Optimal. Leaf size=28 \[ -\frac {4 a^2 \log (a-b x)}{b}-3 a x-\frac {b x^2}{2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {627, 43} \[ -\frac {4 a^2 \log (a-b x)}{b}-3 a x-\frac {b x^2}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 627
Rubi steps
\begin {align*} \int \frac {(a+b x)^3}{a^2-b^2 x^2} \, dx &=\int \frac {(a+b x)^2}{a-b x} \, dx\\ &=\int \left (-3 a-b x+\frac {4 a^2}{a-b x}\right ) \, dx\\ &=-3 a x-\frac {b x^2}{2}-\frac {4 a^2 \log (a-b x)}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 28, normalized size = 1.00 \[ -\frac {4 a^2 \log (a-b x)}{b}-3 a x-\frac {b x^2}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.84, size = 31, normalized size = 1.11 \[ -\frac {b^{2} x^{2} + 6 \, a b x + 8 \, a^{2} \log \left (b x - a\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 38, normalized size = 1.36 \[ -\frac {4 \, a^{2} \log \left ({\left | b x - a \right |}\right )}{b} - \frac {b^{3} x^{2} + 6 \, a b^{2} x}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 28, normalized size = 1.00 \[ -\frac {b \,x^{2}}{2}-\frac {4 a^{2} \ln \left (b x -a \right )}{b}-3 a x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.37, size = 27, normalized size = 0.96 \[ -\frac {1}{2} \, b x^{2} - 3 \, a x - \frac {4 \, a^{2} \log \left (b x - a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.41, size = 27, normalized size = 0.96 \[ -3\,a\,x-\frac {b\,x^2}{2}-\frac {4\,a^2\,\ln \left (b\,x-a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.15, size = 26, normalized size = 0.93 \[ - \frac {4 a^{2} \log {\left (- a + b x \right )}}{b} - 3 a x - \frac {b x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________