Optimal. Leaf size=1210 \[ -\frac {2 a b d g m n x}{e}-\frac {3 a b g i m n x}{2 j}-\frac {2 b^2 d f n^2 x}{e}+\frac {15 b^2 d g m n^2 x}{4 e}+\frac {7 b^2 g i m n^2 x}{4 j}-\frac {1}{4} b^2 g m n^2 x^2+\frac {b^2 f n^2 (d+e x)^2}{4 e^2}-\frac {b^2 g m n^2 (d+e x)^2}{8 e^2}-\frac {b^2 d^2 g m n^2 \log (d+e x)}{4 e^2}+\frac {b^2 d^2 f n^2 \log ^2(d+e x)}{2 e^2}-\frac {2 b^2 d g m n (d+e x) \log \left (c (d+e x)^n\right )}{e^2}-\frac {3 b^2 g i m n (d+e x) \log \left (c (d+e x)^n\right )}{2 e j}+\frac {1}{4} b g m n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {2 b d f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}-\frac {b f n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {b g m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2}-\frac {b d^2 f n \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}+\frac {d g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac {g i m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e j}-\frac {g m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}-\frac {b^2 g i^2 m n^2 \log (i+j x)}{4 j^2}+\frac {b g i^2 m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{2 j^2}+\frac {b d g i m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{e j}+\frac {d^2 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (i+j x)}{e i-d j}\right )}{2 e^2}-\frac {g i^2 m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (i+j x)}{e i-d j}\right )}{2 j^2}+\frac {1}{4} b^2 g n^2 x^2 \log \left (h (i+j x)^m\right )-\frac {3 b^2 d g n^2 (i+j x) \log \left (h (i+j x)^m\right )}{2 e j}+\frac {3 b^2 d^2 g n^2 \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \log \left (h (i+j x)^m\right )}{2 e^2}+\frac {b d g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right )}{e}-\frac {1}{2} b g n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right )-\frac {d^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (i+j x)^m\right )}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac {b^2 g i^2 m n^2 \text {Li}_2\left (-\frac {j (d+e x)}{e i-d j}\right )}{2 j^2}+\frac {b^2 d g i m n^2 \text {Li}_2\left (-\frac {j (d+e x)}{e i-d j}\right )}{e j}+\frac {b d^2 g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {j (d+e x)}{e i-d j}\right )}{e^2}-\frac {b g i^2 m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {j (d+e x)}{e i-d j}\right )}{j^2}+\frac {3 b^2 d^2 g m n^2 \text {Li}_2\left (\frac {e (i+j x)}{e i-d j}\right )}{2 e^2}-\frac {b^2 d^2 g m n^2 \text {Li}_3\left (-\frac {j (d+e x)}{e i-d j}\right )}{e^2}+\frac {b^2 g i^2 m n^2 \text {Li}_3\left (-\frac {j (d+e x)}{e i-d j}\right )}{j^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.95, antiderivative size = 1210, normalized size of antiderivative = 1.00, number of steps
used = 73, number of rules used = 27, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.844, Rules used = {2489, 2463,
2436, 2333, 2332, 2448, 2437, 2342, 2341, 2443, 2481, 2421, 6724, 6874, 2458, 45, 2372, 12, 14,
2338, 2479, 2441, 2440, 2438, 2442, 2422, 2354} \begin {gather*} -\frac {1}{4} g m n^2 x^2 b^2+\frac {f n^2 (d+e x)^2 b^2}{4 e^2}-\frac {g m n^2 (d+e x)^2 b^2}{8 e^2}+\frac {d^2 f n^2 \log ^2(d+e x) b^2}{2 e^2}-\frac {2 d f n^2 x b^2}{e}+\frac {15 d g m n^2 x b^2}{4 e}+\frac {7 g i m n^2 x b^2}{4 j}-\frac {d^2 g m n^2 \log (d+e x) b^2}{4 e^2}-\frac {2 d g m n (d+e x) \log \left (c (d+e x)^n\right ) b^2}{e^2}-\frac {3 g i m n (d+e x) \log \left (c (d+e x)^n\right ) b^2}{2 e j}-\frac {g i^2 m n^2 \log (i+j x) b^2}{4 j^2}+\frac {1}{4} g n^2 x^2 \log \left (h (i+j x)^m\right ) b^2-\frac {3 d g n^2 (i+j x) \log \left (h (i+j x)^m\right ) b^2}{2 e j}+\frac {3 d^2 g n^2 \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \log \left (h (i+j x)^m\right ) b^2}{2 e^2}+\frac {d g i m n^2 \text {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right ) b^2}{e j}+\frac {g i^2 m n^2 \text {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right ) b^2}{2 j^2}+\frac {3 d^2 g m n^2 \text {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right ) b^2}{2 e^2}-\frac {d^2 g m n^2 \text {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right ) b^2}{e^2}+\frac {g i^2 m n^2 \text {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right ) b^2}{j^2}-\frac {2 a d g m n x