Optimal. Leaf size=127 \[ -\frac {b d^2 n}{25 x^5}-\frac {2 b d e n x^{-5+r}}{(5-r)^2}-\frac {b e^2 n x^{-5+2 r}}{(5-2 r)^2}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {2 d e x^{-5+r} \left (a+b \log \left (c x^n\right )\right )}{5-r}-\frac {e^2 x^{-5+2 r} \left (a+b \log \left (c x^n\right )\right )}{5-2 r} \]
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Rubi [A]
time = 0.12, antiderivative size = 127, 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 = {276, 2372, 12,
14} \begin {gather*} -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {2 d e x^{r-5} \left (a+b \log \left (c x^n\right )\right )}{5-r}-\frac {e^2 x^{2 r-5} \left (a+b \log \left (c x^n\right )\right )}{5-2 r}-\frac {b d^2 n}{25 x^5}-\frac {2 b d e n x^{r-5}}{(5-r)^2}-\frac {b e^2 n x^{2 r-5}}{(5-2 r)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 276
Rule 2372
Rubi steps
\begin {align*} \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx &=-\frac {1}{5} \left (\frac {d^2}{x^5}+\frac {10 d e x^{-5+r}}{5-r}+\frac {5 e^2 x^{-5+2 r}}{5-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-d^2+\frac {10 d e x^r}{-5+r}+\frac {5 e^2 x^{2 r}}{-5+2 r}}{5 x^6} \, dx\\ &=-\frac {1}{5} \left (\frac {d^2}{x^5}+\frac {10 d e x^{-5+r}}{5-r}+\frac {5 e^2 x^{-5+2 r}}{5-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{5} (b n) \int \frac {-d^2+\frac {10 d e x^r}{-5+r}+\frac {5 e^2 x^{2 r}}{-5+2 r}}{x^6} \, dx\\ &=-\frac {1}{5} \left (\frac {d^2}{x^5}+\frac {10 d e x^{-5+r}}{5-r}+\frac {5 e^2 x^{-5+2 r}}{5-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{5} (b n) \int \left (-\frac {d^2}{x^6}+\frac {10 d e x^{-6+r}}{-5+r}+\frac {5 e^2 x^{2 (-3+r)}}{-5+2 r}\right ) \, dx\\ &=-\frac {b d^2 n}{25 x^5}-\frac {2 b d e n x^{-5+r}}{(5-r)^2}-\frac {b e^2 n x^{-5+2 r}}{(5-2 r)^2}-\frac {1}{5} \left (\frac {d^2}{x^5}+\frac {10 d e x^{-5+r}}{5-r}+\frac {5 e^2 x^{-5+2 r}}{5-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 119, normalized size = 0.94 \begin {gather*} \frac {-5 b d^2 n \log (x)-d^2 \left (5 a+b n-5 b n \log (x)+5 b \log \left (c x^n\right )\right )+\frac {50 d e x^r \left (-b n+a (-5+r)+b (-5+r) \log \left (c x^n\right )\right )}{(-5+r)^2}+\frac {25 e^2 x^{2 r} \left (-b n+a (-5+2 r)+b (-5+2 r) \log \left (c x^n\right )\right )}{(5-2 r)^2}}{25 x^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.23, size = 1930, normalized size = 15.20
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1930\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 422 vs.
\(2 (118) = 236\).
time = 0.36, size = 422, normalized size = 3.32 \begin {gather*} -\frac {4 \, {\left (b d^{2} n + 5 \, a d^{2}\right )} r^{4} + 625 \, b d^{2} n - 60 \, {\left (b d^{2} n + 5 \, a d^{2}\right )} r^{3} + 3125 \, a d^{2} + 325 \, {\left (b d^{2} n + 5 \, a d^{2}\right )} r^{2} - 750 \, {\left (b d^{2} n + 5 \, a d^{2}\right )} r - 25 \, {\left ({\left (2 \, b r^{3} - 25 \, b r^{2} + 100 \, b r - 125 \, b\right )} e^{2} \log \left (c\right ) + {\left (2 \, b n r^{3} - 25 \, b n r^{2} + 100 \, b n r - 125 \, b n\right )} e^{2} \log \left (x\right ) + {\left (2 \, a r^{3} - {\left (b n + 25 \, a\right )} r^{2} - 25 \, b n + 10 \, {\left (b n + 10 \, a\right )} r - 125 \, a\right )} e^{2}\right )} x^{2 \, r} - 50 \, {\left ({\left (4 \, b d r^{3} - 40 \, b d r^{2} + 125 \, b d r - 125 \, b d\right )} e \log \left (c\right ) + {\left (4 \, b d n r^{3} - 40 \, b d n r^{2} + 125 \, b d n r - 125 \, b d n\right )} e \log \left (x\right ) + {\left (4 \, a d r^{3} - 25 \, b d n - 4 \, {\left (b d n + 10 \, a d\right )} r^{2} - 125 \, a d + 5 \, {\left (4 \, b d n + 25 \, a d\right )} r\right )} e\right )} x^{r} + 5 \, {\left (4 \, b d^{2} r^{4} - 60 \, b d^{2} r^{3} + 325 \, b d^{2} r^{2} - 750 \, b d^{2} r + 625 \, b d^{2}\right )} \log \left (c\right ) + 5 \, {\left (4 \, b d^{2} n r^{4} - 60 \, b d^{2} n r^{3} + 325 \, b d^{2} n r^{2} - 750 \, b d^{2} n r + 625 \, b d^{2} n\right )} \log \left (x\right )}{25 \, {\left (4 \, r^{4} - 60 \, r^{3} + 325 \, r^{2} - 750 \, r + 625\right )} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = -0.01 \begin {gather*} \int \frac {{\left (d+e\,x^r\right )}^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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