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

Optimal. Leaf size=35 \[ 3 a^2 b x+a^3 \log (x)+\frac{3}{2} a b^2 x^2+\frac{b^3 x^3}{3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0105865, antiderivative size = 35, 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} \[ 3 a^2 b x+a^3 \log (x)+\frac{3}{2} a b^2 x^2+\frac{b^3 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/x,x]

[Out]

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

Mathematica [A]  time = 0.0050428, size = 35, normalized size = 1. \[ 3 a^2 b x+a^3 \log (x)+\frac{3}{2} a b^2 x^2+\frac{b^3 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/x,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 32, normalized size = 0.9 \begin{align*} 3\,{a}^{2}bx+{\frac{3\,a{b}^{2}{x}^{2}}{2}}+{\frac{{b}^{3}{x}^{3}}{3}}+{a}^{3}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.04152, size = 42, normalized size = 1.2 \begin{align*} \frac{1}{3} \, b^{3} x^{3} + \frac{3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.50323, size = 73, normalized size = 2.09 \begin{align*} \frac{1}{3} \, b^{3} x^{3} + \frac{3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.310695, size = 34, normalized size = 0.97 \begin{align*} a^{3} \log{\left (x \right )} + 3 a^{2} b x + \frac{3 a b^{2} x^{2}}{2} + \frac{b^{3} x^{3}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.16073, size = 43, normalized size = 1.23 \begin{align*} \frac{1}{3} \, b^{3} x^{3} + \frac{3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3} \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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