Optimal. Leaf size=308 \[ -\frac {6 b^2 n^2 (g h-f i) \text {Li}_3\left (-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac {6 a b^2 i n^2 x}{g}+\frac {3 b n (g h-f i) \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^2}+\frac {(g h-f i) \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g^2}-\frac {3 b i n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}+\frac {i (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g}+\frac {6 b^3 i n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e g}+\frac {6 b^3 n^3 (g h-f i) \text {Li}_4\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^2}-\frac {6 b^3 i n^3 x}{g} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.36, antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2418, 2389, 2296, 2295, 2396, 2433, 2374, 2383, 6589} \[ -\frac {6 b^2 n^2 (g h-f i) \text {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac {3 b n (g h-f i) \text {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^2}+\frac {6 b^3 n^3 (g h-f i) \text {PolyLog}\left (4,-\frac {g (d+e x)}{e f-d g}\right )}{g^2}+\frac {6 a b^2 i n^2 x}{g}+\frac {(g h-f i) \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g^2}-\frac {3 b i n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}+\frac {i (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g}+\frac {6 b^3 i n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e g}-\frac {6 b^3 i n^3 x}{g} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2295
Rule 2296
Rule 2374
Rule 2383
Rule 2389
Rule 2396
Rule 2418
Rule 2433
Rule 6589
Rubi steps
\begin {align*} \int \frac {(h+230 x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx &=\int \left (\frac {230 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g}+\frac {(-230 f+g h) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g (f+g x)}\right ) \, dx\\ &=\frac {230 \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{g}+\frac {(-230 f+g h) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx}{g}\\ &=-\frac {(230 f-g h) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}+\frac {230 \operatorname {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e g}+\frac {(3 b e (230 f-g h) n) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{g^2}\\ &=\frac {230 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g}-\frac {(230 f-g h) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}-\frac {(690 b n) \operatorname {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e g}+\frac {(3 b (230 f-g h) n) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {e \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g^2}\\ &=-\frac {690 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}+\frac {230 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g}-\frac {(230 f-g h) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}-\frac {3 b (230 f-g h) n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^2}+\frac {\left (1380 b^2 n^2\right ) \operatorname {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e g}+\frac {\left (6 b^2 (230 f-g h) n^2\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g^2}\\ &=\frac {1380 a b^2 n^2 x}{g}-\frac {690 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}+\frac {230 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g}-\frac {(230 f-g h) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}-\frac {3 b (230 f-g h) n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^2}+\frac {6 b^2 (230 f-g h) n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^2}+\frac {\left (1380 b^3 n^2\right ) \operatorname {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e g}-\frac {\left (6 b^3 (230 f-g h) n^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g^2}\\ &=\frac {1380 a b^2 n^2 x}{g}-\frac {1380 b^3 n^3 x}{g}+\frac {1380 b^3 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e g}-\frac {690 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}+\frac {230 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g}-\frac {(230 f-g h) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}-\frac {3 b (230 f-g h) n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^2}+\frac {6 b^2 (230 f-g h) n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^2}-\frac {6 b^3 (230 f-g h) n^3 \text {Li}_4\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.44, size = 799, normalized size = 2.59 \[ \frac {b^3 e g h \left (\log \left (\frac {e (f+g x)}{e f-d g}\right ) \log ^3(d+e x)+3 \text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right ) \log ^2(d+e x)-6 \text {Li}_3\left (\frac {g (d+e x)}{d g-e f}\right ) \log (d+e x)+6 \text {Li}_4\left (\frac {g (d+e x)}{d g-e f}\right )\right ) n^3-b^3 i \left (g \left (-\left ((d+e x) \log ^3(d+e x)\right )+3 (d+e x) \log ^2(d+e x)-6 (d+e x) \log (d+e x)+6 e x\right )+e f \left (\log \left (\frac {e (f+g x)}{e f-d g}\right ) \log ^3(d+e x)+3 \text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right ) \log ^2(d+e x)-6 \text {Li}_3\left (\frac {g (d+e x)}{d g-e f}\right ) \log (d+e x)+6 \text {Li}_4\left (\frac {g (d+e x)}{d g-e f}\right )\right )\right ) n^3+3 b^2 i \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (g \left ((d+e x) \log ^2(d+e x)-2 (d+e x) \log (d+e x)+2 e x\right )-e f \left (\log \left (\frac {e (f+g x)}{e f-d g}\right ) \log ^2(d+e x)+2 \text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right ) \log (d+e x)-2 \text {Li}_3\left (\frac {g (d+e x)}{d g-e f}\right )\right )\right ) n^2+6 b^2 e g h \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\frac {1}{2} \log \left (\frac {e (f+g x)}{e f-d g}\right ) \log ^2(d+e x)+\text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right ) \log (d+e x)-\text {Li}_3\left (\frac {g (d+e x)}{d g-e f}\right )\right ) n^2+3 b e g h \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \left (\log (d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+\text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right )\right ) n-3 b i \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \left (e f \left (\log (d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+\text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right )\right )-g (d+e x) (\log (d+e x)-1)\right ) n+e g i x \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3+e (g h-f i) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3 \log (f+g x)}{e g^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{3} i x + a^{3} h + {\left (b^{3} i x + b^{3} h\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{3} + 3 \, {\left (a b^{2} i x + a b^{2} h\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 3 \, {\left (a^{2} b i x + a^{2} b h\right )} \log \left ({\left (e x + d\right )}^{n} c\right )}{g x + f}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (i x + h\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{g x + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 2.21, size = 0, normalized size = 0.00 \[ \int \frac {\left (i x +h \right ) \left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{3}}{g x +f}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{3} i {\left (\frac {x}{g} - \frac {f \log \left (g x + f\right )}{g^{2}}\right )} + \frac {a^{3} h \log \left (g x + f\right )}{g} + \int \frac {b^{3} h \log \relax (c)^{3} + 3 \, a b^{2} h \log \relax (c)^{2} + 3 \, a^{2} b h \log \relax (c) + {\left (b^{3} i x + b^{3} h\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{3} + 3 \, {\left (b^{3} h \log \relax (c) + a b^{2} h + {\left (b^{3} i \log \relax (c) + a b^{2} i\right )} x\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + {\left (b^{3} i \log \relax (c)^{3} + 3 \, a b^{2} i \log \relax (c)^{2} + 3 \, a^{2} b i \log \relax (c)\right )} x + 3 \, {\left (b^{3} h \log \relax (c)^{2} + 2 \, a b^{2} h \log \relax (c) + a^{2} b h + {\left (b^{3} i \log \relax (c)^{2} + 2 \, a b^{2} i \log \relax (c) + a^{2} b i\right )} x\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{g x + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (h+i\,x\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3}{f+g\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{3} \left (h + i x\right )}{f + g x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________