Optimal. Leaf size=742 \[ \frac {a g i^3 m x}{4 j^3}+\frac {b d^3 f n x}{4 e^3}-\frac {5 b d^3 g m n x}{16 e^3}-\frac {5 b g i^3 m n x}{16 j^3}-\frac {5 b d g i^2 m n x}{24 e j^2}-\frac {5 b d^2 g i m n x}{24 e^2 j}+\frac {3 b d^2 g m n x^2}{32 e^2}+\frac {3 b g i^2 m n x^2}{32 j^2}+\frac {b d g i m n x^2}{12 e j}-\frac {7 b d g m n x^3}{144 e}-\frac {7 b g i m n x^3}{144 j}+\frac {1}{32} b g m n x^4+\frac {b d^4 g m n \log (d+e x)}{16 e^4}+\frac {b d^2 g i^2 m n \log (d+e x)}{8 e^2 j^2}+\frac {b d^3 g i m n \log (d+e x)}{12 e^3 j}+\frac {b g i^3 m (d+e x) \log \left (c (d+e x)^n\right )}{4 e j^3}-\frac {g i^2 m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{8 j^2}+\frac {g i m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{12 j}-\frac {1}{16} g m x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b g i^4 m n \log (i+j x)}{16 j^4}+\frac {b d g i^3 m n \log (i+j x)}{12 e j^3}+\frac {b d^2 g i^2 m n \log (i+j x)}{8 e^2 j^2}-\frac {g i^4 m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{4 j^4}+\frac {b d^3 g n (i+j x) \log \left (h (i+j x)^m\right )}{4 e^3 j}-\frac {b d^2 n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{8 e^2}+\frac {b d n x^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{12 e}-\frac {1}{16} b n x^4 \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {b d^4 n \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{4 e^4}+\frac {1}{4} x^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {b g i^4 m n \text {Li}_2\left (-\frac {j (d+e x)}{e i-d j}\right )}{4 j^4}-\frac {b d^4 g m n \text {Li}_2\left (\frac {e (i+j x)}{e i-d j}\right )}{4 e^4} \]
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Rubi [A]
time = 0.61, antiderivative size = 742, normalized size of antiderivative = 1.00, number of steps
used = 35, number of rules used = 9, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2489, 45,
2463, 2436, 2332, 2442, 2441, 2440, 2438} \begin {gather*} -\frac {b d^4 g m n \text {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{4 e^4}-\frac {b g i^4 m n \text {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{4 j^4}+\frac {1}{4} x^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {g i^4 m \log \left (\frac {e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 j^4}-\frac {g i^2 m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{8 j^2}+\frac {g i m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{12 j}-\frac {1}{16} g m x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {a g i^3 m x}{4 j^3}+\frac {b g i^3 m (d+e x) \log \left (c (d+e x)^n\right )}{4 e j^3}-\frac {b d^4 n \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{4 e^4}+\frac {b d^4 g m n \log (d+e x)}{16 e^4}+\frac {b d^3 f n x}{4 e^3}+\frac {b d^3 g n (i+j x) \log \left (h (i+j x)^m\right )}{4 e^3 j}+\frac {b d^3 g i m n \log (d+e x)}{12 e^3 j}-\frac {5 b d^3 g m n x}{16 e^3}-\frac {b d^2 n x^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{8 e^2}+\frac {b d^2 g i^2 m n \log (d+e x)}{8 e^2 j^2}+\frac {b d^2 g i^2 m n \log (i+j x)}{8 e^2 j^2}-\frac {5 b d^2 g i m n x}{24 e^2 j}+\frac {3 b d^2 g m n x^2}{32 e^2}+\frac {b d n x^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{12 e}+\frac {b d g i^3 m n \log (i+j x)}{12 e j^3}-\frac {5 b d g i^2 m n x}{24 e j^2}+\frac {b d g i m n x^2}{12 e j}-\frac {7 b d g m n x^3}{144 e}-\frac {1}{16} b n x^4 \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac {b g i^4 m n \log (i+j x)}{16 j^4}-\frac {5 b g i^3 m n x}{16 j^3}+\frac {3 b g i^2 m n x^2}{32 j^2}-\frac {7 b g i m n x^3}{144 j}+\frac {1}{32} b g m n x^4 \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2332
