Optimal. Leaf size=43 \[ \frac{4 a^2 c^2 \log (a+b x)}{b}+\frac{c^2 (a-b x)^2}{2 b}-2 a c^2 x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0143916, antiderivative size = 43, 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} \[ \frac{4 a^2 c^2 \log (a+b x)}{b}+\frac{c^2 (a-b x)^2}{2 b}-2 a c^2 x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rubi steps
\begin{align*} \int \frac{(a c-b c x)^2}{a+b x} \, dx &=\int \left (-2 a c^2+\frac{4 a^2 c^2}{a+b x}-c (a c-b c x)\right ) \, dx\\ &=-2 a c^2 x+\frac{c^2 (a-b x)^2}{2 b}+\frac{4 a^2 c^2 \log (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0061941, size = 31, normalized size = 0.72 \[ c^2 \left (\frac{4 a^2 \log (a+b x)}{b}-3 a x+\frac{b x^2}{2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.001, size = 35, normalized size = 0.8 \begin{align*}{\frac{{c}^{2}b{x}^{2}}{2}}-3\,a{c}^{2}x+4\,{\frac{{a}^{2}{c}^{2}\ln \left ( bx+a \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.02392, size = 46, normalized size = 1.07 \begin{align*} \frac{1}{2} \, b c^{2} x^{2} - 3 \, a c^{2} x + \frac{4 \, a^{2} c^{2} \log \left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.4938, size = 81, normalized size = 1.88 \begin{align*} \frac{b^{2} c^{2} x^{2} - 6 \, a b c^{2} x + 8 \, a^{2} c^{2} \log \left (b x + a\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.296108, size = 34, normalized size = 0.79 \begin{align*} \frac{4 a^{2} c^{2} \log{\left (a + b x \right )}}{b} - 3 a c^{2} x + \frac{b c^{2} x^{2}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.0569, size = 61, normalized size = 1.42 \begin{align*} \frac{4 \, a^{2} c^{2} \log \left ({\left | b x + a \right |}\right )}{b} + \frac{b^{3} c^{2} x^{2} - 6 \, a b^{2} c^{2} x}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]