Optimal. Leaf size=229 \[ \frac {10 b d n x}{e^5}-\frac {d (60 a+47 b n) x}{6 e^5}-\frac {5 b n x^2}{2 e^4}-\frac {10 b d x \log \left (c x^n\right )}{e^5}-\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}-\frac {x^4 \left (5 a+b n+5 b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^2}-\frac {x^3 \left (20 a+9 b n+20 b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)}+\frac {x^2 \left (60 a+47 b n+60 b \log \left (c x^n\right )\right )}{12 e^4}+\frac {d^2 \left (60 a+47 b n+60 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{6 e^6}+\frac {10 b d^2 n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^6} \]
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Rubi [A]
time = 0.28, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2384, 45,
2393, 2332, 2341, 2354, 2438} \begin {gather*} \frac {10 b d^2 n \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^6}+\frac {d^2 \log \left (\frac {e x}{d}+1\right ) \left (60 a+60 b \log \left (c x^n\right )+47 b n\right )}{6 e^6}-\frac {x^3 \left (20 a+20 b \log \left (c x^n\right )+9 b n\right )}{6 e^3 (d+e x)}-\frac {x^4 \left (5 a+5 b \log \left (c x^n\right )+b n\right )}{6 e^2 (d+e x)^2}-\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}+\frac {x^2 \left (60 a+60 b \log \left (c x^n\right )+47 b n\right )}{12 e^4}-\frac {d x (60 a+47 b n)}{6 e^5}-\frac {10 b d x \log \left (c x^n\right )}{e^5}+\frac {10 b d n x}{e^5}-\frac {5 b n x^2}{2 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2332
Rule 2341
Rule 2354
Rule 2384
Rule 2393
Rule 2438
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx &=\int \left (-\frac {4 d \left (a+b \log \left (c x^n\right )\right )}{e^5}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac {d^5 \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)^4}+\frac {5 d^4 \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)^3}-\frac {10 d^3 \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)^2}+\frac {10 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)}\right ) \, dx\\ &=-\frac {(4 d) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^5}+\frac {\left (10 d^2\right ) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^5}-\frac {\left (10 d^3\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^5}+\frac {\left (5 d^4\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{e^5}-\frac {d^5 \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^4} \, dx}{e^5}+\frac {\int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^4}\\ &=-\frac {4 a d x}{e^5}-\frac {b n x^2}{4 e^4}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^4}+\frac {d^5 \left (a+b \log \left (c x^n\right )\right )}{3 e^6 (d+e x)^3}-\frac {5 d^4 \left (a+b \log \left (c x^n\right )\right )}{2 e^6 (d+e x)^2}-\frac {10 d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)}+\frac {10 d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^6}-\frac {(4 b d) \int \log \left (c x^n\right ) \, dx}{e^5}-\frac {\left (10 b d^2 n\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^6}+\frac {\left (5 b d^4 n\right ) \int \frac {1}{x (d+e x)^2} \, dx}{2 e^6}-\frac {\left (b d^5 n\right ) \int \frac {1}{x (d+e x)^3} \, dx}{3 e^6}+\frac {\left (10 b d^2 n\right ) \int \frac {1}{d+e x} \, dx}{e^5}\\ &=-\frac {4 a d x}{e^5}+\frac {4 b d n x}{e^5}-\frac {b n x^2}{4 e^4}-\frac {4 b d x \log \left (c x^n\right )}{e^5}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^4}+\frac {d^5 \left (a+b \log \left (c x^n\right )\right )}{3 e^6 (d+e x)^3}-\frac {5 d^4 \left (a+b \log \left (c x^n\right )\right )}{2 e^6 (d+e x)^2}-\frac {10 d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)}+\frac {10 b d^2 n \log (d+e x)}{e^6}+\frac {10 d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^6}+\frac {10 b d^2 n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^6}+\frac {\left (5 b d^4 n\right ) \int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx}{2 e^6}-\frac {\left (b d^5 n\right ) \int \left (\frac {1}{d^3 x}-\frac {e}{d (d+e x)^3}-\frac {e}{d^2 (d+e x)^2}-\frac {e}{d^3 (d+e x)}\right ) \, dx}{3 e^6}\\ &=-\frac {4 a d x}{e^5}+\frac {4 b d n x}{e^5}-\frac {b n x^2}{4 e^4}-\frac {b d^4 n}{6 e^6 (d+e x)^2}+\frac {13 b d^3 n}{6 e^6 (d+e x)}+\frac {13 b d^2 n \log (x)}{6 e^6}-\frac {4 b d x \log \left (c x^n\right )}{e^5}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^4}+\frac {d^5 \left (a+b \log \left (c x^n\right )\right )}{3 e^6 (d+e x)^3}-\frac {5 d^4 \left (a+b \log \left (c x^n\right )\right )}{2 e^6 (d+e x)^2}-\frac {10 d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)}+\frac {47 b d^2 n \log (d+e x)}{6 e^6}+\frac {10 d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^6}+\frac {10 b d^2 n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^6}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 249, normalized size = 1.09 \begin {gather*} \frac {-48 a d e x+48 b d e n x-3 b e^2 n x^2-48 b d e x \log \left (c x^n\right )+6 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {4 d^5 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3}-\frac {30 d^4 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}+\frac {120 d^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x}-2 b d^2 n \left (\frac {d (3 d+2 e x)}{(d+e x)^2}+2 \log (x)-2 \log (d+e x)\right )-120 b d^2 n (\log (x)-\log (d+e x))+30 b d^2 n \left (\frac {d}{d+e x}+\log (x)-\log (d+e x)\right )+120 d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )+120 b d^2 n \text {Li}_2\left (-\frac {e x}{d}\right )}{12 e^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.12, size = 1153, normalized size = 5.03
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1153\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [A]
time = 65.64, size = 617, normalized size = 2.69 \begin {gather*} - \frac {a d^{5} \left (\begin {cases} \frac {x}{d^{4}} & \text {for}\: e = 0 \\- \frac {1}{3 e \left (d + e x\right )^{3}} & \text {otherwise} \end {cases}\right )}{e^{5}} + \frac {5 a d^{4} \left (\begin {cases} \frac {x}{d^{3}} & \text {for}\: e = 0 \\- \frac {1}{2 e \left (d + e x\right )^{2}} & \text {otherwise} \end {cases}\right )}{e^{5}} - \frac {10 a d^{3} \left (\begin {cases} \frac {x}{d^{2}} & \text {for}\: e = 0 \\- \frac {1}{d e + e^{2} x} & \text {otherwise} \end {cases}\right )}{e^{5}} + \frac {10 a d^{2} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{5}} - \frac {4 a d x}{e^{5}} + \frac {a x^{2}}{2 e^{4}} + \frac {b d^{5} n \left (\begin {cases} \frac {x}{d^{4}} & \text {for}\: e = 0 \\- \frac {3 d}{6 d^{4} e + 12 d^{3} e^{2} x + 6 d^{2} e^{3} x^{2}} - \frac {2 e x}{6 d^{4} e + 12 d^{3} e^{2} x + 6 d^{2} e^{3} x^{2}} - \frac {\log {\left (x \right )}}{3 d^{3} e} + \frac {\log {\left (\frac {d}{e} + x \right )}}{3 d^{3} e} & \text {otherwise} \end {cases}\right )}{e^{5}} - \frac {b d^{5} \left (\begin {cases} \frac {x}{d^{4}} & \text {for}\: e = 0 \\- \frac {1}{3 e \left (d + e x\right )^{3}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{e^{5}} - \frac {5 b d^{4} n \left (\begin {cases} \frac {x}{d^{3}} & \text {for}\: e = 0 \\- \frac {1}{2 d^{2} e + 2 d e^{2} x} - \frac {\log {\left (x \right )}}{2 d^{2} e} + \frac {\log {\left (\frac {d}{e} + x \right )}}{2 d^{2} e} & \text {otherwise} \end {cases}\right )}{e^{5}} + \frac {5 b d^{4} \left (\begin {cases} \frac {x}{d^{3}} & \text {for}\: e = 0 \\- \frac {1}{2 e \left (d + e x\right )^{2}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{e^{5}} + \frac {10 b d^{3} n \left (\begin {cases} \frac {x}{d^{2}} & \text {for}\: e = 0 \\- \frac {\log {\left (x \right )}}{d e} + \frac {\log {\left (\frac {d}{e} + x \right )}}{d e} & \text {otherwise} \end {cases}\right )}{e^{5}} - \frac {10 b d^{3} \left (\begin {cases} \frac {x}{d^{2}} & \text {for}\: e = 0 \\- \frac {1}{d e + e^{2} x} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{e^{5}} - \frac {10 b d^{2} n \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\begin {cases} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (d \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (d \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (d \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right )}{e^{5}} + \frac {10 b d^{2} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{e^{5}} + \frac {4 b d n x}{e^{5}} - \frac {4 b d x \log {\left (c x^{n} \right )}}{e^{5}} - \frac {b n x^{2}}{4 e^{4}} + \frac {b x^{2} \log {\left (c x^{n} \right )}}{2 e^{4}} \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 = -0.00 \begin {gather*} \int \frac {x^5\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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