Optimal. Leaf size=178 \[ -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x^4}-\frac {b d^2 n \left (a+b \log \left (c x^n\right )\right )}{8 x^4}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac {4 b d e n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {b e^2 n \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b^2 d^2 n^2}{32 x^4}-\frac {4 b^2 d e n^2}{27 x^3}-\frac {b^2 e^2 n^2}{4 x^2} \]
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Rubi [A] time = 0.21, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2353, 2305, 2304} \[ -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x^4}-\frac {b d^2 n \left (a+b \log \left (c x^n\right )\right )}{8 x^4}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac {4 b d e n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {b e^2 n \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b^2 d^2 n^2}{32 x^4}-\frac {4 b^2 d e n^2}{27 x^3}-\frac {b^2 e^2 n^2}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2305
Rule 2353
Rubi steps
\begin {align*} \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^5} \, dx &=\int \left (\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^5}+\frac {2 d e \left (a+b \log \left (c x^n\right )\right )^2}{x^4}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^3}\right ) \, dx\\ &=d^2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^5} \, dx+(2 d e) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx+e^2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx\\ &=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x^4}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}+\frac {1}{2} \left (b d^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x^5} \, dx+\frac {1}{3} (4 b d e n) \int \frac {a+b \log \left (c x^n\right )}{x^4} \, dx+\left (b e^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx\\ &=-\frac {b^2 d^2 n^2}{32 x^4}-\frac {4 b^2 d e n^2}{27 x^3}-\frac {b^2 e^2 n^2}{4 x^2}-\frac {b d^2 n \left (a+b \log \left (c x^n\right )\right )}{8 x^4}-\frac {4 b d e n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac {b e^2 n \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x^4}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 134, normalized size = 0.75 \[ -\frac {216 d^2 \left (a+b \log \left (c x^n\right )\right )^2+27 b d^2 n \left (4 a+4 b \log \left (c x^n\right )+b n\right )+576 d e x \left (a+b \log \left (c x^n\right )\right )^2+128 b d e n x \left (3 a+3 b \log \left (c x^n\right )+b n\right )+432 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2+216 b e^2 n x^2 \left (2 a+2 b \log \left (c x^n\right )+b n\right )}{864 x^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 332, normalized size = 1.87 \[ -\frac {27 \, b^{2} d^{2} n^{2} + 108 \, a b d^{2} n + 216 \, a^{2} d^{2} + 216 \, {\left (b^{2} e^{2} n^{2} + 2 \, a b e^{2} n + 2 \, a^{2} e^{2}\right )} x^{2} + 72 \, {\left (6 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d e x + 3 \, b^{2} d^{2}\right )} \log \relax (c)^{2} + 72 \, {\left (6 \, b^{2} e^{2} n^{2} x^{2} + 8 \, b^{2} d e n^{2} x + 3 \, b^{2} d^{2} n^{2}\right )} \log \relax (x)^{2} + 64 \, {\left (2 \, b^{2} d e n^{2} + 6 \, a b d e n + 9 \, a^{2} d e\right )} x + 12 \, {\left (9 \, b^{2} d^{2} n + 36 \, a b d^{2} + 36 \, {\left (b^{2} e^{2} n + 2 \, a b e^{2}\right )} x^{2} + 32 \, {\left (b^{2} d e n + 3 \, a b d e\right )} x\right )} \log \relax (c) + 12 \, {\left (9 \, b^{2} d^{2} n^{2} + 36 \, a b d^{2} n + 36 \, {\left (b^{2} e^{2} n^{2} + 2 \, a b e^{2} n\right )} x^{2} + 32 \, {\left (b^{2} d e n^{2} + 3 \, a b d e n\right )} x + 12 \, {\left (6 \, b^{2} e^{2} n x^{2} + 8 \, b^{2} d e n x + 3 \, b^{2} d^{2} n\right )} \log \relax (c)\right )} \log \relax (x)}{864 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 366, normalized size = 2.06 \[ -\frac {432 \, b^{2} n^{2} x^{2} e^{2} \log \relax (x)^{2} + 576 \, b^{2} d n^{2} x e \log \relax (x)^{2} + 432 \, b^{2} n^{2} x^{2} e^{2} \log \relax (x) + 384 \, b^{2} d n^{2} x e \log \relax (x) + 864 \, b^{2} n x^{2} e^{2} \log \relax (c) \log \relax (x) + 1152 \, b^{2} d n x e \log \relax (c) \log \relax (x) + 216 \, b^{2} d^{2} n^{2} \log \relax (x)^{2} + 216 \, b^{2} n^{2} x^{2} e^{2} + 128 \, b^{2} d n^{2} x e + 432 \, b^{2} n x^{2} e^{2} \log \relax (c) + 384 \, b^{2} d n x e \log \relax (c) + 432 \, b^{2} x^{2} e^{2} \log \relax (c)^{2} + 576 \, b^{2} d x e \log \relax (c)^{2} + 108 \, b^{2} d^{2} n^{2} \log \relax (x) + 864 \, a b n x^{2} e^{2} \log \relax (x) + 1152 \, a b d n x e \log \relax (x) + 432 \, b^{2} d^{2} n \log \relax (c) \log \relax (x) + 27 \, b^{2} d^{2} n^{2} + 432 \, a b n x^{2} e^{2} + 384 \, a b d n x e + 108 \, b^{2} d^{2} n \log \relax (c) + 864 \, a b x^{2} e^{2} \log \relax (c) + 1152 \, a b d x e \log \relax (c) + 216 \, b^{2} d^{2} \log \relax (c)^{2} + 432 \, a b d^{2} n \log \relax (x) + 108 \, a b d^{2} n + 432 \, a^{2} x^{2} e^{2} + 576 \, a^{2} d x e + 432 \, a b d^{2} \log \relax (c) + 216 \, a^{2} d^{2}}{864 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.33, size = 2475, normalized size = 13.90 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.80, size = 251, normalized size = 1.41 \[ -\frac {1}{4} \, b^{2} e^{2} {\left (\frac {n^{2}}{x^{2}} + \frac {2 \, n \log \left (c x^{n}\right )}{x^{2}}\right )} - \frac {4}{27} \, b^{2} d e {\left (\frac {n^{2}}{x^{3}} + \frac {3 \, n \log \left (c x^{n}\right )}{x^{3}}\right )} - \frac {1}{32} \, b^{2} d^{2} {\left (\frac {n^{2}}{x^{4}} + \frac {4 \, n \log \left (c x^{n}\right )}{x^{4}}\right )} - \frac {b^{2} e^{2} \log \left (c x^{n}\right )^{2}}{2 \, x^{2}} - \frac {a b e^{2} n}{2 \, x^{2}} - \frac {a b e^{2} \log \left (c x^{n}\right )}{x^{2}} - \frac {2 \, b^{2} d e \log \left (c x^{n}\right )^{2}}{3 \, x^{3}} - \frac {4 \, a b d e n}{9 \, x^{3}} - \frac {a^{2} e^{2}}{2 \, x^{2}} - \frac {4 \, a b d e \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {b^{2} d^{2} \log \left (c x^{n}\right )^{2}}{4 \, x^{4}} - \frac {a b d^{2} n}{8 \, x^{4}} - \frac {2 \, a^{2} d e}{3 \, x^{3}} - \frac {a b d^{2} \log \left (c x^{n}\right )}{2 \, x^{4}} - \frac {a^{2} d^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.67, size = 188, normalized size = 1.06 \[ -\frac {x\,\left (48\,d\,e\,a^2+32\,d\,e\,a\,b\,n+\frac {32\,d\,e\,b^2\,n^2}{3}\right )+x^2\,\left (36\,a^2\,e^2+36\,a\,b\,e^2\,n+18\,b^2\,e^2\,n^2\right )+18\,a^2\,d^2+\frac {9\,b^2\,d^2\,n^2}{4}+9\,a\,b\,d^2\,n}{72\,x^4}-\frac {{\ln \left (c\,x^n\right )}^2\,\left (\frac {b^2\,d^2}{4}+\frac {2\,b^2\,d\,e\,x}{3}+\frac {b^2\,e^2\,x^2}{2}\right )}{x^4}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {3\,b\,\left (4\,a+b\,n\right )\,d^2}{4}+\frac {8\,b\,\left (3\,a+b\,n\right )\,d\,e\,x}{3}+3\,b\,\left (2\,a+b\,n\right )\,e^2\,x^2\right )}{6\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.50, size = 512, normalized size = 2.88 \[ - \frac {a^{2} d^{2}}{4 x^{4}} - \frac {2 a^{2} d e}{3 x^{3}} - \frac {a^{2} e^{2}}{2 x^{2}} - \frac {a b d^{2} n \log {\relax (x )}}{2 x^{4}} - \frac {a b d^{2} n}{8 x^{4}} - \frac {a b d^{2} \log {\relax (c )}}{2 x^{4}} - \frac {4 a b d e n \log {\relax (x )}}{3 x^{3}} - \frac {4 a b d e n}{9 x^{3}} - \frac {4 a b d e \log {\relax (c )}}{3 x^{3}} - \frac {a b e^{2} n \log {\relax (x )}}{x^{2}} - \frac {a b e^{2} n}{2 x^{2}} - \frac {a b e^{2} \log {\relax (c )}}{x^{2}} - \frac {b^{2} d^{2} n^{2} \log {\relax (x )}^{2}}{4 x^{4}} - \frac {b^{2} d^{2} n^{2} \log {\relax (x )}}{8 x^{4}} - \frac {b^{2} d^{2} n^{2}}{32 x^{4}} - \frac {b^{2} d^{2} n \log {\relax (c )} \log {\relax (x )}}{2 x^{4}} - \frac {b^{2} d^{2} n \log {\relax (c )}}{8 x^{4}} - \frac {b^{2} d^{2} \log {\relax (c )}^{2}}{4 x^{4}} - \frac {2 b^{2} d e n^{2} \log {\relax (x )}^{2}}{3 x^{3}} - \frac {4 b^{2} d e n^{2} \log {\relax (x )}}{9 x^{3}} - \frac {4 b^{2} d e n^{2}}{27 x^{3}} - \frac {4 b^{2} d e n \log {\relax (c )} \log {\relax (x )}}{3 x^{3}} - \frac {4 b^{2} d e n \log {\relax (c )}}{9 x^{3}} - \frac {2 b^{2} d e \log {\relax (c )}^{2}}{3 x^{3}} - \frac {b^{2} e^{2} n^{2} \log {\relax (x )}^{2}}{2 x^{2}} - \frac {b^{2} e^{2} n^{2} \log {\relax (x )}}{2 x^{2}} - \frac {b^{2} e^{2} n^{2}}{4 x^{2}} - \frac {b^{2} e^{2} n \log {\relax (c )} \log {\relax (x )}}{x^{2}} - \frac {b^{2} e^{2} n \log {\relax (c )}}{2 x^{2}} - \frac {b^{2} e^{2} \log {\relax (c )}^{2}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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