Optimal. Leaf size=24 \[ -\frac {a^2}{2 x^2}-\frac {2 a b}{x}+b^2 \log (x) \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 24, 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} \begin {gather*} -\frac {a^2}{2 x^2}-\frac {2 a b}{x}+b^2 \log (x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x)^2}{x^3} \, dx &=\int \left (\frac {a^2}{x^3}+\frac {2 a b}{x^2}+\frac {b^2}{x}\right ) \, dx\\ &=-\frac {a^2}{2 x^2}-\frac {2 a b}{x}+b^2 \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 24, normalized size = 1.00 \begin {gather*} -\frac {a^2}{2 x^2}-\frac {2 a b}{x}+b^2 \log (x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^2}{x^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.91, size = 26, normalized size = 1.08 \begin {gather*} \frac {2 \, b^{2} x^{2} \log \relax (x) - 4 \, a b x - a^{2}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.18, size = 22, normalized size = 0.92 \begin {gather*} b^{2} \log \left ({\left | x \right |}\right ) - \frac {4 \, a b x + a^{2}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 23, normalized size = 0.96 \begin {gather*} b^{2} \ln \relax (x )-\frac {2 a b}{x}-\frac {a^{2}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.18, size = 21, normalized size = 0.88 \begin {gather*} b^{2} \log \relax (x) - \frac {4 \, a b x + a^{2}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 23, normalized size = 0.96 \begin {gather*} b^2\,\ln \relax (x)-\frac {\frac {a^2}{2}+2\,b\,x\,a}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.17, size = 22, normalized size = 0.92 \begin {gather*} b^{2} \log {\relax (x )} + \frac {- a^{2} - 4 a b x}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________