b}{e}-\frac {3 a g i m n x b}{2 j}+\frac {1}{4} g m n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) b-\frac {f n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) b}{2 e^2}+\frac {g m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) b}{4 e^2}+\frac {2 d f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) b}{e^2}-\frac {d^2 f n \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) b}{e^2}+\frac {d g i m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right ) b}{e j}+\frac {g i^2 m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right ) b}{2 j^2}-\frac {1}{2} g n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right ) b+\frac {d g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right ) b}{e}+\frac {d^2 g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right ) b}{e^2}-\frac {g i^2 m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right ) b}{j^2}-\frac {g m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac {d g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac {g i m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e j}+\frac {d^2 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (i+j x)}{e i-d j}\right )}{2 e^2}-\frac {g i^2 m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (i+j x)}{e i-d j}\right )}{2 j^2}-\frac {d^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (i+j x)^m\right )}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 45
Rule 2332
Rule 2333
Rule 2338
Rule 2341
Rule 2342
Rule 2354
Rule 2372
Rule 2421
Rule 2422
Rule 2436
Rule 2437
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2443
Rule 2448
Rule 2458
Rule 2463
Rule 2479
Rule 2481
Rule 2489
Rule 6724
Rule 6874
Rubi steps
\begin {align*} \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right ) \, dx &=\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right )-\frac {1}{2} (g j m) \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{393+j x} \, dx-(b e n) \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (393+j x)^m\right )\right )}{d+e x} \, dx\\ &=\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right )-\frac {1}{2} (g j m) \int \left (-\frac {393 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j^2}+\frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j}+\frac {154449 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j^2 (393+j x)}\right ) \, dx-(b e n) \int \left (\frac {f x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}+\frac {g x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{d+e x}\right ) \, dx\\ &=\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right )-\frac {1}{2} (g m) \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx+\frac {(393 g m) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{2 j}-\frac {(154449 g m) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{393+j x} \, dx}{2 j}-(b e f n) \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx-(b e g n) \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{d+e x} \, dx\\ &=-\frac {154449 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (393+j x)}{393 e-d j}\right )}{2 j^2}+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right )-\frac {1}{2} (g m) \int \left (-\frac {d \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}\right ) \, dx+\frac {(393 g m) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{2 e j}-(b f n) \text {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )-(b e g n) \int \left (-\frac {d \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{e^2}+\frac {x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{e}+\frac {d^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{e^2 (d+e x)}\right ) \, dx+\frac {(154449 b e g m n) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (393+j x)}{393 e-d j}\right )}{d+e x} \, dx}{j^2}\\ &=\frac {1}{2} b f n \left (\frac {4 d (d+e x)}{e^2}-\frac {(d+e x)^2}{e^2}-\frac {2 d^2 \log (d+e x)}{e^2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {393 g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e j}-\frac {154449 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (393+j x)}{393 e-d j}\right )}{2 j^2}+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right )-\frac {(g m) \int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{2 