Optimal. Leaf size=165 \[ \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h i-g j}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (i+j x)}{f i-e j}\right )}{h i-g j}+\frac {b p q \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h i-g j}-\frac {b p q \text {Li}_2\left (-\frac {j (e+f x)}{f i-e j}\right )}{h i-g j} \]
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Rubi [A]
time = 0.31, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2465, 2441,
2440, 2438, 2495} \begin {gather*} \frac {b p q \text {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h i-g j}-\frac {b p q \text {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right )}{h i-g j}+\frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j}-\frac {\log \left (\frac {f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j} \end {gather*}
Antiderivative was successfully verified.
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Rule 2438
Rule 2440
Rule 2441
Rule 2465
Rule 2495
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (527+j x)} \, dx &=\text {Subst}\left (\int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(g+h x) (527+j x)} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\text {Subst}\left (\int \left (\frac {h \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(527 h-g j) (g+h x)}-\frac {j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(527 h-g j) (527+j x)}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\text {Subst}\left (\frac {h \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{g+h x} \, dx}{527 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {j \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{527+j x} \, dx}{527 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{527 h-g j}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (527+j x)}{527 f-e j}\right )}{527 h-g j}-\text {Subst}\left (\frac {(b f p q) \int \frac {\log \left (\frac {f (g+h x)}{f g-e h}\right )}{e+f x} \, dx}{527 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(b f p q) \int \frac {\log \left (\frac {f (527+j x)}{527 f-e j}\right )}{e+f x} \, dx}{527 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{527 h-g j}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (527+j x)}{527 f-e j}\right )}{527 h-g j}-\text {Subst}\left (\frac {(b p q) \text {Subst}\left (\int \frac {\log \left (1+\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{527 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(b p q) \text {Subst}\left (\int \frac {\log \left (1+\frac {j x}{527 f-e j}\right )}{x} \, dx,x,e+f x\right )}{527 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{527 h-g j}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (527+j x)}{527 f-e j}\right )}{527 h-g j}+\frac {b p q \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{527 h-g j}-\frac {b p q \text {Li}_2\left (-\frac {j (e+f x)}{527 f-e j}\right )}{527 h-g j}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 117, normalized size = 0.71 \begin {gather*} \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \left (\log \left (\frac {f (g+h x)}{f g-e h}\right )-\log \left (\frac {f (i+j x)}{f i-e j}\right )\right )+b p q \text {Li}_2\left (\frac {h (e+f x)}{-f g+e h}\right )-b p q \text {Li}_2\left (\frac {j (e+f x)}{-f i+e j}\right )}{h i-g j} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.40, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{\left (h x +g \right ) \left (j x +i \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{\left (g + h x\right ) \left (i + j x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{\left (g+h\,x\right )\,\left (i+j\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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