Optimal. Leaf size=168 \[ -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac {2 b d^2 n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac {d e \left (a+b \log \left (c x^n\right )\right )^2}{x^2}-\frac {b d e n \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {2 b e^2 n \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {2 b^2 d^2 n^2}{27 x^3}-\frac {b^2 d e n^2}{2 x^2}-\frac {2 b^2 e^2 n^2}{x} \]
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Rubi [A] time = 0.21, antiderivative size = 168, 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} \[ -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac {2 b d^2 n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac {d e \left (a+b \log \left (c x^n\right )\right )^2}{x^2}-\frac {b d e n \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {2 b e^2 n \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {2 b^2 d^2 n^2}{27 x^3}-\frac {b^2 d e n^2}{2 x^2}-\frac {2 b^2 e^2 n^2}{x} \]
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^4} \, dx &=\int \left (\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^4}+\frac {2 d e \left (a+b \log \left (c x^n\right )\right )^2}{x^3}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^2}\right ) \, dx\\ &=d^2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx+(2 d e) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx+e^2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx\\ &=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac {d e \left (a+b \log \left (c x^n\right )\right )^2}{x^2}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {1}{3} \left (2 b d^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x^4} \, dx+(2 b d e n) \int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx+\left (2 b e^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx\\ &=-\frac {2 b^2 d^2 n^2}{27 x^3}-\frac {b^2 d e n^2}{2 x^2}-\frac {2 b^2 e^2 n^2}{x}-\frac {2 b d^2 n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac {b d e n \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac {2 b e^2 n \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac {d e \left (a+b \log \left (c x^n\right )\right )^2}{x^2}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{x}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 131, normalized size = 0.78 \[ -\frac {18 d^2 \left (a+b \log \left (c x^n\right )\right )^2+4 b d^2 n \left (3 a+3 b \log \left (c x^n\right )+b n\right )+54 d e x \left (a+b \log \left (c x^n\right )\right )^2+27 b d e n x \left (2 a+2 b \log \left (c x^n\right )+b n\right )+54 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2+108 b e^2 n x^2 \left (a+b \log \left (c x^n\right )+b n\right )}{54 x^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.81, size = 326, normalized size = 1.94 \[ -\frac {4 \, b^{2} d^{2} n^{2} + 12 \, a b d^{2} n + 18 \, a^{2} d^{2} + 54 \, {\left (2 \, b^{2} e^{2} n^{2} + 2 \, a b e^{2} n + a^{2} e^{2}\right )} x^{2} + 18 \, {\left (3 \, b^{2} e^{2} x^{2} + 3 \, b^{2} d e x + b^{2} d^{2}\right )} \log \relax (c)^{2} + 18 \, {\left (3 \, b^{2} e^{2} n^{2} x^{2} + 3 \, b^{2} d e n^{2} x + b^{2} d^{2} n^{2}\right )} \log \relax (x)^{2} + 27 \, {\left (b^{2} d e n^{2} + 2 \, a b d e n + 2 \, a^{2} d e\right )} x + 6 \, {\left (2 \, b^{2} d^{2} n + 6 \, a b d^{2} + 18 \, {\left (b^{2} e^{2} n + a b e^{2}\right )} x^{2} + 9 \, {\left (b^{2} d e n + 2 \, a b d e\right )} x\right )} \log \relax (c) + 6 \, {\left (2 \, b^{2} d^{2} n^{2} + 6 \, a b d^{2} n + 18 \, {\left (b^{2} e^{2} n^{2} + a b e^{2} n\right )} x^{2} + 9 \, {\left (b^{2} d e n^{2} + 2 \, a b d e n\right )} x + 6 \, {\left (3 \, b^{2} e^{2} n x^{2} + 3 \, b^{2} d e n x + b^{2} d^{2} n\right )} \log \relax (c)\right )} \log \relax (x)}{54 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.44, size = 366, normalized size = 2.18 \[ -\frac {54 \, b^{2} n^{2} x^{2} e^{2} \log \relax (x)^{2} + 54 \, b^{2} d n^{2} x e \log \relax (x)^{2} + 108 \, b^{2} n^{2} x^{2} e^{2} \log \relax (x) + 54 \, b^{2} d n^{2} x e \log \relax (x) + 108 \, b^{2} n x^{2} e^{2} \log \relax (c) \log \relax (x) + 108 \, b^{2} d n x e \log \relax (c) \log \relax (x) + 18 \, b^{2} d^{2} n^{2} \log \relax (x)^{2} + 108 \, b^{2} n^{2} x^{2} e^{2} + 27 \, b^{2} d n^{2} x e + 108 \, b^{2} n x^{2} e^{2} \log \relax (c) + 54 \, b^{2} d n x e \log \relax (c) + 54 \, b^{2} x^{2} e^{2} \log \relax (c)^{2} + 54 \, b^{2} d x e \log \relax (c)^{2} + 12 \, b^{2} d^{2} n^{2} \log \relax (x) + 108 \, a b n x^{2} e^{2} \log \relax (x) + 108 \, a b d n x e \log \relax (x) + 36 \, b^{2} d^{2} n \log \relax (c) \log \relax (x) + 4 \, b^{2} d^{2} n^{2} + 108 \, a b n x^{2} e^{2} + 54 \, a b d n x e + 12 \, b^{2} d^{2} n \log \relax (c) + 108 \, a b x^{2} e^{2} \log \relax (c) + 108 \, a b d x e \log \relax (c) + 18 \, b^{2} d^{2} \log \relax (c)^{2} + 36 \, a b d^{2} n \log \relax (x) + 12 \, a b d^{2} n + 54 \, a^{2} x^{2} e^{2} + 54 \, a^{2} d x e + 36 \, a b d^{2} \log \relax (c) + 18 \, a^{2} d^{2}}{54 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.30, size = 2473, normalized size = 14.72 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 250, normalized size = 1.49 \[ -2 \, b^{2} e^{2} {\left (\frac {n^{2}}{x} + \frac {n \log \left (c x^{n}\right )}{x}\right )} - \frac {1}{2} \, b^{2} d e {\left (\frac {n^{2}}{x^{2}} + \frac {2 \, n \log \left (c x^{n}\right )}{x^{2}}\right )} - \frac {2}{27} \, b^{2} d^{2} {\left (\frac {n^{2}}{x^{3}} + \frac {3 \, n \log \left (c x^{n}\right )}{x^{3}}\right )} - \frac {b^{2} e^{2} \log \left (c x^{n}\right )^{2}}{x} - \frac {2 \, a b e^{2} n}{x} - \frac {2 \, a b e^{2} \log \left (c x^{n}\right )}{x} - \frac {b^{2} d e \log \left (c x^{n}\right )^{2}}{x^{2}} - \frac {a b d e n}{x^{2}} - \frac {a^{2} e^{2}}{x} - \frac {2 \, a b d e \log \left (c x^{n}\right )}{x^{2}} - \frac {b^{2} d^{2} \log \left (c x^{n}\right )^{2}}{3 \, x^{3}} - \frac {2 \, a b d^{2} n}{9 \, x^{3}} - \frac {a^{2} d e}{x^{2}} - \frac {2 \, a b d^{2} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {a^{2} d^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.90, size = 184, normalized size = 1.10 \[ -\frac {x\,\left (9\,d\,e\,a^2+9\,d\,e\,a\,b\,n+\frac {9\,d\,e\,b^2\,n^2}{2}\right )+x^2\,\left (9\,a^2\,e^2+18\,a\,b\,e^2\,n+18\,b^2\,e^2\,n^2\right )+3\,a^2\,d^2+\frac {2\,b^2\,d^2\,n^2}{3}+2\,a\,b\,d^2\,n}{9\,x^3}-\frac {{\ln \left (c\,x^n\right )}^2\,\left (\frac {b^2\,d^2}{3}+b^2\,d\,e\,x+b^2\,e^2\,x^2\right )}{x^3}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {2\,b\,\left (3\,a+b\,n\right )\,d^2}{3}+3\,b\,\left (2\,a+b\,n\right )\,d\,e\,x+6\,b\,\left (a+b\,n\right )\,e^2\,x^2\right )}{3\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.41, size = 479, normalized size = 2.85 \[ - \frac {a^{2} d^{2}}{3 x^{3}} - \frac {a^{2} d e}{x^{2}} - \frac {a^{2} e^{2}}{x} - \frac {2 a b d^{2} n \log {\relax (x )}}{3 x^{3}} - \frac {2 a b d^{2} n}{9 x^{3}} - \frac {2 a b d^{2} \log {\relax (c )}}{3 x^{3}} - \frac {2 a b d e n \log {\relax (x )}}{x^{2}} - \frac {a b d e n}{x^{2}} - \frac {2 a b d e \log {\relax (c )}}{x^{2}} - \frac {2 a b e^{2} n \log {\relax (x )}}{x} - \frac {2 a b e^{2} n}{x} - \frac {2 a b e^{2} \log {\relax (c )}}{x} - \frac {b^{2} d^{2} n^{2} \log {\relax (x )}^{2}}{3 x^{3}} - \frac {2 b^{2} d^{2} n^{2} \log {\relax (x )}}{9 x^{3}} - \frac {2 b^{2} d^{2} n^{2}}{27 x^{3}} - \frac {2 b^{2} d^{2} n \log {\relax (c )} \log {\relax (x )}}{3 x^{3}} - \frac {2 b^{2} d^{2} n \log {\relax (c )}}{9 x^{3}} - \frac {b^{2} d^{2} \log {\relax (c )}^{2}}{3 x^{3}} - \frac {b^{2} d e n^{2} \log {\relax (x )}^{2}}{x^{2}} - \frac {b^{2} d e n^{2} \log {\relax (x )}}{x^{2}} - \frac {b^{2} d e n^{2}}{2 x^{2}} - \frac {2 b^{2} d e n \log {\relax (c )} \log {\relax (x )}}{x^{2}} - \frac {b^{2} d e n \log {\relax (c )}}{x^{2}} - \frac {b^{2} d e \log {\relax (c )}^{2}}{x^{2}} - \frac {b^{2} e^{2} n^{2} \log {\relax (x )}^{2}}{x} - \frac {2 b^{2} e^{2} n^{2} \log {\relax (x )}}{x} - \frac {2 b^{2} e^{2} n^{2}}{x} - \frac {2 b^{2} e^{2} n \log {\relax (c )} \log {\relax (x )}}{x} - \frac {2 b^{2} e^{2} n \log {\relax (c )}}{x} - \frac {b^{2} e^{2} \log {\relax (c )}^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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