Optimal. Leaf size=191 \[ -\frac {b d^3 n}{9 x^3}-\frac {b e^3 n x^{-3 (1-r)}}{9 (1-r)^2}-\frac {3 b d^2 e n x^{-3+r}}{(3-r)^2}-\frac {3 b d e^2 n x^{-3+2 r}}{(3-2 r)^2}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {e^3 x^{-3 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{3 (1-r)}-\frac {3 d^2 e x^{-3+r} \left (a+b \log \left (c x^n\right )\right )}{3-r}-\frac {3 d e^2 x^{-3+2 r} \left (a+b \log \left (c x^n\right )\right )}{3-2 r} \]
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Rubi [A]
time = 0.27, antiderivative size = 191, 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^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} \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 )^3 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx &=-\frac {1}{3} \left (\frac {d^3}{x^3}+\frac {e^3 x^{-3 (1-r)}}{1-r}+\frac {9 d^2 e x^{-3+r}}{3-r}+\frac {9 d e^2 x^{-3+2 r}}{3-2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(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 {1}{3} \left (\frac {d^3}{x^3}+\frac {e^3 x^{-3 (1-r)}}{1-r}+\frac {9 d^2 e x^{-3+r}}{3-r}+\frac {9 d 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 \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 {1}{3} \left (\frac {d^3}{x^3}+\frac {e^3 x^{-3 (1-r)}}{1-r}+\frac {9 d^2 e x^{-3+r}}{3-r}+\frac {9 d 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 (-\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 {1}{3} \left (\frac {d^3}{x^3}+\frac {e^3 x^{-3 (1-r)}}{1-r}+\frac {9 d^2 e x^{-3+r}}{3-r}+\frac {9 d 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 = 0.16, size = 159, normalized size = 0.83 \begin {gather*} \frac {-3 b d^3 n \log (x)-d^3 \left (3 a+b n-3 b n \log (x)+3 b \log \left (c x^n\right )\right )+\frac {27 d^2 e x^r \left (-b n+a (-3+r)+b (-3+r) \log \left (c x^n\right )\right )}{(-3+r)^2}+\frac {e^3 x^{3 r} \left (-b n+3 a (-1+r)+3 b (-1+r) \log \left (c x^n\right )\right )}{(-1+r)^2}+\frac {27 d e^2 x^{2 r} \left (-b n+a (-3+2 r)+b (-3+2 r) \log \left (c x^n\right )\right )}{(3-2 r)^2}}{9 x^3} \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 = 0.32, size = 4027, normalized size = 21.08
method | result | size |
risch | \(\text {Expression too large to display}\) | \(4027\) |
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. 843 vs.
\(2 (173) = 346\).
time = 0.35, size = 843, normalized size = 4.41 \begin {gather*} -\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 (3 \, {\left (4 \, b r^{5} - 40 \, b r^{4} + 153 \, b r^{3} - 279 \, b r^{2} + 243 \, b r - 81 \, b\right )} e^{3} \log \left (c\right ) + 3 \, {\left (4 \, b n r^{5} - 40 \, b n r^{4} + 153 \, b n r^{3} - 279 \, b n r^{2} + 243 \, b n r - 81 \, b n\right )} e^{3} \log \left (x\right ) + {\left (12 \, a r^{5} - 4 \, {\left (b n + 30 \, a\right )} r^{4} + 9 \, {\left (4 \, b n + 51 \, a\right )} r^{3} - 9 \, {\left (13 \, b n + 93 \, a\right )} r^{2} - 81 \, b n + 81 \, {\left (2 \, b n + 9 \, a\right )} r - 243 \, a\right )} e^{3}\right )} x^{3 \, r} - 27 \, {\left ({\left (2 \, b d r^{5} - 19 \, b d r^{4} + 68 \, b d r^{3} - 114 \, b d r^{2} + 90 \, b d r - 27 \, b d\right )} e^{2} \log \left (c\right ) + {\left (2 \, b d n r^{5} - 19 \, b d n r^{4} + 68 \, b d n r^{3} - 114 \, b d n r^{2} + 90 \, b d n r - 27 \, b d n\right )} e^{2} \log \left (x\right ) + {\left (2 \, a d r^{5} - {\left (b d n + 19 \, a d\right )} r^{4} + 4 \, {\left (2 \, b d n + 17 \, a d\right )} r^{3} - 9 \, b d n - 2 \, {\left (11 \, b d n + 57 \, a d\right )} r^{2} - 27 \, a d + 6 \, {\left (4 \, b d n + 15 \, a d\right )} r\right )} e^{2}\right )} x^{2 \, r} - 27 \, {\left ({\left (4 \, b d^{2} r^{5} - 32 \, b d^{2} r^{4} + 97 \, b d^{2} r^{3} - 141 \, b d^{2} r^{2} + 99 \, b d^{2} r - 27 \, b d^{2}\right )} e \log \left (c\right ) + {\left (4 \, b d^{2} n r^{5} - 32 \, b d^{2} n r^{4} + 97 \, b d^{2} n r^{3} - 141 \, b d^{2} n r^{2} + 99 \, b d^{2} n r - 27 \, b d^{2} n\right )} e \log \left (x\right ) + {\left (4 \, a d^{2} r^{5} - 4 \, {\left (b d^{2} n + 8 \, a d^{2}\right )} r^{4} - 9 \, b d^{2} n + {\left (20 \, b d^{2} n + 97 \, a d^{2}\right )} r^{3} - 27 \, a d^{2} - {\left (37 \, b d^{2} n + 141 \, a d^{2}\right )} r^{2} + 3 \, {\left (10 \, b d^{2} n + 33 \, a d^{2}\right )} r\right )} e\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 88.39, size = 338, normalized size = 1.77 \begin {gather*} - \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 \\\log {\left (x \right )} & \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} \\\log {\left (x \right )} & \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 \\\log {\left (x \right )} & \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}}{r x^{3} - 3 x^{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}}{2 r x^{3} - 3 x^{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 r x^{3} - 3 x^{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 )} \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 )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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