3.36 \(\int \frac{\left (a+b x^2\right )^3}{x^5} \, dx\)

Optimal. Leaf size=40 \[ -\frac{a^3}{4 x^4}-\frac{3 a^2 b}{2 x^2}+3 a b^2 \log (x)+\frac{b^3 x^2}{2} \]

[Out]

-a^3/(4*x^4) - (3*a^2*b)/(2*x^2) + (b^3*x^2)/2 + 3*a*b^2*Log[x]

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Rubi [A]  time = 0.0546777, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{a^3}{4 x^4}-\frac{3 a^2 b}{2 x^2}+3 a b^2 \log (x)+\frac{b^3 x^2}{2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^3/x^5,x]

[Out]

-a^3/(4*x^4) - (3*a^2*b)/(2*x^2) + (b^3*x^2)/2 + 3*a*b^2*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{3}}{4 x^{4}} - \frac{3 a^{2} b}{2 x^{2}} + \frac{3 a b^{2} \log{\left (x^{2} \right )}}{2} + \frac{\int ^{x^{2}} b^{3}\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**3/x**5,x)

[Out]

-a**3/(4*x**4) - 3*a**2*b/(2*x**2) + 3*a*b**2*log(x**2)/2 + Integral(b**3, (x, x
**2))/2

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Mathematica [A]  time = 0.00762455, size = 40, normalized size = 1. \[ -\frac{a^3}{4 x^4}-\frac{3 a^2 b}{2 x^2}+3 a b^2 \log (x)+\frac{b^3 x^2}{2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^3/x^5,x]

[Out]

-a^3/(4*x^4) - (3*a^2*b)/(2*x^2) + (b^3*x^2)/2 + 3*a*b^2*Log[x]

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Maple [A]  time = 0.01, size = 35, normalized size = 0.9 \[ -{\frac{{a}^{3}}{4\,{x}^{4}}}-{\frac{3\,{a}^{2}b}{2\,{x}^{2}}}+{\frac{{b}^{3}{x}^{2}}{2}}+3\,a{b}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^3/x^5,x)

[Out]

-1/4*a^3/x^4-3/2*a^2*b/x^2+1/2*b^3*x^2+3*a*b^2*ln(x)

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Maxima [A]  time = 1.3458, size = 50, normalized size = 1.25 \[ \frac{1}{2} \, b^{3} x^{2} + \frac{3}{2} \, a b^{2} \log \left (x^{2}\right ) - \frac{6 \, a^{2} b x^{2} + a^{3}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/x^5,x, algorithm="maxima")

[Out]

1/2*b^3*x^2 + 3/2*a*b^2*log(x^2) - 1/4*(6*a^2*b*x^2 + a^3)/x^4

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Fricas [A]  time = 0.200147, size = 53, normalized size = 1.32 \[ \frac{2 \, b^{3} x^{6} + 12 \, a b^{2} x^{4} \log \left (x\right ) - 6 \, a^{2} b x^{2} - a^{3}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/x^5,x, algorithm="fricas")

[Out]

1/4*(2*b^3*x^6 + 12*a*b^2*x^4*log(x) - 6*a^2*b*x^2 - a^3)/x^4

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Sympy [A]  time = 1.353, size = 36, normalized size = 0.9 \[ 3 a b^{2} \log{\left (x \right )} + \frac{b^{3} x^{2}}{2} - \frac{a^{3} + 6 a^{2} b x^{2}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**3/x**5,x)

[Out]

3*a*b**2*log(x) + b**3*x**2/2 - (a**3 + 6*a**2*b*x**2)/(4*x**4)

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GIAC/XCAS [A]  time = 0.207896, size = 62, normalized size = 1.55 \[ \frac{1}{2} \, b^{3} x^{2} + \frac{3}{2} \, a b^{2}{\rm ln}\left (x^{2}\right ) - \frac{9 \, a b^{2} x^{4} + 6 \, a^{2} b x^{2} + a^{3}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/x^5,x, algorithm="giac")

[Out]

1/2*b^3*x^2 + 3/2*a*b^2*ln(x^2) - 1/4*(9*a*b^2*x^4 + 6*a^2*b*x^2 + a^3)/x^4