3.134 \(\int \frac{x^3 \left (a c+b c x^2\right )}{\left (a+b x^2\right )^3} \, dx\)

Optimal. Leaf size=35 \[ \frac{a c}{2 b^2 \left (a+b x^2\right )}+\frac{c \log \left (a+b x^2\right )}{2 b^2} \]

[Out]

(a*c)/(2*b^2*(a + b*x^2)) + (c*Log[a + b*x^2])/(2*b^2)

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Rubi [A]  time = 0.0624562, antiderivative size = 35, 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 \[ \frac{a c}{2 b^2 \left (a+b x^2\right )}+\frac{c \log \left (a+b x^2\right )}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(a*c)/(2*b^2*(a + b*x^2)) + (c*Log[a + b*x^2])/(2*b^2)

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Rubi in Sympy [A]  time = 11.4785, size = 29, normalized size = 0.83 \[ \frac{a c}{2 b^{2} \left (a + b x^{2}\right )} + \frac{c \log{\left (a + b x^{2} \right )}}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*c/(2*b**2*(a + b*x**2)) + c*log(a + b*x**2)/(2*b**2)

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Mathematica [A]  time = 0.0157707, size = 28, normalized size = 0.8 \[ \frac{c \left (\frac{a}{a+b x^2}+\log \left (a+b x^2\right )\right )}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(c*(a/(a + b*x^2) + Log[a + b*x^2]))/(2*b^2)

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Maple [A]  time = 0.007, size = 32, normalized size = 0.9 \[{\frac{ac}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{c\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a*c/b^2/(b*x^2+a)+1/2*c*ln(b*x^2+a)/b^2

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Maxima [A]  time = 1.3366, size = 46, normalized size = 1.31 \[ \frac{a c}{2 \,{\left (b^{3} x^{2} + a b^{2}\right )}} + \frac{c \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a*c/(b^3*x^2 + a*b^2) + 1/2*c*log(b*x^2 + a)/b^2

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Fricas [A]  time = 0.227442, size = 54, normalized size = 1.54 \[ \frac{a c +{\left (b c x^{2} + a c\right )} \log \left (b x^{2} + a\right )}{2 \,{\left (b^{3} x^{2} + a b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(a*c + (b*c*x^2 + a*c)*log(b*x^2 + a))/(b^3*x^2 + a*b^2)

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Sympy [A]  time = 1.39611, size = 31, normalized size = 0.89 \[ c \left (\frac{a}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{\log{\left (a + b x^{2} \right )}}{2 b^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*(a/(2*a*b**2 + 2*b**3*x**2) + log(a + b*x**2)/(2*b**2))

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GIAC/XCAS [A]  time = 0.222955, size = 43, normalized size = 1.23 \[ \frac{c{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{2}} + \frac{a c}{2 \,{\left (b x^{2} + a\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*c*ln(abs(b*x^2 + a))/b^2 + 1/2*a*c/((b*x^2 + a)*b^2)