3.72 \(\int \frac{(a+b x)^3}{x^4} \, dx\)

Optimal. Leaf size=37 \[ -\frac{3 a^2 b}{2 x^2}-\frac{a^3}{3 x^3}-\frac{3 a b^2}{x}+b^3 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0122868, antiderivative size = 37, 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} \[ -\frac{3 a^2 b}{2 x^2}-\frac{a^3}{3 x^3}-\frac{3 a b^2}{x}+b^3 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/x^4,x]

[Out]

-a^3/(3*x^3) - (3*a^2*b)/(2*x^2) - (3*a*b^2)/x + b^3*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^4} \, dx &=\int \left (\frac{a^3}{x^4}+\frac{3 a^2 b}{x^3}+\frac{3 a b^2}{x^2}+\frac{b^3}{x}\right ) \, dx\\ &=-\frac{a^3}{3 x^3}-\frac{3 a^2 b}{2 x^2}-\frac{3 a b^2}{x}+b^3 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0037367, size = 37, normalized size = 1. \[ -\frac{3 a^2 b}{2 x^2}-\frac{a^3}{3 x^3}-\frac{3 a b^2}{x}+b^3 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/x^4,x]

[Out]

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

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Maple [A]  time = 0.006, size = 34, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}}{3\,{x}^{3}}}-{\frac{3\,{a}^{2}b}{2\,{x}^{2}}}-3\,{\frac{{b}^{2}a}{x}}+{b}^{3}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3/x^4,x)

[Out]

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

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Maxima [A]  time = 1.04267, size = 46, normalized size = 1.24 \begin{align*} b^{3} \log \left (x\right ) - \frac{18 \, a b^{2} x^{2} + 9 \, a^{2} b x + 2 \, a^{3}}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^4,x, algorithm="maxima")

[Out]

b^3*log(x) - 1/6*(18*a*b^2*x^2 + 9*a^2*b*x + 2*a^3)/x^3

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Fricas [A]  time = 1.58624, size = 85, normalized size = 2.3 \begin{align*} \frac{6 \, b^{3} x^{3} \log \left (x\right ) - 18 \, a b^{2} x^{2} - 9 \, a^{2} b x - 2 \, a^{3}}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^4,x, algorithm="fricas")

[Out]

1/6*(6*b^3*x^3*log(x) - 18*a*b^2*x^2 - 9*a^2*b*x - 2*a^3)/x^3

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Sympy [A]  time = 0.56435, size = 34, normalized size = 0.92 \begin{align*} b^{3} \log{\left (x \right )} - \frac{2 a^{3} + 9 a^{2} b x + 18 a b^{2} x^{2}}{6 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3/x**4,x)

[Out]

b**3*log(x) - (2*a**3 + 9*a**2*b*x + 18*a*b**2*x**2)/(6*x**3)

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Giac [A]  time = 1.19726, size = 47, normalized size = 1.27 \begin{align*} b^{3} \log \left ({\left | x \right |}\right ) - \frac{18 \, a b^{2} x^{2} + 9 \, a^{2} b x + 2 \, a^{3}}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^4,x, algorithm="giac")

[Out]

b^3*log(abs(x)) - 1/6*(18*a*b^2*x^2 + 9*a^2*b*x + 2*a^3)/x^3