Optimal. Leaf size=109 \[ -\frac {b d n \left (a+b \log \left (c x^n\right )\right )}{8 x^4}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{4 x^4}-\frac {2 b e n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac {b^2 d n^2}{32 x^4}-\frac {2 b^2 e n^2}{27 x^3} \]
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Rubi [A] time = 0.14, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2353, 2305, 2304} \[ -\frac {b d n \left (a+b \log \left (c x^n\right )\right )}{8 x^4}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{4 x^4}-\frac {2 b e n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac {b^2 d n^2}{32 x^4}-\frac {2 b^2 e n^2}{27 x^3} \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2305
Rule 2353
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (a+b \log \left (c x^n\right )\right )^2}{x^5} \, dx &=\int \left (\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{x^5}+\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{x^4}\right ) \, dx\\ &=d \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^5} \, dx+e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx\\ &=-\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{4 x^4}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}+\frac {1}{2} (b d n) \int \frac {a+b \log \left (c x^n\right )}{x^5} \, dx+\frac {1}{3} (2 b e n) \int \frac {a+b \log \left (c x^n\right )}{x^4} \, dx\\ &=-\frac {b^2 d n^2}{32 x^4}-\frac {2 b^2 e n^2}{27 x^3}-\frac {b d n \left (a+b \log \left (c x^n\right )\right )}{8 x^4}-\frac {2 b e n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{4 x^4}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 82, normalized size = 0.75 \[ -\frac {216 d \left (a+b \log \left (c x^n\right )\right )^2+27 b d n \left (4 a+4 b \log \left (c x^n\right )+b n\right )+288 e x \left (a+b \log \left (c x^n\right )\right )^2+64 b e n x \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{864 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 188, normalized size = 1.72 \[ -\frac {27 \, b^{2} d n^{2} + 108 \, a b d n + 216 \, a^{2} d + 72 \, {\left (4 \, b^{2} e x + 3 \, b^{2} d\right )} \log \relax (c)^{2} + 72 \, {\left (4 \, b^{2} e n^{2} x + 3 \, b^{2} d n^{2}\right )} \log \relax (x)^{2} + 32 \, {\left (2 \, b^{2} e n^{2} + 6 \, a b e n + 9 \, a^{2} e\right )} x + 12 \, {\left (9 \, b^{2} d n + 36 \, a b d + 16 \, {\left (b^{2} e n + 3 \, a b e\right )} x\right )} \log \relax (c) + 12 \, {\left (9 \, b^{2} d n^{2} + 36 \, a b d n + 16 \, {\left (b^{2} e n^{2} + 3 \, a b e n\right )} x + 12 \, {\left (4 \, b^{2} e n x + 3 \, b^{2} d n\right )} \log \relax (c)\right )} \log \relax (x)}{864 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 206, normalized size = 1.89 \[ -\frac {288 \, b^{2} n^{2} x e \log \relax (x)^{2} + 192 \, b^{2} n^{2} x e \log \relax (x) + 576 \, b^{2} n x e \log \relax (c) \log \relax (x) + 216 \, b^{2} d n^{2} \log \relax (x)^{2} + 64 \, b^{2} n^{2} x e + 192 \, b^{2} n x e \log \relax (c) + 288 \, b^{2} x e \log \relax (c)^{2} + 108 \, b^{2} d n^{2} \log \relax (x) + 576 \, a b n x e \log \relax (x) + 432 \, b^{2} d n \log \relax (c) \log \relax (x) + 27 \, b^{2} d n^{2} + 192 \, a b n x e + 108 \, b^{2} d n \log \relax (c) + 576 \, a b x e \log \relax (c) + 216 \, b^{2} d \log \relax (c)^{2} + 432 \, a b d n \log \relax (x) + 108 \, a b d n + 288 \, a^{2} x e + 432 \, a b d \log \relax (c) + 216 \, a^{2} d}{864 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.26, size = 1486, normalized size = 13.63 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 151, normalized size = 1.39 \[ -\frac {2}{27} \, b^{2} e {\left (\frac {n^{2}}{x^{3}} + \frac {3 \, n \log \left (c x^{n}\right )}{x^{3}}\right )} - \frac {1}{32} \, b^{2} d {\left (\frac {n^{2}}{x^{4}} + \frac {4 \, n \log \left (c x^{n}\right )}{x^{4}}\right )} - \frac {b^{2} e \log \left (c x^{n}\right )^{2}}{3 \, x^{3}} - \frac {2 \, a b e n}{9 \, x^{3}} - \frac {2 \, a b e \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {b^{2} d \log \left (c x^{n}\right )^{2}}{4 \, x^{4}} - \frac {a b d n}{8 \, x^{4}} - \frac {a^{2} e}{3 \, x^{3}} - \frac {a b d \log \left (c x^{n}\right )}{2 \, x^{4}} - \frac {a^{2} d}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.51, size = 114, normalized size = 1.05 \[ -\frac {x\,\left (24\,e\,a^2+16\,e\,a\,b\,n+\frac {16\,e\,b^2\,n^2}{3}\right )+18\,a^2\,d+\frac {9\,b^2\,d\,n^2}{4}+9\,a\,b\,d\,n}{72\,x^4}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {3\,b\,d\,\left (4\,a+b\,n\right )}{4}+\frac {4\,b\,e\,x\,\left (3\,a+b\,n\right )}{3}\right )}{6\,x^4}-\frac {{\ln \left (c\,x^n\right )}^2\,\left (\frac {b^2\,d}{4}+\frac {b^2\,e\,x}{3}\right )}{x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.07, size = 311, normalized size = 2.85 \[ - \frac {a^{2} d}{4 x^{4}} - \frac {a^{2} e}{3 x^{3}} - \frac {a b d n \log {\relax (x )}}{2 x^{4}} - \frac {a b d n}{8 x^{4}} - \frac {a b d \log {\relax (c )}}{2 x^{4}} - \frac {2 a b e n \log {\relax (x )}}{3 x^{3}} - \frac {2 a b e n}{9 x^{3}} - \frac {2 a b e \log {\relax (c )}}{3 x^{3}} - \frac {b^{2} d n^{2} \log {\relax (x )}^{2}}{4 x^{4}} - \frac {b^{2} d n^{2} \log {\relax (x )}}{8 x^{4}} - \frac {b^{2} d n^{2}}{32 x^{4}} - \frac {b^{2} d n \log {\relax (c )} \log {\relax (x )}}{2 x^{4}} - \frac {b^{2} d n \log {\relax (c )}}{8 x^{4}} - \frac {b^{2} d \log {\relax (c )}^{2}}{4 x^{4}} - \frac {b^{2} e n^{2} \log {\relax (x )}^{2}}{3 x^{3}} - \frac {2 b^{2} e n^{2} \log {\relax (x )}}{9 x^{3}} - \frac {2 b^{2} e n^{2}}{27 x^{3}} - \frac {2 b^{2} e n \log {\relax (c )} \log {\relax (x )}}{3 x^{3}} - \frac {2 b^{2} e n \log {\relax (c )}}{9 x^{3}} - \frac {b^{2} e \log {\relax (c )}^{2}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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