e}+\frac {(d g m) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{2 e}-(b g n) \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right ) \, dx+\frac {(b d g n) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right ) \, dx}{e}-\frac {\left (b d^2 g n\right ) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{d+e x} \, dx}{e}+\frac {(154449 b g m n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {e \left (\frac {393 e-d j}{e}+\frac {j x}{e}\right )}{393 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j^2}-\frac {(393 b g m n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e j}+\left (b^2 f n^2\right ) \text {Subst}\left (\int \frac {x (-4 d+x)+2 d^2 \log (x)}{2 e^2 x} \, dx,x,d+e x\right )\\ &=-\frac {393 a b g m n x}{j}+\frac {1}{2} b f n \left (\frac {4 d (d+e x)}{e^2}-\frac {(d+e x)^2}{e^2}-\frac {2 d^2 \log (d+e x)}{e^2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {393 g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e j}-\frac {154449 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (393+j x)}{393 e-d j}\right )}{2 j^2}+\frac {b d g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{e}-\frac {1}{2} b g n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right )-\frac {154449 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {j (d+e x)}{393 e-d j}\right )}{j^2}-\frac {(g m) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{2 e^2}+\frac {(d g m) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{2 e^2}-\frac {\left (b d^2 g n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (h \left (\frac {393 e-d j}{e}+\frac {j x}{e}\right )^m\right )}{x} \, dx,x,d+e x\right )}{e^2}-\frac {\left (393 b^2 g m n\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e j}+\frac {1}{2} (b g j m n) \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{393+j x} \, dx-\frac {(b d g j m n) \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{393+j x} \, dx}{e}+\frac {\left (b^2 f n^2\right ) \text {Subst}\left (\int \frac {x (-4 d+x)+2 d^2 \log (x)}{x} \, dx,x,d+e x\right )}{2 e^2}-\left (b^2 d g n^2\right ) \int \frac {x \log \left (h (393+j x)^m\right )}{d+e x} \, dx+\frac {1}{2} \left (b^2 e g n^2\right ) \int \frac {x^2 \log \left (h (393+j x)^m\right )}{d+e x} \, dx+\frac {\left (154449 b^2 g m n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {j x}{393 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j^2}\\ &=-\frac {393 a b g m n x}{j}+\frac {393 b^2 g m n^2 x}{j}-\frac {393 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e j}+\frac {1}{2} b f n \left (\frac {4 d (d+e x)}{e^2}-\frac {(d+e x)^2}{e^2}-\frac {2 d^2 \log (d+e x)}{e^2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {d g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac {393 g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e j}-\frac {g m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}-\frac {154449 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (393+j x)}{393 e-d j}\right )}{2 j^2}+\frac {b d g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{e}-\frac {1}{2} b g n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )-\frac {d^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (393+j x)^m\right )}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right )-\frac {154449 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {j (d+e x)}{393 e-d j}\right )}{j^2}+\frac {154449 b^2 g m n^2 \text {Li}_3\left (-\frac {j (d+e x)}{393 e-d j}\right )}{j^2}+\frac {\left (d^2 g j m\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\frac {393 e-d j}{e}+\frac {j x}{e}} \, dx,x,d+e x\right )}{2 e^3}+\frac {(b g m n) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{2 e^2}-\frac {(b d g m n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^2}+\frac {1}{2} (b g j m n) \int \left (-\frac {393 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^2}+\frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{j}+\frac {154449 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^2 (393+j x)}\right ) \, dx-\frac {(b d g j m n) \int \left (\frac {a+b \log \left (c (d+e x)^n\right )}{j}-\frac {393 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j (393+j x)}\right ) \, dx}{e}+\frac {\left (b^2 f