3.71 \(\int \frac{(a+b x)^3}{x^3} \, dx\)

Optimal. Leaf size=33 \[ -\frac{a^3}{2 x^2}-\frac{3 a^2 b}{x}+3 a b^2 \log (x)+b^3 x \]

[Out]

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

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Rubi [A]  time = 0.0296144, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{a^3}{2 x^2}-\frac{3 a^2 b}{x}+3 a b^2 \log (x)+b^3 x \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3/x^3,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00710586, size = 33, normalized size = 1. \[ -\frac{a^3}{2 x^2}-\frac{3 a^2 b}{x}+3 a b^2 \log (x)+b^3 x \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^3/x^3,x]

[Out]

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

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Maple [A]  time = 0.008, size = 32, normalized size = 1. \[ -{\frac{{a}^{3}}{2\,{x}^{2}}}-3\,{\frac{{a}^{2}b}{x}}+{b}^{3}x+3\,a{b}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3/x^3,x)

[Out]

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

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Maxima [A]  time = 1.32131, size = 41, normalized size = 1.24 \[ b^{3} x + 3 \, a b^{2} \log \left (x\right ) - \frac{6 \, a^{2} b x + a^{3}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.207257, size = 50, normalized size = 1.52 \[ \frac{2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} \log \left (x\right ) - 6 \, a^{2} b x - a^{3}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.22046, size = 31, normalized size = 0.94 \[ 3 a b^{2} \log{\left (x \right )} + b^{3} x - \frac{a^{3} + 6 a^{2} b x}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3/x**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.216919, size = 42, normalized size = 1.27 \[ b^{3} x + 3 \, a b^{2}{\rm ln}\left ({\left | x \right |}\right ) - \frac{6 \, a^{2} b x + a^{3}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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