Optimal. Leaf size=105 \[ \frac {1}{3} \left (d^2 x^3+\frac {6 d e x^{r+3}}{r+3}+\frac {3 e^2 x^{2 r+3}}{2 r+3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b d^2 n x^3-\frac {2 b d e n x^{r+3}}{(r+3)^2}-\frac {b e^2 n x^{2 r+3}}{(2 r+3)^2} \]
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Rubi [A] time = 0.16, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {270, 2334, 12, 14} \[ \frac {1}{3} \left (d^2 x^3+\frac {6 d e x^{r+3}}{r+3}+\frac {3 e^2 x^{2 r+3}}{2 r+3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b d^2 n x^3-\frac {2 b d e n x^{r+3}}{(r+3)^2}-\frac {b e^2 n x^{2 r+3}}{(2 r+3)^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 270
Rule 2334
Rubi steps
\begin {align*} \int x^2 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {1}{3} \left (d^2 x^3+\frac {6 d e x^{3+r}}{3+r}+\frac {3 e^2 x^{3+2 r}}{3+2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {1}{3} x^2 \left (d^2+\frac {6 d e x^r}{3+r}+\frac {3 e^2 x^{2 r}}{3+2 r}\right ) \, dx\\ &=\frac {1}{3} \left (d^2 x^3+\frac {6 d e x^{3+r}}{3+r}+\frac {3 e^2 x^{3+2 r}}{3+2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (b n) \int x^2 \left (d^2+\frac {6 d e x^r}{3+r}+\frac {3 e^2 x^{2 r}}{3+2 r}\right ) \, dx\\ &=\frac {1}{3} \left (d^2 x^3+\frac {6 d e x^{3+r}}{3+r}+\frac {3 e^2 x^{3+2 r}}{3+2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} (b n) \int \left (d^2 x^2+\frac {3 e^2 x^{2 (1+r)}}{3+2 r}+\frac {6 d e x^{2+r}}{3+r}\right ) \, dx\\ &=-\frac {1}{9} b d^2 n x^3-\frac {2 b d e n x^{3+r}}{(3+r)^2}-\frac {b e^2 n x^{3+2 r}}{(3+2 r)^2}+\frac {1}{3} \left (d^2 x^3+\frac {6 d e x^{3+r}}{3+r}+\frac {3 e^2 x^{3+2 r}}{3+2 r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.27, size = 124, normalized size = 1.18 \[ \frac {1}{9} x^3 \left (3 a \left (d^2+\frac {6 d e x^r}{r+3}+\frac {3 e^2 x^{2 r}}{2 r+3}\right )+3 b \log \left (c x^n\right ) \left (d^2+\frac {6 d e x^r}{r+3}+\frac {3 e^2 x^{2 r}}{2 r+3}\right )+b n \left (-d^2-\frac {18 d e x^r}{(r+3)^2}-\frac {9 e^2 x^{2 r}}{(2 r+3)^2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 497, normalized size = 4.73 \[ \frac {3 \, {\left (4 \, b d^{2} r^{4} + 36 \, b d^{2} r^{3} + 117 \, b d^{2} r^{2} + 162 \, b d^{2} r + 81 \, b d^{2}\right )} x^{3} \log \relax (c) + 3 \, {\left (4 \, b d^{2} n r^{4} + 36 \, b d^{2} n r^{3} + 117 \, b d^{2} n r^{2} + 162 \, b d^{2} n r + 81 \, b d^{2} n\right )} x^{3} \log \relax (x) - {\left (4 \, {\left (b d^{2} n - 3 \, a d^{2}\right )} r^{4} + 81 \, b d^{2} n + 36 \, {\left (b d^{2} n - 3 \, a d^{2}\right )} r^{3} - 243 \, a d^{2} + 117 \, {\left (b d^{2} n - 3 \, a d^{2}\right )} r^{2} + 162 \, {\left (b d^{2} n - 3 \, a d^{2}\right )} r\right )} x^{3} + 9 \, {\left ({\left (2 \, b e^{2} r^{3} + 15 \, b e^{2} r^{2} + 36 \, b e^{2} r + 27 \, b e^{2}\right )} x^{3} \log \relax (c) + {\left (2 \, b e^{2} n r^{3} + 15 \, b e^{2} n r^{2} + 36 \, b e^{2} n r + 27 \, b e^{2} n\right )} x^{3} \log \relax (x) + {\left (2 \, a e^{2} r^{3} - 9 \, b e^{2} n + 27 \, a e^{2} - {\left (b e^{2} n - 15 \, a e^{2}\right )} r^{2} - 6 \, {\left (b e^{2} n - 6 \, a e^{2}\right )} r\right )} x^{3}\right )} x^{2 \, r} + 18 \, {\left ({\left (4 \, b d e r^{3} + 24 \, b d e r^{2} + 45 \, b d e r + 27 \, b d e\right )} x^{3} \log \relax (c) + {\left (4 \, b d e n r^{3} + 24 \, b d e n r^{2} + 45 \, b