Integrand size = 23, antiderivative size = 191 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {b d^3 n}{9 x^3}-\frac {b e^3 n x^{-3 (1-r)}}{9 (1-r)^2}-\frac {3 b d^2 e n x^{-3+r}}{(3-r)^2}-\frac {3 b d e^2 n x^{-3+2 r}}{(3-2 r)^2}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {e^3 x^{-3 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{3 (1-r)}-\frac {3 d^2 e x^{-3+r} \left (a+b \log \left (c x^n\right )\right )}{3-r}-\frac {3 d e^2 x^{-3+2 r} \left (a+b \log \left (c x^n\right )\right )}{3-2 r} \]
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Time = 0.26 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {276, 2372, 12, 14} \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {3 d^2 e x^{r-3} \left (a+b \log \left (c x^n\right )\right )}{3-r}-\frac {3 d e^2 x^{2 r-3} \left (a+b \log \left (c x^n\right )\right )}{3-2 r}-\frac {e^3 x^{-3 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{3 (1-r)}-\frac {b d^3 n}{9 x^3}-\frac {3 b d^2 e n x^{r-3}}{(3-r)^2}-\frac {3 b d e^2 n x^{2 r-3}}{(3-2 r)^2}-\frac {b e^3 n x^{-3 (1-r)}}{9 (1-r)^2} \]
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Rule 12
Rule 14
Rule 276
Rule 2372
Rubi steps \begin{align*} \text {integral}& = -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {e^3 x^{-3 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{3 (1-r)}-\frac {3 d^2 e x^{-3+r} \left (a+b \log \left (c x^n\right )\right )}{3-r}-\frac {3 d e^2 x^{-3+2 r} \left (a+b \log \left (c x^n\right )\right )}{3-2 r}-(b n) \int \frac {-d^3+\frac {9 d^2 e x^r}{-3+r}+\frac {9 d e^2 x^{2 r}}{-3+2 r}+\frac {e^3 x^{3 r}}{-1+r}}{3 x^4} \, dx \\ & = -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {e^3 x^{-3 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{3 (1-r)}-\frac {3 d^2 e x^{-3+r} \left (a+b \log \left (c x^n\right )\right )}{3-r}-\frac {3 d e^2 x^{-3+2 r} \left (a+b \log \left (c x^n\right )\right )}{3-2 r}-\frac {1}{3} (b n) \int \frac {-d^3+\frac {9 d^2 e x^r}{-3+r}+\frac {9 d e^2 x^{2 r}}{-3+2 r}+\frac {e^3 x^{3 r}}{-1+r}}{x^4} \, dx \\ & = -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {e^3 x^{-3 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{3 (1-r)}-\frac {3 d^2 e x^{-3+r} \left (a+b \log \left (c x^n\right )\right )}{3-r}-\frac {3 d e^2 x^{-3+2 r} \left (a+b \log \left (c x^n\right )\right )}{3-2 r}-\frac {1}{3} (b n) \int \left (-\frac {d^3}{x^4}+\frac {9 d^2 e x^{-4+r}}{-3+r}+\frac {9 d e^2 x^{2 (-2+r)}}{-3+2 r}+\frac {e^3 x^{-4+3 r}}{-1+r}\right ) \, dx \\ & = -\frac {b d^3 n}{9 x^3}-\frac {b e^3 n x^{-3 (1-r)}}{9 (1-r)^2}-\frac {3 b d^2 e n x^{-3+r}}{(3-r)^2}-\frac {3 b d e^2 n x^{-3+2 r}}{(3-2 r)^2}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {e^3 x^{-3 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{3 (1-r)}-\frac {3 d^2 e x^{-3+r} \left (a+b \log \left (c x^n\right )\right )}{3-r}-\frac {3 d e^2 x^{-3+2 r} \left (a+b \log \left (c x^n\right )\right )}{3-2 r} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.94 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\frac {b n \left (-d^3-\frac {27 d^2 e x^r}{(-3+r)^2}-\frac {27 d e^2 x^{2 r}}{(3-2 r)^2}-\frac {e^3 x^{3 r}}{(-1+r)^2}\right )+3 a \left (-d^3+\frac {9 d^2 e x^r}{-3+r}+\frac {9 d e^2 x^{2 r}}{-3+2 r}+\frac {e^3 x^{3 r}}{-1+r}\right )+3 b \left (-d^3+\frac {9 d^2 e x^r}{-3+r}+\frac {9 d e^2 x^{2 r}}{-3+2 r}+\frac {e^3 x^{3 r}}{-1+r}\right ) \log \left (c x^n\right )}{9 x^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1038\) vs. \(2(183)=366\).
