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

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

[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 = 0.0126738, 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, Rules used = {43} \[ -\frac{3 a^2 b}{x}-\frac{a^3}{2 x^2}+3 a b^2 \log (x)+b^3 x \]

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]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(a+b x)^3}{x^3} \, dx &=\int \left (b^3+\frac{a^3}{x^3}+\frac{3 a^2 b}{x^2}+\frac{3 a b^2}{x}\right ) \, dx\\ &=-\frac{a^3}{2 x^2}-\frac{3 a^2 b}{x}+b^3 x+3 a b^2 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0071219, size = 33, normalized size = 1. \[ -\frac{3 a^2 b}{x}-\frac{a^3}{2 x^2}+3 a b^2 \log (x)+b^3 x \]

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 = 0.005, size = 32, normalized size = 1. \begin{align*} -{\frac{{a}^{3}}{2\,{x}^{2}}}-3\,{\frac{{a}^{2}b}{x}}+{b}^{3}x+3\,a{b}^{2}\ln \left ( x \right ) \end{align*}

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 = 1.0208, size = 41, normalized size = 1.24 \begin{align*} b^{3} x + 3 \, a b^{2} \log \left (x\right ) - \frac{6 \, a^{2} b x + a^{3}}{2 \, x^{2}} \end{align*}

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 = 1.57627, size = 81, normalized size = 2.45 \begin{align*} \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}} \end{align*}

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 = 0.44382, size = 31, normalized size = 0.94 \begin{align*} 3 a b^{2} \log{\left (x \right )} + b^{3} x - \frac{a^{3} + 6 a^{2} b x}{2 x^{2}} \end{align*}

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 [A]  time = 1.14986, size = 42, normalized size = 1.27 \begin{align*} b^{3} x + 3 \, a b^{2} \log \left ({\left | x \right |}\right ) - \frac{6 \, a^{2} b x + a^{3}}{2 \, x^{2}} \end{align*}

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