n^2\right ) \text {Subst}\left (\int \left (-4 d+x+\frac {2 d^2 \log (x)}{x}\right ) \, dx,x,d+e x\right )}{2 e^2}-\left (b^2 d g n^2\right ) \int \left (\frac {\log \left (h (393+j x)^m\right )}{e}-\frac {d \log \left (h (393+j x)^m\right )}{e (d+e x)}\right ) \, dx+\frac {1}{2} \left (b^2 e g n^2\right ) \int \left (-\frac {d \log \left (h (393+j x)^m\right )}{e^2}+\frac {x \log \left (h (393+j x)^m\right )}{e}+\frac {d^2 \log \left (h (393+j x)^m\right )}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac {a b d g m n x}{e}-\frac {393 a b g m n x}{j}-\frac {2 b^2 d f n^2 x}{e}+\frac {393 b^2 g m n^2 x}{j}+\frac {b^2 f n^2 (d+e x)^2}{4 e^2}-\frac {b^2 g m n^2 (d+e x)^2}{8 e^2}-\frac {393 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e j}+\frac {b g m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2}+\frac {1}{2} b f n \left (\frac {4 d (d+e x)}{e^2}-\frac {(d+e x)^2}{e^2}-\frac {2 d^2 \log (d+e x)}{e^2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {d g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac {393 g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e j}-\frac {g m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}-\frac {154449 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (393+j x)}{393 e-d j}\right )}{2 j^2}+\frac {b d g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{e}-\frac {1}{2} b g n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )-\frac {d^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (393+j x)^m\right )}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right )+\frac {d^2 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac {j (d+e x)}{393 e-d j}\right )}{2 e^2}-\frac {154449 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {j (d+e x)}{393 e-d j}\right )}{j^2}+\frac {154449 b^2 g m n^2 \text {Li}_3\left (-\frac {j (d+e x)}{393 e-d j}\right )}{j^2}+\frac {1}{2} (b g m n) \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx-\frac {\left (b^2 d g m n\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e^2}-\frac {\left (b d^2 g m n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {j x}{393 e-d j}\right )}{x} \, dx,x,d+e x\right )}{e^2}-\frac {(b d g m n) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{e}+\frac {(393 b d g m n) \int \frac {a+b \log \left (c (d+e x)^n\right )}{393+j x} \, dx}{e}-\frac {(393 b g m n) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{2 j}+\frac {(154449 b g m n) \int \frac {a+b \log \left (c (d+e x)^n\right )}{393+j x} \, dx}{2 j}+\frac {\left (b^2 d^2 f n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+e x\right )}{e^2}+\frac {1}{2} \left (b^2 g n^2\right ) \int x \log \left (h (393+j x)^m\right ) \, dx-\frac {\left (b^2 d g n^2\right ) \int \log \left (h (393+j x)^m\right ) \, dx}{2 e}-\frac {\left (b^2 d g n^2\right ) \int \log \left (h (393+j x)^m\right ) \, dx}{e}+\frac {\left (b^2 d^2 g n^2\right ) \int \frac {\log \left (h (393+j x)^m\right )}{d+e x} \, dx}{2 e}+\frac {\left (b^2 d^2 g n^2\right ) \int \frac {\log \left (h (393+j x)^m\right )}{d+e x} \, dx}{e}\\ &=-\frac {2 a b d g m n x}{e}-\frac {1179 a b g m n x}{2 j}-\frac {2 b^2 d f n^2 x}{e}+\frac {b^2 d g m n^2 x}{e}+\frac {393 b^2 g m n^2 x}{j}+\frac {b^2 f n^2 (d+e x)^2}{4 e^2}-\frac {b^2 g m n^2 (d+e x)^2}{8 e^2}+\frac {b^2 d^2 f n^2 \log ^2(d+e x)}{2 e^2}-\frac {b^2 d g m n (d+e x) \log \left (c (d+e x)^n\right )}{e^2}-\frac {393 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e j}+\frac {1}{4} b g m n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b g m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2}+\frac {1}{2} b f n \left (\frac {4 d (d+e x)}{e^2}-\frac {(d+e x)^2}{e^2}-\frac {2 d^2 \log (d+e x)}{e^2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {d g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac {393 g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e j}-\frac {g m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac {154449 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (393+j x)}{393 e-d j}\right )}{2 j^2}+\frac {393 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (393+j x)}{393 e-d j}\right )}{e j}-\frac {154449 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (393+j x)}{393 e-d j}\right )}{2 