Optimal. Leaf size=131 \[ -\frac{3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}+3 d e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac{1}{2} e^3 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n}{4 x^2}-\frac{b d^3 n}{16 x^4}-\frac{3}{2} b d e^2 n \log ^2(x)-\frac{1}{4} b e^3 n x^2 \]
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Rubi [A] time = 0.122787, antiderivative size = 99, normalized size of antiderivative = 0.76, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {266, 43, 2334, 12, 14, 2301} \[ -\frac{1}{4} \left (\frac{6 d^2 e}{x^2}+\frac{d^3}{x^4}-12 d e^2 \log (x)-2 e^3 x^2\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n}{4 x^2}-\frac{b d^3 n}{16 x^4}-\frac{3}{2} b d e^2 n \log ^2(x)-\frac{1}{4} b e^3 n x^2 \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 2334
Rule 12
Rule 14
Rule 2301
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx &=-\frac{1}{4} \left (\frac{d^3}{x^4}+\frac{6 d^2 e}{x^2}-2 e^3 x^2-12 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{-d^3-6 d^2 e x^2+2 e^3 x^6+12 d e^2 x^4 \log (x)}{4 x^5} \, dx\\ &=-\frac{1}{4} \left (\frac{d^3}{x^4}+\frac{6 d^2 e}{x^2}-2 e^3 x^2-12 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} (b n) \int \frac{-d^3-6 d^2 e x^2+2 e^3 x^6+12 d e^2 x^4 \log (x)}{x^5} \, dx\\ &=-\frac{1}{4} \left (\frac{d^3}{x^4}+\frac{6 d^2 e}{x^2}-2 e^3 x^2-12 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} (b n) \int \left (\frac{-d^3-6 d^2 e x^2+2 e^3 x^6}{x^5}+\frac{12 d e^2 \log (x)}{x}\right ) \, dx\\ &=-\frac{1}{4} \left (\frac{d^3}{x^4}+\frac{6 d^2 e}{x^2}-2 e^3 x^2-12 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} (b n) \int \frac{-d^3-6 d^2 e x^2+2 e^3 x^6}{x^5} \, dx-\left (3 b d e^2 n\right ) \int \frac{\log (x)}{x} \, dx\\ &=-\frac{3}{2} b d e^2 n \log ^2(x)-\frac{1}{4} \left (\frac{d^3}{x^4}+\frac{6 d^2 e}{x^2}-2 e^3 x^2-12 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} (b n) \int \left (-\frac{d^3}{x^5}-\frac{6 d^2 e}{x^3}+2 e^3 x\right ) \, dx\\ &=-\frac{b d^3 n}{16 x^4}-\frac{3 b d^2 e n}{4 x^2}-\frac{1}{4} b e^3 n x^2-\frac{3}{2} b d e^2 n \log ^2(x)-\frac{1}{4} \left (\frac{d^3}{x^4}+\frac{6 d^2 e}{x^2}-2 e^3 x^2-12 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0787547, size = 115, normalized size = 0.88 \[ \frac{1}{16} \left (-\frac{24 d^2 e \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac{4 d^3 \left (a+b \log \left (c x^n\right )\right )}{x^4}+\frac{24 d e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b n}+8 e^3 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{12 b d^2 e n}{x^2}-\frac{b d^3 n}{x^4}-4 b e^3 n x^2\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.256, size = 602, normalized size = 4.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1712, size = 180, normalized size = 1.37 \begin{align*} -\frac{1}{4} \, b e^{3} n x^{2} + \frac{1}{2} \, b e^{3} x^{2} \log \left (c x^{n}\right ) + \frac{1}{2} \, a e^{3} x^{2} + \frac{3 \, b d e^{2} \log \left (c x^{n}\right )^{2}}{2 \, n} + 3 \, a d e^{2} \log \left (x\right ) - \frac{3 \, b d^{2} e n}{4 \, x^{2}} - \frac{3 \, b d^{2} e \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac{3 \, a d^{2} e}{2 \, x^{2}} - \frac{b d^{3} n}{16 \, x^{4}} - \frac{b d^{3} \log \left (c x^{n}\right )}{4 \, x^{4}} - \frac{a d^{3}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53736, size = 356, normalized size = 2.72 \begin{align*} \frac{24 \, b d e^{2} n x^{4} \log \left (x\right )^{2} - 4 \,{\left (b e^{3} n - 2 \, a e^{3}\right )} x^{6} - b d^{3} n - 4 \, a d^{3} - 12 \,{\left (b d^{2} e n + 2 \, a d^{2} e\right )} x^{2} + 4 \,{\left (2 \, b e^{3} x^{6} - 6 \, b d^{2} e x^{2} - b d^{3}\right )} \log \left (c\right ) + 4 \,{\left (2 \, b e^{3} n x^{6} + 12 \, b d e^{2} x^{4} \log \left (c\right ) + 12 \, a d e^{2} x^{4} - 6 \, b d^{2} e n x^{2} - b d^{3} n\right )} \log \left (x\right )}{16 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.15391, size = 209, normalized size = 1.6 \begin{align*} - \frac{a d^{3}}{4 x^{4}} - \frac{3 a d^{2} e}{2 x^{2}} + 3 a d e^{2} \log{\left (x \right )} + \frac{a e^{3} x^{2}}{2} - \frac{b d^{3} n \log{\left (x \right )}}{4 x^{4}} - \frac{b d^{3} n}{16 x^{4}} - \frac{b d^{3} \log{\left (c \right )}}{4 x^{4}} - \frac{3 b d^{2} e n \log{\left (x \right )}}{2 x^{2}} - \frac{3 b d^{2} e n}{4 x^{2}} - \frac{3 b d^{2} e \log{\left (c \right )}}{2 x^{2}} + \frac{3 b d e^{2} n \log{\left (x \right )}^{2}}{2} + 3 b d e^{2} \log{\left (c \right )} \log{\left (x \right )} + \frac{b e^{3} n x^{2} \log{\left (x \right )}}{2} - \frac{b e^{3} n x^{2}}{4} + \frac{b e^{3} x^{2} \log{\left (c \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3287, size = 219, normalized size = 1.67 \begin{align*} \frac{8 \, b n x^{6} e^{3} \log \left (x\right ) + 24 \, b d n x^{4} e^{2} \log \left (x\right )^{2} - 4 \, b n x^{6} e^{3} + 8 \, b x^{6} e^{3} \log \left (c\right ) + 48 \, b d x^{4} e^{2} \log \left (c\right ) \log \left (x\right ) + 8 \, a x^{6} e^{3} + 48 \, a d x^{4} e^{2} \log \left (x\right ) - 24 \, b d^{2} n x^{2} e \log \left (x\right ) - 12 \, b d^{2} n x^{2} e - 24 \, b d^{2} x^{2} e \log \left (c\right ) - 24 \, a d^{2} x^{2} e - 4 \, b d^{3} n \log \left (x\right ) - b d^{3} n - 4 \, b d^{3} \log \left (c\right ) - 4 \, a d^{3}}{16 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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