Optimal. Leaf size=61 \[ \frac{(a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}+\frac{q r (b c-a d) \log (c+d x)}{b d}+r x (-(p+q)) \]
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Rubi [A] time = 0.0148794, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2487, 31, 8} \[ \frac{(a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}+\frac{q r (b c-a d) \log (c+d x)}{b d}+r x (-(p+q)) \]
Antiderivative was successfully verified.
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Rule 2487
Rule 31
Rule 8
Rubi steps
\begin{align*} \int \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx &=\frac{(a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}+\frac{((b c-a d) q r) \int \frac{1}{c+d x} \, dx}{b}-((p+q) r) \int 1 \, dx\\ &=-(p+q) r x+\frac{(b c-a d) q r \log (c+d x)}{b d}+\frac{(a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0628468, size = 57, normalized size = 0.93 \[ x \left (\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-r (p+q)\right )+\frac{a p r \log (a+b x)}{b}+\frac{c q r \log (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.071, size = 61, normalized size = 1. \begin{align*} \ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) x-rpx-rqx+{\frac{rap\ln \left ( bx+a \right ) }{b}}+{\frac{rqc\ln \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15706, size = 101, normalized size = 1.66 \begin{align*} x \log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right ) - \frac{{\left (b f p{\left (\frac{x}{b} - \frac{a \log \left (b x + a\right )}{b^{2}}\right )} + d f q{\left (\frac{x}{d} - \frac{c \log \left (d x + c\right )}{d^{2}}\right )}\right )} r}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.08855, size = 182, normalized size = 2.98 \begin{align*} \frac{b d r x \log \left (f\right ) + b d x \log \left (e\right ) -{\left (b d p + b d q\right )} r x +{\left (b d p r x + a d p r\right )} \log \left (b x + a\right ) +{\left (b d q r x + b c q r\right )} \log \left (d x + c\right )}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23847, size = 242, normalized size = 3.97 \begin{align*} p r x \log \left (b x + a\right ) + q r x \log \left (d x + c\right ) -{\left (p r + q r - r \log \left (f\right ) - 1\right )} x + \frac{{\left (a d p r + b c q r\right )} \log \left ({\left | b d x^{2} + b c x + a d x + a c \right |}\right )}{2 \, b d} + \frac{{\left (a b c d p r - a^{2} d^{2} p r - b^{2} c^{2} q r + a b c d q r\right )} \log \left ({\left | \frac{2 \, b d x + b c + a d -{\left | -b c + a d \right |}}{2 \, b d x + b c + a d +{\left | -b c + a d \right |}} \right |}\right )}{2 \, b d{\left | -b c + a d \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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