Optimal. Leaf size=425 \[ \frac {h^2 \log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^3}-\frac {h^2 \log \left (\frac {f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^3}+\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(i+j x) (h i-g j)^2}+\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{2 (i+j x)^2 (h i-g j)}-\frac {b f^2 p q \log (e+f x)}{2 (f i-e j)^2 (h i-g j)}+\frac {b f^2 p q \log (i+j x)}{2 (f i-e j)^2 (h i-g j)}+\frac {b h^2 p q \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{(h i-g j)^3}-\frac {b h^2 p q \text {Li}_2\left (-\frac {j (e+f x)}{f i-e j}\right )}{(h i-g j)^3}-\frac {b f p q}{2 (i+j x) (f i-e j) (h i-g j)}-\frac {b f h p q \log (e+f x)}{(f i-e j) (h i-g j)^2}+\frac {b f h p q \log (i+j x)}{(f i-e j) (h i-g j)^2} \]
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Rubi [A] time = 0.84, 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 = {2418, 2394, 2393, 2391, 2395, 44, 36, 31, 2445} \[ \frac {b h^2 p q \text {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{(h i-g j)^3}-\frac {b h^2 p q \text {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right )}{(h i-g j)^3}+\frac {h^2 \log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^3}-\frac {h^2 \log \left (\frac {f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^3}+\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(i+j x) (h i-g j)^2}+\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{2 (i+j x)^2 (h i-g j)}-\frac {b f^2 p q \log (e+f x)}{2 (f i-e j)^2 (h i-g j)}+\frac {b f^2 p q \log (i+j x)}{2 (f i-e j)^2 (h i-g j)}-\frac {b f p q}{2 (i+j x) (f i-e j) (h i-g j)}-\frac {b f h p q \log (e+f x)}{(f i-e j) (h i-g j)^2}+\frac {b f h p q \log (i+j x)}{(f i-e j) (h i-g j)^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 44
Rule 2391
Rule 2393
Rule 2394
Rule 2395
Rule 2418
Rule 2445
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 &=\operatorname {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 )\\ &=\operatorname {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 )\\ &=\operatorname {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 )-\operatorname {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 )-\operatorname {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 )-\operatorname {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}-\operatorname {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 )+\operatorname {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 )-\operatorname {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 )-\operatorname {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}-\operatorname {Subst}\left (\frac {\left (b h^2 p q\right ) \operatorname {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 )+\operatorname {Subst}\left (\frac {\left (b h^2 p q\right ) \operatorname {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 )-\operatorname {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 )+\operatorname {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 )-\operatorname {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 = 0.48, size = 363, normalized size = 0.85 \[ \frac {2 h^2 \log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )-2 h^2 \log \left (\frac {f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )+\frac {2 a h (h i-g j)}{i+j x}+\frac {a (h i-g j)^2}{(i+j x)^2}+\frac {2 b h (h i-g j) \log \left (c \left (d (e+f x)^p\right )^q\right )}{i+j x}+\frac {b (h i-g j)^2 \log \left (c \left (d (e+f x)^p\right )^q\right )}{(i+j x)^2}+2 b h^2 p q \text {Li}_2\left (\frac {h (e+f x)}{e h-f g}\right )-\frac {2 b f h p q (h i-g j) (\log (e+f x)-\log (i+j x))}{f i-e j}-\frac {b f p q (h i-g j)^2 (f (i+j x) \log (e+f x)-e j-f (i+j x) \log (i+j x)+f i)}{(i+j x) (f i-e j)^2}-2 b h^2 p q \text {Li}_2\left (\frac {j (e+f x)}{e j-f i}\right )}{2 (h i-g j)^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{h j^{3} x^{4} + g i^{3} + {\left (3 \, h i j^{2} + g j^{3}\right )} x^{3} + 3 \, {\left (h i^{2} j + g i j^{2}\right )} x^{2} + {\left (h i^{3} + 3 \, g i^{2} j\right )} x}, x\right ) \]
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 \[ \int \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{{\left (h x + g\right )} {\left (j x + i\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.52, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a}{\left (h x +g \right ) \left (j x +i \right )^{3}}\, dx \]
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 \[ \frac {1}{2} \, {\left (\frac {2 \, h^{2} \log \left (h x + g\right )}{h^{3} i^{3} - 3 \, g h^{2} i^{2} j + 3 \, g^{2} h i j^{2} - g^{3} j^{3}} - \frac {2 \, h^{2} \log \left (j x + i\right )}{h^{3} i^{3} - 3 \, g h^{2} i^{2} j + 3 \, g^{2} h i j^{2} - g^{3} j^{3}} + \frac {2 \, h j x + 3 \, h i - g j}{h^{2} i^{4} - 2 \, g h i^{3} j + g^{2} i^{2} j^{2} + {\left (h^{2} i^{2} j^{2} - 2 \, g h i j^{3} + g^{2} j^{4}\right )} x^{2} + 2 \, {\left (h^{2} i^{3} j - 2 \, g h i^{2} j^{2} + g^{2} i j^{3}\right )} x}\right )} a + b \int \frac {q \log \relax (d) + \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right ) + \log \relax (c)}{h j^{3} x^{4} + g i^{3} + {\left (3 \, h i j^{2} + g j^{3}\right )} x^{3} + 3 \, {\left (h i^{2} j + g i j^{2}\right )} x^{2} + {\left (h i^{3} + 3 \, g i^{2} j\right )} x}\,{d x} \]
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 \[ \int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{\left (g+h\,x\right )\,{\left (i+j\,x\right )}^3} \,d x \]
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 \[ \int \frac {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{\left (g + h x\right ) \left (i + j x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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