Rule 2436
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2463
Rule 2489
Rubi steps
\begin {align*} \int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (386+j x)^m\right )\right ) \, dx &=\frac {1}{4} x^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (386+j x)^m\right )\right )-\frac {1}{4} (g j m) \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{386+j x} \, dx-\frac {1}{4} (b e n) \int \frac {x^4 \left (f+g \log \left (h (386+j x)^m\right )\right )}{d+e x} \, dx\\ &=\frac {1}{4} x^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (386+j x)^m\right )\right )-\frac {1}{4} (g j m) \int \left (-\frac {57512456 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^4}+\frac {148996 x \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^3}-\frac {386 x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^2}+\frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j}+\frac {22199808016 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^4 (386+j x)}\right ) \, dx-\frac {1}{4} (b e n) \int \left (-\frac {d^3 \left (f+g \log \left (h (386+j x)^m\right )\right )}{e^4}+\frac {d^2 x \left (f+g \log \left (h (386+j x)^m\right )\right )}{e^3}-\frac {d x^2 \left (f+g \log \left (h (386+j x)^m\right )\right )}{e^2}+\frac {x^3 \left (f+g \log \left (h (386+j x)^m\right )\right )}{e}+\frac {d^4 \left (f+g \log \left (h (386+j x)^m\right )\right )}{e^4 (d+e x)}\right ) \, dx\\ &=\frac {1}{4} x^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (386+j x)^m\right )\right )-\frac {1}{4} (g m) \int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx+\frac {(14378114 g m) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{j^3}-\frac {(5549952004 g m) \int \frac {a+b \log \left (c (d+e x)^n\right )}{386+j x} \, dx}{j^3}-\frac {(37249 g m) \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{j^2}+\frac {(193 g m) \int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{2 j}-\frac {1}{4} (b n) \int x^3 \left (f+g \log \left (h (386+j x)^m\right )\right ) \, dx+\frac {\left (b d^3 n\right ) \int \left (f+g \log \left (h (386+j x)^m\right )\right ) \, dx}{4 e^3}-\frac {\left (b d^4 n\right ) \int \frac {f+g \log \left (h (386+j x)^m\right )}{d+e x} \, dx}{4 e^3}-\frac {\left (b d^2 n\right ) \int x \left (f+g \log \left (h (386+j x)^m\right )\right ) \, dx}{4 e^2}+\frac {(b d n) \int x^2 \left (f+g \log \left (h (386+j x)^m\right )\right ) \, dx}{4 e}\\ &=\frac {14378114 a g m x}{j^3}+\frac {b d^3 f n x}{4 e^3}-\frac {37249 g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 j^2}+\frac {193 g m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{6 j}-\frac {1}{16} g m x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {5549952004 g m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (386+j x)}{386 e-d j}\right )}{j^4}-\frac {b d^2 n x^2 \left (f+g \log \left (h (386+j x)^m\right )\right )}{8 e^2}+\frac {b d n x^3 \left (f+g \log \left (h (386+j x)^m\right )\right )}{12 e}-\frac {1}{16} b n x^4 \left (f+g \log \left (h (386+j x)^m\right )\right )-\frac {b d^4 n \log \left (-\frac {j (d+e x)}{386 e-d j}\right ) \left (f+g \log \left (h (386+j x)^m\right )\right )}{4 e^4}+\frac {1}{4} x^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (386+j x)^m\right )\right )+\frac {(14378114 b g m) \int \log \left (c (d+e x)^n\right ) \, dx}{j^3}+\frac {\left (b d^3 g n\right ) \int \log \left (h (386+j x)^m\right ) \, dx}{4 e^3}+\frac {1}{16} (b e g m n) \int \frac {x^4}{d+e x} \, dx+\frac {(5549952004 b e g m n) \int \frac {\log \left (\frac {e (386+j x)}{386 e-d j}\right )}{d+e x} \, dx}{j^4}+\frac {(37249 b e g m n) \int \frac {x^2}{d+e x} \, dx}{2 j^2}-\frac {(193 b e g m n) \int \frac {x^3}{d+e x} \, dx}{6 j}+\frac {1}{16} (b g j m n) \int \frac {x^4}{386+j x} \, dx+\frac {\left (b d^4 g j m n\right ) \int \frac {\log \left (\frac {j (d+e x)}{-386 e+d j}\right )}{386+j x} \, dx}{4 e^4}+\frac {\left (b d^2 g j m n\right ) \int \frac {x^2}{386+j x} \, dx}{8 e^2}-\frac {(b d g j m n) \int \frac {x^3}{386+j x} \, dx}{12 e}\\ &=\frac {14378114 a g m x}{j^3}+\frac {b d^3 f n x}{4 e^3}-\frac {37249 g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 j^2}+\frac {193 g m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{6 j}-\frac {1}{16} g m x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {5549952004 g m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (386+j x)}{386 e-d j}\right )}{j^4}-\frac {b d^2 n x^2 \left (f+g \log \left (h (386+j x)^m\right )\right )}{8 e^2}+\frac {b d n x^3 \left (f+g \log \left (h (386+j x)^m\right )\right )}{12 e}-\frac {1}{16} b n x^4 \left (f+g \log \left (h (386+j x)^m\right )\right )-\frac {b d^4 n \log \left (-\frac {j (d+e x)}{386 e-d j}\right ) \left (f+g \log \left (h (386+j x)^m\right )\right )}{4 e^4}+\frac {1}{4} x^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (386+j x)^m\right )\right )+\frac {(14378114 b g m) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e j^3}+\frac {\left (b d^3 g n\right ) \text {Subst}\left (\int \log \left (h x^m\right ) \, dx,x,386+j x\right )}{4 e^3 j}+\frac {\left (b d^4 g m n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {e x}{-386 e+d j}\right )}{x} \, dx,x,386+j x\right )}{4 e^4}+\frac {1}{16} (b e g m n) \int \left (-\frac {d^3}{e^4}+\frac {d^2 x}{e^3}-\frac {d x^2}{e^2}+\frac {x^3}{e}+\frac {d^4}{e^4 (d+e x)}\right ) \, dx+\frac {(5549952004 b g m n) \text {Subst}\left (\int \frac {\log \left (1+\frac {j x}{386 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j^4}+\frac {(37249 b e g m n) \int \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx}{2 j^2}-\frac {(193 b e g m n) \int \left (\frac {d^2}{e^3}-\frac {d x}{e^2}+\frac {x^2}{e}-\frac {d^3}{e^3 (d+e x)}\right ) \, dx}{6 j}+\frac {1}{16} (b g j m n) \int \left (-\frac {57512456}{j^4}+\frac {148996 x}{j^3}-\frac {386 x^2}{j^2}+\frac {x^3}{j}+\frac {22199808016}{j^4 (386+j x)}\right ) \, dx+\frac {\left (b d^2 g j m n\right ) \int \left (-\frac {386}{j^2}+\frac {x}{j}+\frac {148996}{j^2 (386+j x)}\right ) \, dx}{8 e^2}-\frac {(b d g j m n) \int \left (\frac {148996}{j^3}-\frac {386 x}{j^2}+\frac {x^2}{j}-\frac {57512456}{j^3 (386+j x)}\right ) \, dx}{12 e}\\ &=\frac {14378114 a g m x}{j^3}+\frac {b d^3 f n x}{4 e^3}-\frac {5 b d^3 g m n x}{16 e^3}-\frac {35945285 b g m n x}{2 j^3}-\frac {186245 b d g m n x}{6 e j^2}-\frac {965 b d^2 g m n x}{12 e^2 j}+\frac {3 b d^2 g m n x^2}{32 e^2}+\frac {111747 b g m n x^2}{8 j^2}+\frac {193 b d g m n x^2}{6 e j}-\frac {7 b d g m n x^3}{144 e}-\frac {1351 b g m n x^3}{72 j}+\frac {1}{32} b g m n x^4+\frac {b d^4 g m n \log (d+e x)}{16 e^4}+\frac {37249 b d^2 g m n \log (d+e x)}{2 e^2 j^2}+\frac {193 b d^3 g m n \log (d+e x)}{6 e^3 j}+\frac {14378114 b g m (d+e x) \log \left (c (d+e x)^n\right )}{e j^3}-\frac {37249 g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 j^2}+\frac {193 g m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{6 j}-\frac {1}{16} g m x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {1387488001 b g m n \log (386+j x)}{j^4}+\frac {14378114 b d g m n \log (386+j x)}{3 e j^3}+\frac {37249 b d^2 g m n \log (386+j x)}{2 e^2 j^2}-\frac {5549952004 g m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (386+j x)}{386 e-d j}\right )}{j^4}+\frac {b d^3 g n (386+j x) \log \left (h (386+j x)^m\right )}{4 e^3 j}-\frac {b d^2 n x^2 \left (f+g \log \left (h (386+j x)^m\right )\right )}{8 e^2}+\frac {b d n x^3 \left (f+g \log \left (h (386+j x)^m\right )\right )}{12 e}-\frac {1}{16} b n x^4 \left (f+g \log \left (h (386+j x)^m\right )\right )-\frac {b d^4 n \log \left (-\frac {j (d+e x)}{386 e-d j}\right ) \left (f+g \log \left (h (386+j x)^m\right )\right )}{4 e^4}+\frac {1}{4} x^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (386+j x)^m\right )\right )-\frac {5549952004 b g m n \text {Li}_2\left (-\frac {j (d+e x)}{386 e-d j}\right )}{j^4}-\frac {b d^4 g m n \text {Li}_2\left (\frac {e (386+j x)}{386 e-d j}\right )}{4 e^4}\\ \end {align*}
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Mathematica [A]
time = 0.71, size = 605, normalized size = 0.82 \begin {gather*} \frac {6 b n \log (d+e x) \left (12 e^4 g i^4 m \log (i+j x)-12 g \left (e^4 i^4-d^4 j^4\right ) m \log \left (\frac {e (i+j x)}{e i-d j}\right )+d j \left (12 e^3 g i^3 m+6 d e^2 g i^2 j m+4 d^2 e g i j^2 m+3 d^3 j^3 (-4 f+g m)-12 d^3 g j^3 \log \left (h (i+j x)^m\right )\right )\right )+e \left (6 g i m \left (-12 a e^3 i^3+b \left (3 e^3 i^3+4 d e^2 i^2 j+6 d^2 e i j^2+12 d^3 j^3\right ) n\right ) \log (i+j x)-6 b e^3 \log \left (c (d+e x)^n\right ) \left (-12 f j^4 x^4+g j m x \left (-12 i^3+6 i^2 j x-4 i j^2 x^2+3 j^3 x^3\right )+12 g i^4 m \log (i+j x)-12 g j^4 x^4 \log \left (h (i+j x)^m\right )\right )+j \left (6 a e^3 x \left (12 f j^3 x^3+g m \left (12 i^3-6 i^2 j x+4 i j^2 x^2-3 j^3 x^3\right )\right )-b n \left (18 d^3 j^3 (-4 f+5 g m) x+3 d^2 e j^2 x (12 f j x+g m (20 i-9 j x))+e^3 x \left (18 f j^3 x^3+g m \left (90 i^3-27 i^2 j x+14 i j^2 x^2-9 j^3 x^3\right )\right )+2 d e^2 \left (-12 f j^3 x^3+g m \left (36 i^3+30 i^2 j x-12 i j^2 x^2+7 j^3 x^3\right )\right )\right )-6 g j^3 x \left (-12 a e^3 x^3+b n \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right ) \log \left (h (i+j x)^m\right )\right )\right )-72 b g \left (e^4 i^4-d^4 j^4\right ) m n \text {Li}_2\left (\frac {j (d+e x)}{-e i+d j}\right )}{288 e^4 j^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 2.70, size = 4217, normalized size = 5.68
method | result | size |
risch | \(\text {Expression too large to display}\) | \(4217\) |
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^3\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )\,\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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