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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, 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 = 0.00, size = 34, normalized size = 1.00 \[ -\frac {a^3}{x}+3 a^2 b \log (x)+3 a b^2 x+\frac {b^3 x^2}{2} \]

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]

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 36, normalized size = 1.06 \[ \frac {b^{3} x^{3} + 6 \, a b^{2} x^{2} + 6 \, a^{2} b x \log \relax (x) - 2 \, a^{3}}{2 \, x} \]

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

________________________________________________________________________________________

giac [A]  time = 1.36, size = 33, normalized size = 0.97 \[ \frac {1}{2} \, b^{3} x^{2} + 3 \, a b^{2} x + 3 \, a^{2} b \log \left ({\left | x \right |}\right ) - \frac {a^{3}}{x} \]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 33, normalized size = 0.97 \[ \frac {b^{3} x^{2}}{2}+3 a^{2} b \ln \relax (x )+3 a \,b^{2} x -\frac {a^{3}}{x} \]

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)

________________________________________________________________________________________

maxima [A]  time = 1.29, size = 32, normalized size = 0.94 \[ \frac {1}{2} \, b^{3} x^{2} + 3 \, a b^{2} x + 3 \, a^{2} b \log \relax (x) - \frac {a^{3}}{x} \]

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

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 32, normalized size = 0.94 \[ \frac {b^3\,x^2}{2}-\frac {a^3}{x}+3\,a^2\,b\,\ln \relax (x)+3\,a\,b^2\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^3/x^2,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.13, size = 31, normalized size = 0.91 \[ - \frac {a^{3}}{x} + 3 a^{2} b \log {\relax (x )} + 3 a b^{2} x + \frac {b^{3} x^{2}}{2} \]

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

________________________________________________________________________________________