*(c*d - 2*a*e)*ln(abs(c*x
+ a))/(a*c^4) + 1/4*(c*d + 2*a*e)*ln(abs(c*x - a))/(a*c^4)