Optimal. Leaf size=425 \[ -\frac {b f p q}{2 (f i-e j) (h i-g j) (i+j x)}-\frac {b f h p q \log (e+f x)}{(f i-e j) (h i-g j)^2}-\frac {b f^2 p q \log (e+f x)}{2 (f i-e j)^2 (h i-g j)}+\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{2 (h i-g j) (i+j x)^2}+\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^2 (i+j x)}+\frac {h^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{(h i-g j)^3}+\frac {b f h p q \log (i+j x)}{(f i-e j) (h i-g j)^2}+\frac {b f^2 p q \log (i+j x)}{2 (f i-e j)^2 (h i-g j)}-\frac {h^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (i+j x)}{f i-e j}\right )}{(h i-g j)^3}+\frac {b h^2 p q \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{(h i-g j)^3}-\frac {b h^2 p q \text {Li}_2\left (-\frac {j (e+f x)}{f i-e j}\right )}{(h i-g j)^3} \]
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Rubi [A]
time = 0.56, antiderivative size = 425, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 9, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2465, 2441,
2440, 2438, 2442, 46, 36, 31, 2495} \begin {gather*} \frac {b h^2 p q \text {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{(h i-g j)^3}-\frac {b h^2 p q \text {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right )}{(h i-g j)^3}+\frac {h^2 \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)^3}-\frac {h^2 \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)^3}+\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(i+j x) (h i-g j)^2}+\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{2 (i+j x)^2 (h i-g j)}-\frac {b f^2 p q \log (e+f x)}{2 (f i-e j)^2 (h i-g j)}+\frac {b f^2 p q \log (i+j x)}{2 (f i-e j)^2 (h i-g j)}-\frac {b f p q}{2 (i+j x) (f i-e j) (h i-g j)}-\frac {b f h p q \log (e+f x)}{(f i-e j) (h i-g j)^2}+\frac {b f h p q \log (i+j x)}{(f i-e j) (h i-g j)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 46
Rule 2438
Rule 2440
Rule 2441
Rule 2442
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) (529+j x)^3} \, dx &=\text {Subst}\left (\int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(g+h x) (529+j x)^3} \, 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^3 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(529 h-g j)^3 (g+h x)}-\frac {j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(529 h-g j) (529+j x)^3}-\frac {h j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(529 h-g j)^2 (529+j x)^2}-\frac {h^2 j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{(529 h-g j)^3 (529+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^3 \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{g+h x} \, dx}{(529 h-g j)^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (h^2 j\right ) \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{529+j x} \, dx}{(529 h-g j)^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(h j) \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(529+j x)^2} \, dx}{(529 h-g j)^2},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 )}{(529+j x)^3} \, dx}{529 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{2 (529 h-g j) (529+j x)^2}+\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(529 h-g j)^2 (529+j x)}+\frac {h^2 \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 )}{(529 h-g j)^3}-\frac {h^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (529+j x)}{529 f-e j}\right )}{(529 h-g j)^3}-\text {Subst}\left (\frac {\left (b f h^2 p q\right ) \int \frac {\log \left (\frac {f (g+h x)}{f g-e h}\right )}{e+f x} \, dx}{(529 h-g j)^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (b f h^2 p q\right ) \int \frac {\log \left (\frac {f (529+j x)}{529 f-e j}\right )}{e+f x} \, dx}{(529 h-g j)^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(b f h p q) \int \frac {1}{(e+f x) (529+j x)} \, dx}{(529 h-g j)^2},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 {1}{(e+f x) (529+j x)^2} \, dx}{2 (529 h-g j)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{2 (529 h-g j) (529+j x)^2}+\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(529 h-g j)^2 (529+j x)}+\frac {h^2 \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 )}{(529 h-g j)^3}-\frac {h^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (529+j x)}{529 f-e j}\right )}{(529 h-g j)^3}-\text {Subst}\left (\frac {\left (b h^2 p q\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{(529 h-g j)^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (b h^2 p q\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {j x}{529 f-e j}\right )}{x} \, dx,x,e+f x\right )}{(529 h-g j)^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (b f^2 h p q\right ) \int \frac {1}{e+f x} \, dx}{(529 f-e j) (529 h-g j)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(b f h j p q) \int \frac {1}{529+j x} \, dx}{(529 f-e j) (529 h-g j)^2},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 \left (\frac {f^2}{(529 f-e j)^2 (e+f x)}-\frac {j}{(529 f-e j) (529+j x)^2}-\frac {f j}{(529 f-e j)^2 (529+j x)}\right ) \, dx}{2 (529 h-g j)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {b f p q}{2 (529 f-e j) (529 h-g j) (529+j x)}-\frac {b f h p q \log (e+f x)}{(529 f-e j) (529 h-g j)^2}-\frac {b f^2 p q \log (e+f x)}{2 (529 f-e j)^2 (529 h-g j)}+\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{2 (529 h-g j) (529+j x)^2}+\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(529 h-g j)^2 (529+j x)}+\frac {h^2 \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 )}{(529 h-g j)^3}+\frac {b f h p q \log (529+j x)}{(529 f-e j) (529 h-g j)^2}+\frac {b f^2 p q \log (529+j x)}{2 (529 f-e j)^2 (529 h-g j)}-\frac {h^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (529+j x)}{529 f-e j}\right )}{(529 h-g j)^3}+\frac {b h^2 p q \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{(529 h-g j)^3}-\frac {b h^2 p q \text {Li}_2\left (-\frac {j (e+f x)}{529 f-e j}\right )}{(529 h-g j)^3}\\ \end {align*}
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Mathematica [A]
time = 0.53, size = 434, normalized size = 1.02 \begin {gather*} \frac {\frac {(h i-g j)^2 \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(i+j x)^2}+\frac {2 h (h i-g j) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{i+j x}+2 h^2 \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log (g+h x)-2 h^2 \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log (i+j x)+b p q \left (\frac {2 h (h i-g j) (-j (e+f x) \log (e+f x)+f (i+j x) \log (i+j x))}{(f i-e j) (i+j x)}+\frac {(h i-g j)^2 (j (e+f x) (e j-f (2 i+j x)) \log (e+f x)+f (i+j x) (-f i+e j+f (i+j x) \log (i+j x)))}{(f i-e j)^2 (i+j x)^2}+2 h^2 \left (\log (e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+\text {Li}_2\left (\frac {h (e+f x)}{-f g+e h}\right )\right )-2 h^2 \left (\log (e+f x) \log \left (\frac {f (i+j x)}{f i-e j}\right )+\text {Li}_2\left (\frac {j (e+f x)}{-f i+e j}\right )\right )\right )}{2 (h i-g j)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.35, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{\left (h x +g \right ) \left (j x +i \right )^{3}}\, dx\]
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 )^{3}}\, 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 = -0.00 \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 )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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