Optimal. Leaf size=258 \[ \frac {d^3 n \log (d+e x) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )}{3 e^3}-\frac {d^2 n (d+e x) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )}{e^3}+\frac {d n (d+e x)^2 \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )}{2 e^3}-\frac {n (d+e x)^3 \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )}{9 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )-\frac {b d^3 g n^2 \log ^2(d+e x)}{3 e^3}+\frac {2 b d^2 g n^2 x}{e^2}-\frac {b d g n^2 (d+e x)^2}{2 e^3}+\frac {2 b g n^2 (d+e x)^3}{27 e^3} \]
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Rubi [A] time = 0.44, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2439, 2411, 43, 2334, 12, 14, 2301} \[ -\frac {1}{18} g n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )-\frac {1}{18} b n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )+\frac {2 b d^2 g n^2 x}{e^2}-\frac {b d^3 g n^2 \log ^2(d+e x)}{3 e^3}-\frac {b d g n^2 (d+e x)^2}{2 e^3}+\frac {2 b g n^2 (d+e x)^3}{27 e^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 43
Rule 2301
Rule 2334
Rule 2411
Rule 2439
Rubi steps
\begin {align*} \int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right ) \, dx &=\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-\frac {1}{3} (b e n) \int \frac {x^3 \left (f+g \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx-\frac {1}{3} (e g n) \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx\\ &=\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-\frac {1}{3} (b n) \operatorname {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right )^3 \left (f+g \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )-\frac {1}{3} (g n) \operatorname {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )\\ &=-\frac {1}{18} g n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {1}{18} b n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+2 \left (\frac {1}{3} \left (b g n^2\right ) \operatorname {Subst}\left (\int \frac {18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{6 e^3 x} \, dx,x,d+e x\right )\right )\\ &=-\frac {1}{18} g n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {1}{18} b n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+2 \frac {\left (b g n^2\right ) \operatorname {Subst}\left (\int \frac {18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{x} \, dx,x,d+e x\right )}{18 e^3}\\ &=-\frac {1}{18} g n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {1}{18} b n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+2 \frac {\left (b g n^2\right ) \operatorname {Subst}\left (\int \left (18 d^2-9 d x+2 x^2-\frac {6 d^3 \log (x)}{x}\right ) \, dx,x,d+e x\right )}{18 e^3}\\ &=-\frac {1}{18} g n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {1}{18} b n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+2 \left (\frac {b d^2 g n^2 x}{e^2}-\frac {b d g n^2 (d+e x)^2}{4 e^3}+\frac {b g n^2 (d+e x)^3}{27 e^3}-\frac {\left (b d^3 g n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+e x\right )}{3 e^3}\right )\\ &=2 \left (\frac {b d^2 g n^2 x}{e^2}-\frac {b d g n^2 (d+e x)^2}{4 e^3}+\frac {b g n^2 (d+e x)^3}{27 e^3}-\frac {b d^3 g n^2 \log ^2(d+e x)}{6 e^3}\right )-\frac {1}{18} g n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {1}{18} b n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 342, normalized size = 1.33 \[ \frac {1}{3} a g x^3 \log \left (c (d+e x)^n\right )+\frac {a d^3 g n \log (d+e x)}{3 e^3}-\frac {a d^2 g n x}{3 e^2}+\frac {a d g n x^2}{6 e}+\frac {1}{3} a f x^3-\frac {1}{9} a g n x^3+\frac {b d^3 g \log ^2\left (c (d+e x)^n\right )}{3 e^3}-\frac {11 b d^3 g n \log \left (c (d+e x)^n\right )}{9 e^3}-\frac {2 b d^2 g n x \log \left (c (d+e x)^n\right )}{3 e^2}+\frac {1}{3} b f x^3 \log \left (c (d+e x)^n\right )+\frac {1}{3} b g x^3 \log ^2\left (c (d+e x)^n\right )-\frac {2}{9} b g n x^3 \log \left (c (d+e x)^n\right )+\frac {b d g n x^2 \log \left (c (d+e x)^n\right )}{3 e}+\frac {b d^3 f n \log (d+e x)}{3 e^3}-\frac {b d^2 f n x}{3 e^2}+\frac {11 b d^2 g n^2 x}{9 e^2}+\frac {b d f n x^2}{6 e}-\frac {5 b d g n^2 x^2}{18 e}-\frac {1}{9} b f n x^3+\frac {2}{27} b g n^2 x^3 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 329, normalized size = 1.28 \[ \frac {18 \, b e^{3} g x^{3} \log \relax (c)^{2} + 2 \, {\left (2 \, b e^{3} g n^{2} + 9 \, a e^{3} f - 3 \, {\left (b e^{3} f + a e^{3} g\right )} n\right )} x^{3} - 3 \, {\left (5 \, b d e^{2} g n^{2} - 3 \, {\left (b d e^{2} f + a d e^{2} g\right )} n\right )} x^{2} + 18 \, {\left (b e^{3} g n^{2} x^{3} + b d^{3} g n^{2}\right )} \log \left (e x + d\right )^{2} + 6 \, {\left (11 \, b d^{2} e g n^{2} - 3 \, {\left (b d^{2} e f + a d^{2} e g\right )} n\right )} x + 6 \, {\left (3 \, b d e^{2} g n^{2} x^{2} - 6 \, b d^{2} e g n^{2} x - 11 \, b d^{3} g n^{2} - {\left (2 \, b e^{3} g n^{2} - 3 \, {\left (b e^{3} f + a e^{3} g\right )} n\right )} x^{3} + 3 \, {\left (b d^{3} f + a d^{3} g\right )} n + 6 \, {\left (b e^{3} g n x^{3} + b d^{3} g n\right )} \log \relax (c)\right )} \log \left (e x + d\right ) + 6 \, {\left (3 \, b d e^{2} g n x^{2} - 6 \, b d^{2} e g n x - {\left (2 \, b e^{3} g n - 3 \, b e^{3} f - 3 \, a e^{3} g\right )} x^{3}\right )} \log \relax (c)}{54 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 756, normalized size = 2.93 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.53, size = 1785, normalized size = 6.92 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 274, normalized size = 1.06 \[ \frac {1}{3} \, b g x^{3} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + \frac {1}{3} \, b f x^{3} \log \left ({\left (e x + d\right )}^{n} c\right ) + \frac {1}{3} \, a g x^{3} \log \left ({\left (e x + d\right )}^{n} c\right ) + \frac {1}{3} \, a f x^{3} + \frac {1}{18} \, b e f n {\left (\frac {6 \, d^{3} \log \left (e x + d\right )}{e^{4}} - \frac {2 \, e^{2} x^{3} - 3 \, d e x^{2} + 6 \, d^{2} x}{e^{3}}\right )} + \frac {1}{18} \, a e g n {\left (\frac {6 \, d^{3} \log \left (e x + d\right )}{e^{4}} - \frac {2 \, e^{2} x^{3} - 3 \, d e x^{2} + 6 \, d^{2} x}{e^{3}}\right )} + \frac {1}{54} \, {\left (6 \, e n {\left (\frac {6 \, d^{3} \log \left (e x + d\right )}{e^{4}} - \frac {2 \, e^{2} x^{3} - 3 \, d e x^{2} + 6 \, d^{2} x}{e^{3}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right ) + \frac {{\left (4 \, e^{3} x^{3} - 15 \, d e^{2} x^{2} - 18 \, d^{3} \log \left (e x + d\right )^{2} + 66 \, d^{2} e x - 66 \, d^{3} \log \left (e x + d\right )\right )} n^{2}}{e^{3}}\right )} b g \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 323, normalized size = 1.25 \[ \ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (\frac {x^3\,\left (a\,g+b\,f-\frac {2\,b\,g\,n}{3}\right )}{3}+\frac {x^2\,\left (\frac {3\,d\,\left (a\,g+b\,f\right )}{2\,e}-\frac {d\,\left (9\,a\,g+9\,b\,f-6\,b\,g\,n\right )}{6\,e}\right )}{3}-\frac {d\,x\,\left (\frac {9\,d\,\left (a\,g+b\,f\right )}{e}-\frac {d\,\left (9\,a\,g+9\,b\,f-6\,b\,g\,n\right )}{e}\right )}{9\,e}\right )+x^2\,\left (\frac {d\,\left (3\,a\,f-b\,g\,n^2\right )}{6\,e}-\frac {d\,\left (a\,f-\frac {a\,g\,n}{3}-\frac {b\,f\,n}{3}+\frac {2\,b\,g\,n^2}{9}\right )}{2\,e}\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2\,\left (\frac {b\,g\,x^3}{3}+\frac {b\,d^3\,g}{3\,e^3}\right )-x\,\left (\frac {d\,\left (\frac {d\,\left (3\,a\,f-b\,g\,n^2\right )}{3\,e}-\frac {d\,\left (a\,f-\frac {a\,g\,n}{3}-\frac {b\,f\,n}{3}+\frac {2\,b\,g\,n^2}{9}\right )}{e}\right )}{e}-\frac {2\,b\,d^2\,g\,n^2}{3\,e^2}\right )+x^3\,\left (\frac {a\,f}{3}-\frac {a\,g\,n}{9}-\frac {b\,f\,n}{9}+\frac {2\,b\,g\,n^2}{27}\right )+\frac {\ln \left (d+e\,x\right )\,\left (3\,a\,d^3\,g\,n+3\,b\,d^3\,f\,n-11\,b\,d^3\,g\,n^2\right )}{9\,e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.63, size = 508, normalized size = 1.97 \[ \begin {cases} \frac {a d^{3} g n \log {\left (d + e x \right )}}{3 e^{3}} - \frac {a d^{2} g n x}{3 e^{2}} + \frac {a d g n x^{2}}{6 e} + \frac {a f x^{3}}{3} + \frac {a g n x^{3} \log {\left (d + e x \right )}}{3} - \frac {a g n x^{3}}{9} + \frac {a g x^{3} \log {\relax (c )}}{3} + \frac {b d^{3} f n \log {\left (d + e x \right )}}{3 e^{3}} + \frac {b d^{3} g n^{2} \log {\left (d + e x \right )}^{2}}{3 e^{3}} - \frac {11 b d^{3} g n^{2} \log {\left (d + e x \right )}}{9 e^{3}} + \frac {2 b d^{3} g n \log {\relax (c )} \log {\left (d + e x \right )}}{3 e^{3}} - \frac {b d^{2} f n x}{3 e^{2}} - \frac {2 b d^{2} g n^{2} x \log {\left (d + e x \right )}}{3 e^{2}} + \frac {11 b d^{2} g n^{2} x}{9 e^{2}} - \frac {2 b d^{2} g n x \log {\relax (c )}}{3 e^{2}} + \frac {b d f n x^{2}}{6 e} + \frac {b d g n^{2} x^{2} \log {\left (d + e x \right )}}{3 e} - \frac {5 b d g n^{2} x^{2}}{18 e} + \frac {b d g n x^{2} \log {\relax (c )}}{3 e} + \frac {b f n x^{3} \log {\left (d + e x \right )}}{3} - \frac {b f n x^{3}}{9} + \frac {b f x^{3} \log {\relax (c )}}{3} + \frac {b g n^{2} x^{3} \log {\left (d + e x \right )}^{2}}{3} - \frac {2 b g n^{2} x^{3} \log {\left (d + e x \right )}}{9} + \frac {2 b g n^{2} x^{3}}{27} + \frac {2 b g n x^{3} \log {\relax (c )} \log {\left (d + e x \right )}}{3} - \frac {2 b g n x^{3} \log {\relax (c )}}{9} + \frac {b g x^{3} \log {\relax (c )}^{2}}{3} & \text {for}\: e \neq 0 \\\frac {x^{3} \left (a + b \log {\left (c d^{n} \right )}\right ) \left (f + g \log {\left (c d^{n} \right )}\right )}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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