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 = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, 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 8
Rule 31
Rule 2487
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*}
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Mathematica [A] time = 0.07, 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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fricas [A] time = 0.41, size = 72, normalized size = 1.18 \[ \frac {b d r x \log \relax (f) + b d x \log \relax (e) - {\left (b d p + b d q\right )} r x + {\left (b d p r x + a d p r\right )} \log \left (b x + a\right ) + {\left (b d q r x + b c q r\right )} \log \left (d x + c\right )}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 66, normalized size = 1.08 \[ p r x \log \left (b x + a\right ) + q r x \log \left (d x + c\right ) + \frac {a p r \log \left (b x + a\right )}{b} + \frac {c q r \log \left (-d x - c\right )}{d} - {\left (p r + q r - r \log \relax (f) - 1\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 61, normalized size = 1.00 \[ \frac {a p r \ln \left (b x +a \right )}{b}+\frac {c q r \ln \left (d x +c \right )}{d}-p r x -q r x +x \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 75, normalized size = 1.23 \[ x \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) - \frac {{\left (b f p {\left (\frac {x}{b} - \frac {a \log \left (b x + a\right )}{b^{2}}\right )} + d f q {\left (\frac {x}{d} - \frac {c \log \left (d x + c\right )}{d^{2}}\right )}\right )} r}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 60, normalized size = 0.98 \[ x\,\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )-p\,r\,x-q\,r\,x+\frac {a\,p\,r\,\ln \left (a+b\,x\right )}{b}+\frac {c\,q\,r\,\ln \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.64, size = 187, normalized size = 3.07 \[ \begin {cases} x \log {\left (e \left (a^{p} c^{q} f\right )^{r} \right )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {c q r \log {\left (c + d x \right )}}{d} + p r x \log {\relax (a )} + q r x \log {\left (c + d x \right )} - q r x + r x \log {\relax (f )} + x \log {\relax (e )} & \text {for}\: b = 0 \\\frac {a p r \log {\left (a + b x \right )}}{b} + p r x \log {\left (a + b x \right )} - p r x + q r x \log {\relax (c )} + r x \log {\relax (f )} + x \log {\relax (e )} & \text {for}\: d = 0 \\\frac {a p r \log {\left (a + b x \right )}}{b} + \frac {c q r \log {\left (c + d x \right )}}{d} + p r x \log {\left (a + b x \right )} - p r x + q r x \log {\left (c + d x \right )} - q r x + r x \log {\relax (f )} + x \log {\relax (e )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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