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

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

[Out]

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

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Rubi [A]  time = 0.0576987, 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 \[ -\frac{a^3}{2 x^2}+3 a^2 b \log (x)+\frac{3}{2} a b^2 x^2+\frac{b^3 x^4}{4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^3/x^3,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0120422, size = 40, normalized size = 1. \[ -\frac{a^3}{2 x^2}+3 a^2 b \log (x)+\frac{3}{2} a b^2 x^2+\frac{b^3 x^4}{4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^3/x^3,x]

[Out]

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

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Maple [A]  time = 0.009, size = 35, normalized size = 0.9 \[ -{\frac{{a}^{3}}{2\,{x}^{2}}}+{\frac{3\,a{b}^{2}{x}^{2}}{2}}+{\frac{{b}^{3}{x}^{4}}{4}}+3\,{a}^{2}b\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^3/x^3,x)

[Out]

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

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Maxima [A]  time = 1.34689, size = 49, normalized size = 1.22 \[ \frac{1}{4} \, b^{3} x^{4} + \frac{3}{2} \, a b^{2} x^{2} + \frac{3}{2} \, a^{2} b \log \left (x^{2}\right ) - \frac{a^{3}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.204462, size = 51, normalized size = 1.27 \[ \frac{b^{3} x^{6} + 6 \, a b^{2} x^{4} + 12 \, a^{2} b x^{2} \log \left (x\right ) - 2 \, a^{3}}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.18794, size = 37, normalized size = 0.92 \[ - \frac{a^{3}}{2 x^{2}} + 3 a^{2} b \log{\left (x \right )} + \frac{3 a b^{2} x^{2}}{2} + \frac{b^{3} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**3/x**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.209, size = 62, normalized size = 1.55 \[ \frac{1}{4} \, b^{3} x^{4} + \frac{3}{2} \, a b^{2} x^{2} + \frac{3}{2} \, a^{2} b{\rm ln}\left (x^{2}\right ) - \frac{3 \, a^{2} b x^{2} + a^{3}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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