Optimal. Leaf size=24 \[ \frac {(a+b x) \log \left (c (a+b x)^p\right )}{b}-p x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2389, 2295} \[ \frac {(a+b x) \log \left (c (a+b x)^p\right )}{b}-p x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2295
Rule 2389
Rubi steps
\begin {align*} \int \log \left (c (a+b x)^p\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+b x\right )}{b}\\ &=-p x+\frac {(a+b x) \log \left (c (a+b x)^p\right )}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 24, normalized size = 1.00 \[ \frac {(a+b x) \log \left (c (a+b x)^p\right )}{b}-p x \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 32, normalized size = 1.33 \[ -\frac {b p x - b x \log \relax (c) - {\left (b p x + a p\right )} \log \left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 39, normalized size = 1.62 \[ \frac {{\left (b x + a\right )} p \log \left (b x + a\right )}{b} - \frac {{\left (b x + a\right )} p}{b} + \frac {{\left (b x + a\right )} \log \relax (c)}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 30, normalized size = 1.25 \[ \frac {a p \ln \left (b x +a \right )}{b}-p x +x \ln \left (c \left (b x +a \right )^{p}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.43, size = 35, normalized size = 1.46 \[ -b p {\left (\frac {x}{b} - \frac {a \log \left (b x + a\right )}{b^{2}}\right )} + x \log \left ({\left (b x + a\right )}^{p} c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.07, size = 29, normalized size = 1.21 \[ x\,\ln \left (c\,{\left (a+b\,x\right )}^p\right )-p\,x+\frac {a\,p\,\ln \left (a+b\,x\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.46, size = 37, normalized size = 1.54 \[ \begin {cases} \frac {a p \log {\left (a + b x \right )}}{b} + p x \log {\left (a + b x \right )} - p x + x \log {\relax (c )} & \text {for}\: b \neq 0 \\x \log {\left (a^{p} c \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________