d e n r + 27 \, b d e n\right )} x^{3} \log \relax (x) + {\left (4 \, a d e r^{3} - 9 \, b d e n + 27 \, a d e - 4 \, {\left (b d e n - 6 \, a d e\right )} r^{2} - 3 \, {\left (4 \, b d e n - 15 \, a d e\right )} r\right )} x^{3}\right )} x^{r}}{9 \, {\left (4 \, r^{4} + 36 \, r^{3} + 117 \, r^{2} + 162 \, r + 81\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.38, size = 746, normalized size = 7.10 \[ \frac {12 \, b d^{2} n r^{4} x^{3} \log \relax (x) + 72 \, b d n r^{3} x^{3} x^{r} e \log \relax (x) - 4 \, b d^{2} n r^{4} x^{3} + 12 \, b d^{2} r^{4} x^{3} \log \relax (c) + 72 \, b d r^{3} x^{3} x^{r} e \log \relax (c) + 108 \, b d^{2} n r^{3} x^{3} \log \relax (x) + 18 \, b n r^{3} x^{3} x^{2 \, r} e^{2} \log \relax (x) + 432 \, b d n r^{2} x^{3} x^{r} e \log \relax (x) - 36 \, b d^{2} n r^{3} x^{3} + 12 \, a d^{2} r^{4} x^{3} - 72 \, b d n r^{2} x^{3} x^{r} e + 72 \, a d r^{3} x^{3} x^{r} e + 108 \, b d^{2} r^{3} x^{3} \log \relax (c) + 18 \, b r^{3} x^{3} x^{2 \, r} e^{2} \log \relax (c) + 432 \, b d r^{2} x^{3} x^{r} e \log \relax (c) + 351 \, b d^{2} n r^{2} x^{3} \log \relax (x) + 135 \, b n r^{2} x^{3} x^{2 \, r} e^{2} \log \relax (x) + 810 \, b d n r x^{3} x^{r} e \log \relax (x) - 117 \, b d^{2} n r^{2} x^{3} + 108 \, a d^{2} r^{3} x^{3} - 9 \, b n r^{2} x^{3} x^{2 \, r} e^{2} + 18 \, a r^{3} x^{3} x^{2 \, r} e^{2} - 216 \, b d n r x^{3} x^{r} e + 432 \, a d r^{2} x^{3} x^{r} e + 351 \, b d^{2} r^{2} x^{3} \log \relax (c) + 135 \, b r^{2} x^{3} x^{2 \, r} e^{2} \log \relax (c) + 810 \, b d r x^{3} x^{r} e \log \relax (c) + 486 \, b d^{2} n r x^{3} \log \relax (x) + 324 \, b n r x^{3} x^{2 \, r} e^{2} \log \relax (x) + 486 \, b d n x^{3} x^{r} e \log \relax (x) - 162 \, b d^{2} n r x^{3} + 351 \, a d^{2} r^{2} x^{3} - 54 \, b n r x^{3} x^{2 \, r} e^{2} + 135 \, a r^{2} x^{3} x^{2 \, r} e^{2} - 162 \, b d n x^{3} x^{r} e + 810 \, a d r x^{3} x^{r} e + 486 \, b d^{2} r x^{3} \log \relax (c) + 324 \, b r x^{3} x^{2 \, r} e^{2} \log \relax (c) + 486 \, b d x^{3} x^{r} e \log \relax (c) + 243 \, b d^{2} n x^{3} \log \relax (x) + 243 \, b n x^{3} x^{2 \, r} e^{2} \log \relax (x) - 81 \, b d^{2} n x^{3} + 486 \, a d^{2} r x^{3} - 81 \, b n x^{3} x^{2 \, r} e^{2} + 324 \, a r x^{3} x^{2 \, r} e^{2} + 486 \, a d x^{3} x^{r} e + 243 \, b d^{2} x^{3} \log \relax (c) + 243 \, b x^{3} x^{2 \, r} e^{2} \log \relax (c) + 243 \, a d^{2} x^{3} + 243 \, a x^{3} x^{2 \, r} e^{2}}{9 \, {\left (4 \, r^{4} + 36 \, r^{3} + 117 \, r^{2} + 162 \, r + 81\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.36, size = 1930, normalized size = 18.38 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 152, normalized size = 1.45 \[ -\frac {1}{9} \, b d^{2} n x^{3} + \frac {1}{3} \, b d^{2} x^{3} \log \left (c x^{n}\right ) + \frac {1}{3} \, a d^{2} x^{3} + \frac {b e^{2} x^{2 \, r + 3} \log \left (c x^{n}\right )}{2 \, r + 3} + \frac {2 \, b d e x^{r + 3} \log \left (c x^{n}\right )}{r + 3} - \frac {b e^{2} n x^{2 \, r + 3}}{{\left (2 \, r + 3\right )}^{2}} + \frac {a e^{2} x^{2 \, r + 3}}{2 \, r + 3} - \frac {2 \, b d e n x^{r + 3}}{{\left (r + 3\right )}^{2}} + \frac {2 \, a d e x^{r + 3}}{r + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\left (d+e\,x^r\right )}^2\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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