Optimal. Leaf size=410 \[ -\frac {6 b^2 p^2 q^2 \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j}+\frac {6 b^2 p^2 q^2 \text {Li}_3\left (-\frac {j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j}+\frac {3 b p q \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h i-g j}-\frac {3 b p q \text {Li}_2\left (-\frac {j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h i-g j}+\frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h i-g j}-\frac {\log \left (\frac {f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h i-g j}+\frac {6 b^3 p^3 q^3 \text {Li}_4\left (-\frac {h (e+f x)}{f g-e h}\right )}{h i-g j}-\frac {6 b^3 p^3 q^3 \text {Li}_4\left (-\frac {j (e+f x)}{f i-e j}\right )}{h i-g j} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.26, antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2418, 2396, 2433, 2374, 2383, 6589, 2445} \[ -\frac {6 b^2 p^2 q^2 \text {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j}+\frac {6 b^2 p^2 q^2 \text {PolyLog}\left (3,-\frac {j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j}+\frac {3 b p q \text {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h i-g j}-\frac {3 b p q \text {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h i-g j}+\frac {6 b^3 p^3 q^3 \text {PolyLog}\left (4,-\frac {h (e+f x)}{f g-e h}\right )}{h i-g j}-\frac {6 b^3 p^3 q^3 \text {PolyLog}\left (4,-\frac {j (e+f x)}{f i-e j}\right )}{h i-g j}+\frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h i-g j}-\frac {\log \left (\frac {f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h i-g j} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2374
Rule 2383
Rule 2396
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 )^3}{(g+h x) (538+j x)} \, dx &=\operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{(g+h x) (538+j x)} \, 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 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{(538 h-g j) (g+h x)}-\frac {j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{(538 h-g j) (538+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 \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{g+h x} \, dx}{538 h-g j},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 )^3}{538+j x} \, dx}{538 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 )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{538 h-g j}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac {f (538+j x)}{538 f-e j}\right )}{538 h-g j}-\operatorname {Subst}\left (\frac {(3 b f p q) \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{e+f x} \, dx}{538 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {(3 b f p q) \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2 \log \left (\frac {f (538+j x)}{538 f-e j}\right )}{e+f x} \, dx}{538 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 )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{538 h-g j}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac {f (538+j x)}{538 f-e j}\right )}{538 h-g j}-\operatorname {Subst}\left (\frac {(3 b p q) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right )^2 \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 )}{538 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {(3 b p q) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right )^2 \log \left (\frac {f \left (\frac {538 f-e j}{f}+\frac {j x}{f}\right )}{538 f-e j}\right )}{x} \, dx,x,e+f x\right )}{538 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 )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{538 h-g j}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac {f (538+j x)}{538 f-e j}\right )}{538 h-g j}+\frac {3 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{538 h-g j}-\frac {3 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_2\left (-\frac {j (e+f x)}{538 f-e j}\right )}{538 h-g j}-\operatorname {Subst}\left (\frac {\left (6 b^2 p^2 q^2\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right ) \text {Li}_2\left (-\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{538 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {\left (6 b^2 p^2 q^2\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c d^q x^{p q}\right )\right ) \text {Li}_2\left (-\frac {j x}{538 f-e j}\right )}{x} \, dx,x,e+f x\right )}{538 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 )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{538 h-g j}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac {f (538+j x)}{538 f-e j}\right )}{538 h-g j}+\frac {3 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{538 h-g j}-\frac {3 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_2\left (-\frac {j (e+f x)}{538 f-e j}\right )}{538 h-g j}-\frac {6 b^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{538 h-g j}+\frac {6 b^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_3\left (-\frac {j (e+f x)}{538 f-e j}\right )}{538 h-g j}+\operatorname {Subst}\left (\frac {\left (6 b^3 p^3 q^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{538 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {\left (6 b^3 p^3 q^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {j x}{538 f-e j}\right )}{x} \, dx,x,e+f x\right )}{538 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 )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{538 h-g j}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac {f (538+j x)}{538 f-e j}\right )}{538 h-g j}+\frac {3 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{538 h-g j}-\frac {3 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_2\left (-\frac {j (e+f x)}{538 f-e j}\right )}{538 h-g j}-\frac {6 b^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{538 h-g j}+\frac {6 b^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_3\left (-\frac {j (e+f x)}{538 f-e j}\right )}{538 h-g j}+\frac {6 b^3 p^3 q^3 \text {Li}_4\left (-\frac {h (e+f x)}{f g-e h}\right )}{538 h-g j}-\frac {6 b^3 p^3 q^3 \text {Li}_4\left (-\frac {j (e+f x)}{538 f-e j}\right )}{538 h-g j}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.56, size = 1350, normalized size = 3.29 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{3} + 3 \, a b^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + 3 \, a^{2} b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a^{3}}{h j x^{2} + g i + {\left (h i + g j\right )} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{3}}{{\left (h x + g\right )} {\left (j x + i\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.54, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a \right )^{3}}{\left (h x +g \right ) \left (j x +i \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{3} {\left (\frac {\log \left (h x + g\right )}{h i - g j} - \frac {\log \left (j x + i\right )}{h i - g j}\right )} + \int \frac {b^{3} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )^{3} + 3 \, {\left (q \log \relax (d) + \log \relax (c)\right )} a^{2} b + 3 \, {\left (q^{2} \log \relax (d)^{2} + 2 \, q \log \relax (c) \log \relax (d) + \log \relax (c)^{2}\right )} a b^{2} + {\left (q^{3} \log \relax (d)^{3} + 3 \, q^{2} \log \relax (c) \log \relax (d)^{2} + 3 \, q \log \relax (c)^{2} \log \relax (d) + \log \relax (c)^{3}\right )} b^{3} + 3 \, {\left ({\left (q \log \relax (d) + \log \relax (c)\right )} b^{3} + a b^{2}\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )^{2} + 3 \, {\left (2 \, {\left (q \log \relax (d) + \log \relax (c)\right )} a b^{2} + {\left (q^{2} \log \relax (d)^{2} + 2 \, q \log \relax (c) \log \relax (d) + \log \relax (c)^{2}\right )} b^{3} + a^{2} b\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )}{h j x^{2} + g i + {\left (h i + g j\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^3}{\left (g+h\,x\right )\,\left (i+j\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{3}}{\left (g + h x\right ) \left (i + j x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________