Optimal. Leaf size=210 \[ \frac {b^2 n^2}{3 d e^2 (d+e x)}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^2}+\frac {b n \left (a+b \log \left (c x^n\right )\right )}{3 d e^2 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{6 d^2 e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 (d+e x)^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{3 d^2 e^2}-\frac {b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{3 d^2 e^2} \]
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Rubi [A]
time = 0.16, antiderivative size = 222, normalized size of antiderivative = 1.06, number of steps
used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2383, 2381,
2384, 2354, 2438, 2373, 45} \begin {gather*} -\frac {b^2 n^2 \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{3 d^2 e^2}-\frac {b n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )+b n\right )}{3 d^2 e^2}+\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 e (d+e x)}-\frac {b n x^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^2 (d+e x)^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 d^2 (d+e x)^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}+\frac {b^2 n^2 \log (d+e x)}{3 d^2 e^2}+\frac {b^2 n^2}{3 d e^2 (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2354
Rule 2373
Rule 2381
Rule 2383
Rule 2384
Rule 2438
Rubi steps
\begin {align*} \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx &=\int \left (-\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{e (d+e x)^4}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{e (d+e x)^3}\right ) \, dx\\ &=\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx}{e}-\frac {d \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx}{e}\\ &=\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 (d+e x)^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}+\frac {(b n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{e^2}-\frac {(2 b d n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^3} \, dx}{3 e^2}\\ &=\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 (d+e x)^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}-\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{3 e^2}+\frac {(b n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{d e^2}+\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{3 e}-\frac {(b n) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d e}\\ &=-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^2}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{d^2 e (d+e x)}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 (d+e x)^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}+\frac {(b n) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{d^2 e^2}-\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{3 d e^2}-\frac {(b n) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^2 e}+\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{3 d e}+\frac {\left (b^2 n^2\right ) \int \frac {1}{x (d+e x)^2} \, dx}{3 e^2}+\frac {\left (b^2 n^2\right ) \int \frac {1}{d+e x} \, dx}{d^2 e}\\ &=-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^2}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 e (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 (d+e x)^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}+\frac {b^2 n^2 \log (d+e x)}{d^2 e^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^2 e^2}-\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{3 d^2 e^2}+\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{3 d^2 e}+\frac {\left (b^2 n^2\right ) \int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx}{3 e^2}+\frac {\left (b^2 n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^2 e^2}-\frac {\left (2 b^2 n^2\right ) \int \frac {1}{d+e x} \, dx}{3 d^2 e}\\ &=\frac {b^2 n^2}{3 d e^2 (d+e x)}+\frac {b^2 n^2 \log (x)}{3 d^2 e^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^2}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 e (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{6 d^2 e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 (d+e x)^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{3 d^2 e^2}-\frac {b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d^2 e^2}-\frac {\left (2 b^2 n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{3 d^2 e^2}\\ &=\frac {b^2 n^2}{3 d e^2 (d+e x)}+\frac {b^2 n^2 \log (x)}{3 d^2 e^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^2}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 e (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{6 d^2 e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 (d+e x)^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{3 d^2 e^2}-\frac {b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{3 d^2 e^2}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 281, normalized size = 1.34 \begin {gather*} \frac {2 b^2 d^3 n^2+2 a b d^2 e n x+4 b^2 d^2 e n^2 x+3 a^2 d e^2 x^2+2 a b d e^2 n x^2+2 b^2 d e^2 n^2 x^2+a^2 e^3 x^3+b^2 e^2 x^2 (3 d+e x) \log ^2\left (c x^n\right )-2 a b d^3 n \log \left (1+\frac {e x}{d}\right )-6 a b d^2 e n x \log \left (1+\frac {e x}{d}\right )-6 a b d e^2 n x^2 \log \left (1+\frac {e x}{d}\right )-2 a b e^3 n x^3 \log \left (1+\frac {e x}{d}\right )-2 b \log \left (c x^n\right ) \left (-e x (b d n (d+e x)+a e x (3 d+e x))+b n (d+e x)^3 \log \left (1+\frac {e x}{d}\right )\right )-2 b^2 n^2 (d+e x)^3 \text {Li}_2\left (-\frac {e x}{d}\right )}{6 d^2 e^2 (d+e 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
4.
time = 0.16, size = 1400, normalized size = 6.67
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1400\) |
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{\left (d + e x\right )^{4}}\, dx \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\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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