Optimal. Leaf size=463 \[ -\frac {j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(f i-e j) (h i-g j) (i+j x)}+\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{(h i-g j)^2}+\frac {2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (i+j x)}{f i-e j}\right )}{(f i-e j) (h i-g j)}-\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (i+j x)}{f i-e j}\right )}{(h i-g j)^2}+\frac {2 b h p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{(h i-g j)^2}+\frac {2 b^2 f p^2 q^2 \text {Li}_2\left (-\frac {j (e+f x)}{f i-e j}\right )}{(f i-e j) (h i-g j)}-\frac {2 b h p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {j (e+f x)}{f i-e j}\right )}{(h i-g j)^2}-\frac {2 b^2 h p^2 q^2 \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{(h i-g j)^2}+\frac {2 b^2 h p^2 q^2 \text {Li}_3\left (-\frac {j (e+f x)}{f i-e j}\right )}{(h i-g j)^2} \]
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Rubi [A]
time = 0.81, antiderivative size = 463, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 10, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2465, 2443,
2481, 2421, 6724, 2444, 2441, 2440, 2438, 2495} \begin {gather*} \frac {2 b h p q \text {PolyLog}\left (2,-\frac {h (e+f 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)^2}-\frac {2 b h p q \text {PolyLog}\left (2,-\frac {j (e+f 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)^2}+\frac {2 b^2 f p^2 q^2 \text {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right )}{(f i-e j) (h i-g j)}-\frac {2 b^2 h p^2 q^2 \text {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right )}{(h i-g j)^2}+\frac {2 b^2 h p^2 q^2 \text {PolyLog}\left (3,-\frac {j (e+f x)}{f i-e j}\right )}{(h i-g j)^2}+\frac {2 b f p q \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 )}{(f i-e j) (h i-g j)}-\frac {j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(i+j x) (f i-e j) (h i-g j)}+\frac {h \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 )^2}{(h i-g j)^2}-\frac {h \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 )^2}{(h i-g j)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2421
Rule 2438
Rule 2440
Rule 2441
Rule 2443
Rule 2444
Rule 2465
Rule 2481
Rule 2495
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x) (534+j x)^2} \, dx &=\text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(g+h x) (534+j x)^2} \, 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^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(534 h-g j)^2 (g+h x)}-\frac {j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(534 h-g j) (534+j x)^2}-\frac {h j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(534 h-g j)^2 (534+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^2 \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{g+h x} \, dx}{(534 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 {(h j) \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{534+j x} \, dx}{(534 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 {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(534+j x)^2} \, dx}{534 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(534 f-e j) (534 h-g j) (534+j x)}+\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{(534 h-g j)^2}-\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (534+j x)}{534 f-e j}\right )}{(534 h-g j)^2}-\text {Subst}\left (\frac {(2 b f h p q) \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{e+f x} \, dx}{(534 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 {(2 b f h p q) \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \log \left (\frac {f (534+j x)}{534 f-e j}\right )}{e+f x} \, dx}{(534 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 {(2 b f j p q) \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{534+j x} \, dx}{(534 f-e j) (534 h-g j)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(534 f-e j) (534 h-g j) (534+j x)}+\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{(534 h-g j)^2}+\frac {2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (534+j x)}{534 f-e j}\right )}{(534 f-e j) (534 h-g j)}-\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (534+j x)}{534 f-e j}\right )}{(534 h-g j)^2}-\text {Subst}\left (\frac {(2 b h p q) \text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right ) \log \left (\frac {f \left (\frac {f g-e h}{f}+\frac {h x}{f}\right )}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{(534 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 {(2 b h p q) \text {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right ) \log \left (\frac {f \left (\frac {534 f-e j}{f}+\frac {j x}{f}\right )}{534 f-e j}\right )}{x} \, dx,x,e+f x\right )}{(534 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 {\left (2 b^2 f^2 p^2 q^2\right ) \int \frac {\log \left (\frac {f (534+j x)}{534 f-e j}\right )}{e+f x} \, dx}{(534 f-e j) (534 h-g j)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(534 f-e j) (534 h-g j) (534+j x)}+\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{(534 h-g j)^2}+\frac {2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (534+j x)}{534 f-e j}\right )}{(534 f-e j) (534 h-g j)}-\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (534+j x)}{534 f-e j}\right )}{(534 h-g j)^2}+\frac {2 b h p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{(534 h-g j)^2}-\frac {2 b h p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {j (e+f x)}{534 f-e j}\right )}{(534 h-g j)^2}-\text {Subst}\left (\frac {\left (2 b^2 h p^2 q^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{(534 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 {\left (2 b^2 h p^2 q^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {j x}{534 f-e j}\right )}{x} \, dx,x,e+f x\right )}{(534 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 {\left (2 b^2 f p^2 q^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {j x}{534 f-e j}\right )}{x} \, dx,x,e+f x\right )}{(534 f-e j) (534 h-g j)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(534 f-e j) (534 h-g j) (534+j x)}+\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{(534 h-g j)^2}+\frac {2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (534+j x)}{534 f-e j}\right )}{(534 f-e j) (534 h-g j)}-\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (534+j x)}{534 f-e j}\right )}{(534 h-g j)^2}+\frac {2 b h p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{(534 h-g j)^2}+\frac {2 b^2 f p^2 q^2 \text {Li}_2\left (-\frac {j (e+f x)}{534 f-e j}\right )}{(534 f-e j) (534 h-g j)}-\frac {2 b h p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_2\left (-\frac {j (e+f x)}{534 f-e j}\right )}{(534 h-g j)^2}-\frac {2 b^2 h p^2 q^2 \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{(534 h-g j)^2}+\frac {2 b^2 h p^2 q^2 \text {Li}_3\left (-\frac {j (e+f x)}{534 f-e j}\right )}{(534 h-g j)^2}\\ \end {align*}
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Mathematica [A]
time = 0.46, size = 654, normalized size = 1.41 \begin {gather*} \frac {(f i-e j) (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 )^2+h (f i-e j) (i+j x) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log (g+h x)-h (f i-e j) (i+j x) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log (i+j x)-2 b p q \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \left ((h i-g j) (j (e+f x) \log (e+f x)-f (i+j x) \log (i+j x))-h (f i-e j) (i+j x) \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 )+h (f i-e j) (i+j x) \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 )-b^2 p^2 q^2 \left ((h i-g j) \left (\log (e+f x) \left (j (e+f x) \log (e+f x)-2 f (i+j x) \log \left (\frac {f (i+j x)}{f i-e j}\right )\right )-2 f (i+j x) \text {Li}_2\left (\frac {j (e+f x)}{-f i+e j}\right )\right )-h (f i-e j) (i+j x) \left (\log ^2(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+2 \log (e+f x) \text {Li}_2\left (\frac {h (e+f x)}{-f g+e h}\right )-2 \text {Li}_3\left (\frac {h (e+f x)}{-f g+e h}\right )\right )+h (f i-e j) (i+j x) \left (\log ^2(e+f x) \log \left (\frac {f (i+j x)}{f i-e j}\right )+2 \log (e+f x) \text {Li}_2\left (\frac {j (e+f x)}{-f i+e j}\right )-2 \text {Li}_3\left (\frac {j (e+f x)}{-f i+e j}\right )\right )\right )}{(f i-e j) (h i-g j)^2 (i+j x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.42, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )^{2}}{\left (h x +g \right ) \left (j x +i \right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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 {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{2}}{\left (g + h x\right ) \left (i + j x\right )^{2}}\, 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 {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^2}{\left (g+h\,x\right )\,{\left (i+j\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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