Optimal. Leaf size=23 \[ \frac{c \sqrt{c x^2} \log (a+b x)}{b x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0038999, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 31} \[ \frac{c \sqrt{c x^2} \log (a+b x)}{b x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 31
Rubi steps
\begin{align*} \int \frac{\left (c x^2\right )^{3/2}}{x^3 (a+b x)} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int \frac{1}{a+b x} \, dx}{x}\\ &=\frac{c \sqrt{c x^2} \log (a+b x)}{b x}\\ \end{align*}
Mathematica [A] time = 0.0032357, size = 22, normalized size = 0.96 \[ \frac{\left (c x^2\right )^{3/2} \log (a+b x)}{b x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 21, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( bx+a \right ) }{b{x}^{3}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.05648, size = 18, normalized size = 0.78 \begin{align*} \frac{c^{\frac{3}{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 = 1.57137, size = 46, normalized size = 2. \begin{align*} \frac{\sqrt{c x^{2}} c \log \left (b x + a\right )}{b x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x^{2}\right )^{\frac{3}{2}}}{x^{3} \left (a + b x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.05631, size = 38, normalized size = 1.65 \begin{align*} c^{\frac{3}{2}}{\left (\frac{\log \left ({\left | b x + a \right |}\right ) \mathrm{sgn}\left (x\right )}{b} - \frac{\log \left ({\left | a \right |}\right ) \mathrm{sgn}\left (x\right )}{b}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]