Optimal. Leaf size=61 \[ \frac{8 a^3 c^3 \log (a+b x)}{b}-4 a^2 c^3 x+\frac{c^3 (a-b x)^3}{3 b}+\frac{a c^3 (a-b x)^2}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0212865, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {43} \[ \frac{8 a^3 c^3 \log (a+b x)}{b}-4 a^2 c^3 x+\frac{c^3 (a-b x)^3}{3 b}+\frac{a c^3 (a-b x)^2}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rubi steps
\begin{align*} \int \frac{(a c-b c x)^3}{a+b x} \, dx &=\int \left (-4 a^2 c^3+\frac{8 a^3 c^3}{a+b x}-2 a c^2 (a c-b c x)-c (a c-b c x)^2\right ) \, dx\\ &=-4 a^2 c^3 x+\frac{a c^3 (a-b x)^2}{b}+\frac{c^3 (a-b x)^3}{3 b}+\frac{8 a^3 c^3 \log (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0056005, size = 42, normalized size = 0.69 \[ c^3 \left (\frac{8 a^3 \log (a+b x)}{b}-7 a^2 x+2 a b x^2-\frac{b^2 x^3}{3}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 49, normalized size = 0.8 \begin{align*} -{\frac{{c}^{3}{x}^{3}{b}^{2}}{3}}+2\,{c}^{3}b{x}^{2}a-7\,{a}^{2}{c}^{3}x+8\,{\frac{{a}^{3}{c}^{3}\ln \left ( bx+a \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.05212, size = 65, normalized size = 1.07 \begin{align*} -\frac{1}{3} \, b^{2} c^{3} x^{3} + 2 \, a b c^{3} x^{2} - 7 \, a^{2} c^{3} x + \frac{8 \, a^{3} c^{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 = 1.56054, size = 112, normalized size = 1.84 \begin{align*} -\frac{b^{3} c^{3} x^{3} - 6 \, a b^{2} c^{3} x^{2} + 21 \, a^{2} b c^{3} x - 24 \, a^{3} c^{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 = 0.317511, size = 49, normalized size = 0.8 \begin{align*} \frac{8 a^{3} c^{3} \log{\left (a + b x \right )}}{b} - 7 a^{2} c^{3} x + 2 a b c^{3} x^{2} - \frac{b^{2} c^{3} x^{3}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.04772, size = 80, normalized size = 1.31 \begin{align*} \frac{8 \, a^{3} c^{3} \log \left ({\left | b x + a \right |}\right )}{b} - \frac{b^{5} c^{3} x^{3} - 6 \, a b^{4} c^{3} x^{2} + 21 \, a^{2} b^{3} c^{3} x}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]