Optimal. Leaf size=463 \[ \frac {2 b h p q \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^2}-\frac {2 b h p q \text {Li}_2\left (-\frac {j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^2}+\frac {2 b f p q \log \left (\frac {f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(f i-e j) (h i-g j)}-\frac {j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(i+j x) (f i-e j) (h i-g j)}+\frac {h \log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(h i-g j)^2}-\frac {h \log \left (\frac {f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(h i-g j)^2}+\frac {2 b^2 f p^2 q^2 \text {Li}_2\left (-\frac {j (e+f x)}{f i-e j}\right )}{(f i-e j) (h i-g j)}-\frac {2 b^2 h p^2 q^2 \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{(h i-g j)^2}+\frac {2 b^2 h p^2 q^2 \text {Li}_3\left (-\frac {j (e+f x)}{f i-e j}\right )}{(h i-g j)^2} \]
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Rubi [A] time = 1.18, 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 = {2418, 2396, 2433, 2374, 6589, 2397, 2394, 2393, 2391, 2445} \[ \frac {2 b h p q \text {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^2}-\frac {2 b h p q \text {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^2}+\frac {2 b^2 f p^2 q^2 \text {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right )}{(f i-e j) (h i-g j)}-\frac {2 b^2 h p^2 q^2 \text {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right )}{(h i-g j)^2}+\frac {2 b^2 h p^2 q^2 \text {PolyLog}\left (3,-\frac {j (e+f x)}{f i-e j}\right )}{(h i-g j)^2}+\frac {2 b f p q \log \left (\frac {f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(f i-e j) (h i-g j)}-\frac {j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(i+j x) (f i-e j) (h i-g j)}+\frac {h \log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(h i-g j)^2}-\frac {h \log \left (\frac {f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(h i-g j)^2} \]
Antiderivative was successfully verified.
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Rule 2374
Rule 2391
Rule 2393
Rule 2394
Rule 2396
Rule 2397
Rule 2418
Rule 2433
Rule 2445
Rule 6589
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 &=\operatorname {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 )\\ &=\operatorname {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 )\\ &=\operatorname {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 )-\operatorname {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 )-\operatorname {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}-\operatorname {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 )+\operatorname {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 )+\operatorname {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}-\operatorname {Subst}\left (\frac {(2 b h p q) \operatorname {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 )+\operatorname {Subst}\left (\frac {(2 b h p q) \operatorname {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 )-\operatorname {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}-\operatorname {Subst}\left (\frac {\left (2 b^2 h p^2 q^2\right ) \operatorname {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 )+\operatorname {Subst}\left (\frac {\left (2 b^2 h p^2 q^2\right ) \operatorname {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 )-\operatorname {Subst}\left (\frac {\left (2 b^2 f p^2 q^2\right ) \operatorname {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 = 0.96, size = 654, normalized size = 1.41 \[ \frac {-2 b p q \left (-h (i+j x) (f i-e j) \left (\text {Li}_2\left (\frac {h (e+f x)}{e h-f g}\right )+\log (e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )\right )+(h i-g j) (j (e+f x) \log (e+f x)-f (i+j x) \log (i+j x))+h (i+j x) (f i-e j) \left (\text {Li}_2\left (\frac {j (e+f x)}{e j-f i}\right )+\log (e+f x) \log \left (\frac {f (i+j x)}{f i-e j}\right )\right )\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-b p q \log (e+f x)\right )+(f i-e j) (h i-g j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-b p q \log (e+f x)\right )^2+h (i+j x) (f i-e j) \log (g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-b p q \log (e+f x)\right )^2-h (i+j x) (f i-e j) \log (i+j x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-b p q \log (e+f x)\right )^2-b^2 p^2 q^2 \left (-h (i+j x) (f i-e j) \left (-2 \text {Li}_3\left (\frac {h (e+f x)}{e h-f g}\right )+2 \log (e+f x) \text {Li}_2\left (\frac {h (e+f x)}{e h-f g}\right )+\log ^2(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )\right )+(h i-g j) \left (\log (e+f x) \left (j (e+f x) \log (e+f x)-2 f (i+j x) \log \left (\frac {f (i+j x)}{f i-e j}\right )\right )-2 f (i+j x) \text {Li}_2\left (\frac {j (e+f x)}{e j-f i}\right )\right )+h (i+j x) (f i-e j) \left (-2 \text {Li}_3\left (\frac {j (e+f x)}{e j-f i}\right )+2 \log (e+f x) \text {Li}_2\left (\frac {j (e+f x)}{e j-f i}\right )+\log ^2(e+f x) \log \left (\frac {f (i+j x)}{f i-e j}\right )\right )\right )}{(i+j x) (f i-e j) (h i-g j)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + 2 \, a b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a^{2}}{h j^{2} x^{3} + g i^{2} + {\left (2 \, h i j + g j^{2}\right )} x^{2} + {\left (h i^{2} + 2 \, g i j\right )} x}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{{\left (h x + g\right )} {\left (j x + i\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.53, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a \right )^{2}}{\left (h x +g \right ) \left (j x +i \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{2} {\left (\frac {h \log \left (h x + g\right )}{h^{2} i^{2} - 2 \, g h i j + g^{2} j^{2}} - \frac {h \log \left (j x + i\right )}{h^{2} i^{2} - 2 \, g h i j + g^{2} j^{2}} + \frac {1}{h i^{2} - g i j + {\left (h i j - g j^{2}\right )} x}\right )} + \int \frac {b^{2} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )^{2} + 2 \, {\left (q \log \relax (d) + \log \relax (c)\right )} a b + {\left (q^{2} \log \relax (d)^{2} + 2 \, q \log \relax (c) \log \relax (d) + \log \relax (c)^{2}\right )} b^{2} + 2 \, {\left ({\left (q \log \relax (d) + \log \relax (c)\right )} b^{2} + a b\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )}{h j^{2} x^{3} + g i^{2} + {\left (2 \, h i j + g j^{2}\right )} x^{2} + {\left (h i^{2} + 2 \, g i j\right )} x}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^2}{\left (g+h\,x\right )\,{\left (i+j\,x\right )}^2} \,d x \]
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 \[ \int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{2}}{\left (g + h x\right ) \left (i + j x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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