j^2}+\frac {1}{4} b^2 g n^2 x^2 \log \left (h (393+j x)^m\right )+\frac {3 b^2 d^2 g n^2 \log \left (-\frac {j (d+e x)}{393 e-d j}\right ) \log \left (h (393+j x)^m\right )}{2 e^2}+\frac {b d g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{e}-\frac {1}{2} b g n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )-\frac {d^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (393+j x)^m\right )}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right )+\frac {d^2 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac {j (d+e x)}{393 e-d j}\right )}{2 e^2}+\frac {b d^2 g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {j (d+e x)}{393 e-d j}\right )}{e^2}-\frac {154449 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {j (d+e x)}{393 e-d j}\right )}{j^2}+\frac {154449 b^2 g m n^2 \text {Li}_3\left (-\frac {j (d+e x)}{393 e-d j}\right )}{j^2}-\frac {\left (b^2 d g m n\right ) \int \log \left (c (d+e x)^n\right ) \, dx}{e}-\frac {\left (393 b^2 g m n\right ) \int \log \left (c (d+e x)^n\right ) \, dx}{2 j}-\frac {\left (b^2 d g n^2\right ) \text {Subst}\left (\int \log \left (h x^m\right ) \, dx,x,393+j x\right )}{2 e j}-\frac {\left (b^2 d g n^2\right ) \text {Subst}\left (\int \log \left (h x^m\right ) \, dx,x,393+j x\right )}{e j}-\frac {\left (b^2 d^2 g m n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {j x}{393 e-d j}\right )}{x} \, dx,x,d+e x\right )}{e^2}-\frac {1}{4} \left (b^2 e g m n^2\right ) \int \frac {x^2}{d+e x} \, dx-\frac {\left (154449 b^2 e g m n^2\right ) \int \frac {\log \left (\frac {e (393+j x)}{393 e-d j}\right )}{d+e x} \, dx}{2 j^2}-\frac {\left (393 b^2 d g m n^2\right ) \int \frac {\log \left (\frac {e (393+j x)}{393 e-d j}\right )}{d+e x} \, dx}{j}-\frac {1}{4} \left (b^2 g j m n^2\right ) \int \frac {x^2}{393+j x} \, dx-\frac {\left (b^2 d^2 g j m n^2\right ) \int \frac {\log \left (\frac {j (d+e x)}{-393 e+d j}\right )}{393+j x} \, dx}{2 e^2}-\frac {\left (b^2 d^2 g j m n^2\right ) \int \frac {\log \left (\frac {j (d+e x)}{-393 e+d j}\right )}{393+j x} \, dx}{e^2}\\ &=-\frac {2 a b d g m n x}{e}-\frac {1179 a b g m n x}{2 j}-\frac {2 b^2 d f n^2 x}{e}+\frac {5 b^2 d g m n^2 x}{2 e}+\frac {393 b^2 g m n^2 x}{j}+\frac {b^2 f n^2 (d+e x)^2}{4 e^2}-\frac {b^2 g m n^2 (d+e x)^2}{8 e^2}+\frac {b^2 d^2 f n^2 \log ^2(d+e x)}{2 e^2}-\frac {b^2 d g m n (d+e x) \log \left (c (d+e x)^n\right )}{e^2}-\frac {393 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e j}+\frac {1}{4} b g m n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b g m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2}+\frac {1}{2} b f n \left (\frac {4 d (d+e x)}{e^2}-\frac {(d+e x)^2}{e^2}-\frac {2 d^2 \log (d+e x)}{e^2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {d g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac {393 g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e j}-\frac {g m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac {154449 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (393+j x)}{393 e-d j}\right )}{2 j^2}+\frac {393 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (393+j x)}{393 e-d j}\right )}{e j}-\frac {154449 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (393+j x)}{393 e-d j}\right )}{2 j^2}+\frac {1}{4} b^2 g n^2 x^2 \log \left (h (393+j x)^m\right )-\frac {3 b^2 d g n^2 (393+j x) \log \left (h (393+j x)^m\right )}{2 e j}+\frac {3 b^2 d^2 g n^2 \log \left (-\frac {j (d+e x)}{393 e-d j}\right ) \log \left (h (393+j x)^m\right )}{2 e^2}+\frac {b d g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{e}-\frac {1}{2} b g n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )-\frac {d^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (393+j x)^m\right )}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right )+\frac {d^2 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac {j (d+e x)}{393 e-d j}\right )}{2 e^2}+\frac {b d^2 g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {j (d+e x)}{393 e-d j}\right )}{e^2}-\frac {154449 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {j (d+e x)}{393 e-d j}\right )}{j^2}-\frac {b^2 d^2 g m n^2 \text {Li}_3\left (-\frac {j (d+e x)}{393 e-d j}\right )}{e^2}+\frac {154449 b^2 g m n^2 \text {Li}_3\left (-\frac {j (d+e