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

Optimal. Leaf size=29 \[ \frac{c x^2}{2 b}-\frac{a c \log \left (a+b x^2\right )}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0531508, antiderivative size = 29, 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 \[ \frac{c x^2}{2 b}-\frac{a c \log \left (a+b x^2\right )}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a c \log{\left (a + b x^{2} \right )}}{2 b^{2}} + \frac{c \int ^{x^{2}} \frac{1}{b}\, dx}{2} \]

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)**2,x)

[Out]

-a*c*log(a + b*x**2)/(2*b**2) + c*Integral(1/b, (x, x**2))/2

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Mathematica [A]  time = 0.00728249, size = 29, normalized size = 1. \[ c \left (\frac{x^2}{2 b}-\frac{a \log \left (a+b x^2\right )}{2 b^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 26, normalized size = 0.9 \[{\frac{c{x}^{2}}{2\,b}}-{\frac{ac\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.36612, size = 34, normalized size = 1.17 \[ \frac{c x^{2}}{2 \, b} - \frac{a c \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.20791, size = 32, normalized size = 1.1 \[ \frac{b c x^{2} - a c \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.18829, size = 22, normalized size = 0.76 \[ c \left (- \frac{a \log{\left (a + b x^{2} \right )}}{2 b^{2}} + \frac{x^{2}}{2 b}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.243218, size = 63, normalized size = 2.17 \[ \frac{\frac{a c{\rm ln}\left (\frac{{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2}{\left | b \right |}}\right )}{b} + \frac{{\left (b x^{2} + a\right )} c}{b}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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