Time = 3.47 (sec) , antiderivative size = 1039, normalized size of antiderivative = 5.44
method | result | size |
parallelrisch | \(\text {Expression too large to display}\) | \(1039\) |
risch | \(\text {Expression too large to display}\) | \(4027\) |
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Leaf count of result is larger than twice the leaf count of optimal. 980 vs. \(2 (175) = 350\).
Time = 0.34 (sec) , antiderivative size = 980, normalized size of antiderivative = 5.13 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {4 \, {\left (b d^{3} n + 3 \, a d^{3}\right )} r^{6} - 44 \, {\left (b d^{3} n + 3 \, a d^{3}\right )} r^{5} + 81 \, b d^{3} n + 193 \, {\left (b d^{3} n + 3 \, a d^{3}\right )} r^{4} + 243 \, a d^{3} - 432 \, {\left (b d^{3} n + 3 \, a d^{3}\right )} r^{3} + 522 \, {\left (b d^{3} n + 3 \, a d^{3}\right )} r^{2} - 324 \, {\left (b d^{3} n + 3 \, a d^{3}\right )} r - {\left (12 \, a e^{3} r^{5} - 81 \, b e^{3} n - 4 \, {\left (b e^{3} n + 30 \, a e^{3}\right )} r^{4} - 243 \, a e^{3} + 9 \, {\left (4 \, b e^{3} n + 51 \, a e^{3}\right )} r^{3} - 9 \, {\left (13 \, b e^{3} n + 93 \, a e^{3}\right )} r^{2} + 81 \, {\left (2 \, b e^{3} n + 9 \, a e^{3}\right )} r + 3 \, {\left (4 \, b e^{3} r^{5} - 40 \, b e^{3} r^{4} + 153 \, b e^{3} r^{3} - 279 \, b e^{3} r^{2} + 243 \, b e^{3} r - 81 \, b e^{3}\right )} \log \left (c\right ) + 3 \, {\left (4 \, b e^{3} n r^{5} - 40 \, b e^{3} n r^{4} + 153 \, b e^{3} n r^{3} - 279 \, b e^{3} n r^{2} + 243 \, b e^{3} n r - 81 \, b e^{3} n\right )} \log \left (x\right )\right )} x^{3 \, r} - 27 \, {\left (2 \, a d e^{2} r^{5} - 9 \, b d e^{2} n - {\left (b d e^{2} n + 19 \, a d e^{2}\right )} r^{4} - 27 \, a d e^{2} + 4 \, {\left (2 \, b d e^{2} n + 17 \, a d e^{2}\right )} r^{3} - 2 \, {\left (11 \, b d e^{2} n + 57 \, a d e^{2}\right )} r^{2} + 6 \, {\left (4 \, b d e^{2} n + 15 \, a d e^{2}\right )} r + {\left (2 \, b d e^{2} r^{5} - 19 \, b d e^{2} r^{4} + 68 \, b d e^{2} r^{3} - 114 \, b d e^{2} r^{2} + 90 \, b d e^{2} r - 27 \, b d e^{2}\right )} \log \left (c\right ) + {\left (2 \, b d e^{2} n r^{5} - 19 \, b d e^{2} n r^{4} + 68 \, b d e^{2} n r^{3} - 114 \, b d e^{2} n r^{2} + 90 \, b d e^{2} n r - 27 \, b d e^{2} n\right )} \log \left (x\right )\right )} x^{2 \, r} - 27 \, {\left (4 \, a d^{2} e r^{5} - 9 \, b d^{2} e n - 4 \, {\left (b d^{2} e n + 8 \, a d^{2} e\right )} r^{4} - 27 \, a d^{2} e + {\left (20 \, b d^{2} e n + 97 \, a d^{2} e\right )} r^{3} - {\left (37 \, b d^{2} e n + 141 \, a d^{2} e\right )} r^{2} + 3 \, {\left (10 \, b d^{2} e n + 33 \, a d^{2} e\right )} r + {\left (4 \, b d^{2} e r^{5} - 32 \, b d^{2} e r^{4} + 97 \, b d^{2} e r^{3} - 141 \, b d^{2} e r^{2} + 99 \, b d^{2} e r - 27 \, b d^{2} e\right )} \log \left (c\right ) + {\left (4 \, b d^{2} e n r^{5} - 32 \, b d^{2} e n r^{4} + 97 \, b d^{2} e n r^{3} - 141 \, b d^{2} e n r^{2} + 99 \, b d^{2} e n r - 27 \, b d^{2} e n\right )} \log \left (x\right )\right )} x^{r} + 3 \, {\left (4 \, b d^{3} r^{6} - 44 \, b d^{3} r^{5} + 193 \, b d^{3} r^{4} - 432 \, b d^{3} r^{3} + 522 \, b d^{3} r^{2} - 324 \, b d^{3} r + 81 \, b d^{3}\right )} \log \left (c\right ) + 3 \, {\left (4 \, b d^{3} n r^{6} - 44 \, b d^{3} n r^{5} + 193 \, b d^{3} n r^{4} - 432 \, b d^{3} n r^{3} + 522 \, b d^{3} n r^{2} - 324 \, b d^{3} n r + 81 \, b d^{3} n\right )} \log \left (x\right )}{9 \, {\left (4 \, r^{6} - 44 \, r^{5} + 193 \, r^{4} - 432 \, r^{3} + 522 \, r^{2} - 324 \, r + 81\right )} x^{3}} \]
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Time = 49.68 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.82 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=- \frac {a d^{3}}{3 x^{3}} + 3 a d^{2} e \left (\begin {cases} \frac {x^{r}}{r x^{3} - 3 x^{3}} & \text {for}\: r \neq 3 \\\frac {x^{r} \log {\left (x \right )}}{x^{3}} & \text {otherwise} \end {cases}\right ) + 3 a d e^{2} \left (\begin {cases} \frac {x^{2 r}}{2 r x^{3} - 3 x^{3}} & \text {for}\: r \neq \frac {3}{2} \\\frac {x^{2 r} \log {\left (x \right )}}{x^{3}} & \text {otherwise} \end {cases}\right ) + a e^{3} \left (\begin {cases} \frac {x^{3 r}}{3 r x^{3} - 3 x^{3}} & \text {for}\: r \neq 1 \\\frac {x^{3 r} \log {\left (x \right )}}{x^{3}} & \text {otherwise} \end {cases}\right ) - \frac {b d^{3} n}{9 x^{3}} - \frac {b d^{3} \log {\left (c x^{n} \right )}}{3 x^{3}} - 3 b d^{2} e n \left (\begin {cases} \frac {\begin {cases} \frac {x^{r - 3}}{r - 3} & \text {for}\: r \neq 3 \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{r - 3} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq 3 \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + 3 b d^{2} e \left (\begin {cases} \frac {x^{r - 3}}{r - 3} & \text {for}\: r \neq 3 \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} - 3 b d e^{2} n \left (\begin {cases} \frac {\begin {cases} \frac {x^{2 r - 3}}{2 r - 3} & \text {for}\: r \neq \frac {3}{2} \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{2 r - 3} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq \frac {3}{2} \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + 3 b d e^{2} \left (\begin {cases} \frac {x^{2 r - 3}}{2 r - 3} & \text {for}\: r \neq \frac {3}{2} \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} - b e^{3} n \left (\begin {cases} \frac {\begin {cases} \frac {x^{3 r - 3}}{3 r - 3} & \text {for}\: r \neq 1 \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{3 r - 3} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq 1 \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + b e^{3} \left (\begin {cases} \frac {x^{3 r - 3}}{3 r - 3} & \text {for}\: r \neq 1 \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \]
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Exception generated. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\int { \frac {{\left (e x^{r} + d\right )}^{3} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\int \frac {{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^4} \,d x \]
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