Optimal. Leaf size=49 \[ -\frac{8 a^3 \log (a-b x)}{b}-4 a^2 x-\frac{a (a+b x)^2}{b}-\frac{(a+b x)^3}{3 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0205057, antiderivative size = 49, normalized size of antiderivative = 1., 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} \[ -\frac{8 a^3 \log (a-b x)}{b}-4 a^2 x-\frac{a (a+b x)^2}{b}-\frac{(a+b x)^3}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 627
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^4}{a^2-b^2 x^2} \, dx &=\int \frac{(a+b x)^3}{a-b x} \, dx\\ &=\int \left (-4 a^2+\frac{8 a^3}{a-b x}-2 a (a+b x)-(a+b x)^2\right ) \, dx\\ &=-4 a^2 x-\frac{a (a+b x)^2}{b}-\frac{(a+b x)^3}{3 b}-\frac{8 a^3 \log (a-b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0054987, size = 39, normalized size = 0.8 \[ -\frac{8 a^3 \log (a-b x)}{b}-7 a^2 x-2 a b x^2-\frac{b^2 x^3}{3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.041, size = 39, normalized size = 0.8 \begin{align*} -{\frac{{b}^{2}{x}^{3}}{3}}-2\,ab{x}^{2}-7\,{a}^{2}x-8\,{\frac{{a}^{3}\ln \left ( bx-a \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.23004, size = 51, normalized size = 1.04 \begin{align*} -\frac{1}{3} \, b^{2} x^{3} - 2 \, a b x^{2} - 7 \, a^{2} x - \frac{8 \, a^{3} \log \left (b x - a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.71158, size = 90, normalized size = 1.84 \begin{align*} -\frac{b^{3} x^{3} + 6 \, a b^{2} x^{2} + 21 \, a^{2} b x + 24 \, a^{3} \log \left (b x - a\right )}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.315249, size = 37, normalized size = 0.76 \begin{align*} - \frac{8 a^{3} \log{\left (- a + b x \right )}}{b} - 7 a^{2} x - 2 a b x^{2} - \frac{b^{2} x^{3}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.22229, size = 66, normalized size = 1.35 \begin{align*} -\frac{8 \, a^{3} \log \left ({\left | b x - a \right |}\right )}{b} - \frac{b^{5} x^{3} + 6 \, a b^{4} x^{2} + 21 \, a^{2} b^{3} x}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]