x)}{393 e-d j}\right )}{j^2}-\frac {\left (b^2 d g m n\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e^2}-\frac {\left (393 b^2 g m n\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{2 e j}-\frac {\left (b^2 d^2 g m n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {e x}{-393 e+d j}\right )}{x} \, dx,x,393+j x\right )}{2 e^2}-\frac {\left (b^2 d^2 g m n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {e x}{-393 e+d j}\right )}{x} \, dx,x,393+j x\right )}{e^2}-\frac {1}{4} \left (b^2 e g m n^2\right ) \int \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx-\frac {\left (154449 b^2 g m n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {j x}{393 e-d j}\right )}{x} \, dx,x,d+e x\right )}{2 j^2}-\frac {\left (393 b^2 d g m n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {j x}{393 e-d j}\right )}{x} \, dx,x,d+e x\right )}{e j}-\frac {1}{4} \left (b^2 g j m n^2\right ) \int \left (-\frac {393}{j^2}+\frac {x}{j}+\frac {154449}{j^2 (393+j x)}\right ) \, dx\\ &=-\frac {2 a b d g m n x}{e}-\frac {1179 a b g m n x}{2 j}-\frac {2 b^2 d f n^2 x}{e}+\frac {15 b^2 d g m n^2 x}{4 e}+\frac {2751 b^2 g m n^2 x}{4 j}-\frac {1}{4} b^2 g m n^2 x^2+\frac {b^2 f n^2 (d+e x)^2}{4 e^2}-\frac {b^2 g m n^2 (d+e x)^2}{8 e^2}-\frac {b^2 d^2 g m n^2 \log (d+e x)}{4 e^2}+\frac {b^2 d^2 f n^2 \log ^2(d+e x)}{2 e^2}-\frac {2 b^2 d g m n (d+e x) \log \left (c (d+e x)^n\right )}{e^2}-\frac {1179 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{2 e j}+\frac {1}{4} b g m n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b g m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2}+\frac {1}{2} b f n \left (\frac {4 d (d+e x)}{e^2}-\frac {(d+e x)^2}{e^2}-\frac {2 d^2 \log (d+e x)}{e^2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {d g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac {393 g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e j}-\frac {g m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}-\frac {154449 b^2 g m n^2 \log (393+j x)}{4 j^2}+\frac {154449 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (393+j x)}{393 e-d j}\right )}{2 j^2}+\frac {393 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (393+j x)}{393 e-d j}\right )}{e j}-\frac {154449 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (393+j x)}{393 e-d j}\right )}{2 j^2}+\frac {1}{4} b^2 g n^2 x^2 \log \left (h (393+j x)^m\right )-\frac {3 b^2 d g n^2 (393+j x) \log \left (h (393+j x)^m\right )}{2 e j}+\frac {3 b^2 d^2 g n^2 \log \left (-\frac {j (d+e x)}{393 e-d j}\right ) \log \left (h (393+j x)^m\right )}{2 e^2}+\frac {b d g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )}{e}-\frac {1}{2} b g n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (393+j x)^m\right )-\frac {d^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (393+j x)^m\right )}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (393+j x)^m\right )\right )+\frac {d^2 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac {j (d+e x)}{393 e-d j}\right )}{2 e^2}+\frac {154449 b^2 g m n^2 \text {Li}_2\left (-\frac {j (d+e x)}{393 e-d j}\right )}{2 j^2}+\frac {393 b^2 d g m n^2 \text {Li}_2\left (-\frac {j (d+e x)}{393 e-d j}\right )}{e j}+\frac {b d^2 g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {j (d+e x)}{393 e-d j}\right )}{e^2}-\frac {154449 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {j (d+e x)}{393 e-d j}\right )}{j^2}+\frac {3 b^2 d^2 g m n^2 \text {Li}_2\left (\frac {e (393+j x)}{393 e-d j}\right )}{2 e^2}-\frac {b^2 d^2 g m n^2 \text {Li}_3\left (-\frac {j (d+e x)}{393 e-d j}\right )}{e^2}+\frac {154449 b^2 g m n^2 \text {Li}_3\left (-\frac {j (d+e x)}{393 e-d j}\right )}{j^2}\\ \end {align*}
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Mathematica [A]
time = 0.77, size = 2067, normalized size = 1.71 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int x \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{2} \left (f +g \ln \left (h \left (j x +i \right )^{m}\right )\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 x\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2\,\left (f+g\,\ln \left (h\,{\left (i+j\,x\right )}^m\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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