Optimal. Leaf size=34 \[ a x-b p q x+\frac {b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2436, 2332,
2495} \begin {gather*} a x+\frac {b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}-b p q x \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2332
Rule 2436
Rule 2495
Rubi steps
\begin {align*} \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx &=a x+b \int \log \left (c \left (d (e+f x)^p\right )^q\right ) \, dx\\ &=a x+b \text {Subst}\left (\int \log \left (c d^q (e+f x)^{p q}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=a x+b \text {Subst}\left (\frac {\text {Subst}\left (\int \log \left (c d^q x^{p q}\right ) \, dx,x,e+f x\right )}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=a x-b p q x+\frac {b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 34, normalized size = 1.00 \begin {gather*} a x-b p q x+\frac {b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 42, normalized size = 1.24
method | result | size |
default | \(a x +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) x -b p q x +\frac {b q p e \ln \left (f x +e \right )}{f}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 48, normalized size = 1.41 \begin {gather*} -b f p q {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} + b x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 52, normalized size = 1.53 \begin {gather*} \frac {b f q x \log \left (d\right ) + b f x \log \left (c\right ) - {\left (b f p q - a f\right )} x + {\left (b f p q x + b p q e\right )} \log \left (f x + e\right )}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.21, size = 53, normalized size = 1.56 \begin {gather*} a x + b \left (\begin {cases} \frac {e \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} - p q x + x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} & \text {for}\: f \neq 0 \\x \log {\left (c \left (d e^{p}\right )^{q} \right )} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 4.99, size = 64, normalized size = 1.88 \begin {gather*} {\left (\frac {{\left (f x + e\right )} p q \log \left (f x + e\right )}{f} - \frac {{\left (f x + e\right )} p q}{f} + \frac {{\left (f x + e\right )} q \log \left (d\right )}{f} + \frac {{\left (f x + e\right )} \log \left (c\right )}{f}\right )} b + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.22, size = 41, normalized size = 1.21 \begin {gather*} x\,\left (a-b\,p\,q\right )+b\,x\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )+\frac {b\,e\,p\,q\,\ln \left (e+f\,x\right )}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________