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

Optimal. Leaf size=34 \[ 3 a^2 b \log (x)-\frac{a^3}{x}+3 a b^2 x+\frac{b^3 x^2}{2} \]

[Out]

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

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Rubi [A]  time = 0.0132317, antiderivative size = 34, 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} \[ 3 a^2 b \log (x)-\frac{a^3}{x}+3 a b^2 x+\frac{b^3 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/x^2,x]

[Out]

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

Mathematica [A]  time = 0.0064276, size = 34, normalized size = 1. \[ 3 a^2 b \log (x)-\frac{a^3}{x}+3 a b^2 x+\frac{b^3 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/x^2,x]

[Out]

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

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Maple [A]  time = 0.005, size = 33, normalized size = 1. \begin{align*} -{\frac{{a}^{3}}{x}}+3\,a{b}^{2}x+{\frac{{b}^{3}{x}^{2}}{2}}+3\,{a}^{2}b\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3/x^2,x)

[Out]

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

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Maxima [A]  time = 1.07062, size = 43, normalized size = 1.26 \begin{align*} \frac{1}{2} \, b^{3} x^{2} + 3 \, a b^{2} x + 3 \, a^{2} b \log \left (x\right ) - \frac{a^{3}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.50798, size = 78, normalized size = 2.29 \begin{align*} \frac{b^{3} x^{3} + 6 \, a b^{2} x^{2} + 6 \, a^{2} b x \log \left (x\right ) - 2 \, a^{3}}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.316417, size = 31, normalized size = 0.91 \begin{align*} - \frac{a^{3}}{x} + 3 a^{2} b \log{\left (x \right )} + 3 a b^{2} x + \frac{b^{3} x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3/x**2,x)

[Out]

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

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Giac [A]  time = 1.1732, size = 45, normalized size = 1.32 \begin{align*} \frac{1}{2} \, b^{3} x^{2} + 3 \, a b^{2} x + 3 \, a^{2} b \log \left ({\left | x \right |}\right ) - \frac{a^{3}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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