Optimal. Leaf size=91 \[ \frac{\left (d+e x^2\right )^4 \left (a+b \log \left (c x^n\right )\right )}{8 e}-\frac{3}{16} b d^2 e n x^4-\frac{b d^4 n \log (x)}{8 e}-\frac{1}{4} b d^3 n x^2-\frac{1}{12} b d e^2 n x^6-\frac{1}{64} b e^3 n x^8 \]
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Rubi [A] time = 0.0746843, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {261, 2334, 12, 266, 43} \[ \frac{\left (d+e x^2\right )^4 \left (a+b \log \left (c x^n\right )\right )}{8 e}-\frac{3}{16} b d^2 e n x^4-\frac{b d^4 n \log (x)}{8 e}-\frac{1}{4} b d^3 n x^2-\frac{1}{12} b d e^2 n x^6-\frac{1}{64} b e^3 n x^8 \]
Antiderivative was successfully verified.
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Rule 261
Rule 2334
Rule 12
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{\left (d+e x^2\right )^4 \left (a+b \log \left (c x^n\right )\right )}{8 e}-(b n) \int \frac{\left (d+e x^2\right )^4}{8 e x} \, dx\\ &=\frac{\left (d+e x^2\right )^4 \left (a+b \log \left (c x^n\right )\right )}{8 e}-\frac{(b n) \int \frac{\left (d+e x^2\right )^4}{x} \, dx}{8 e}\\ &=\frac{\left (d+e x^2\right )^4 \left (a+b \log \left (c x^n\right )\right )}{8 e}-\frac{(b n) \operatorname{Subst}\left (\int \frac{(d+e x)^4}{x} \, dx,x,x^2\right )}{16 e}\\ &=\frac{\left (d+e x^2\right )^4 \left (a+b \log \left (c x^n\right )\right )}{8 e}-\frac{(b n) \operatorname{Subst}\left (\int \left (4 d^3 e+\frac{d^4}{x}+6 d^2 e^2 x+4 d e^3 x^2+e^4 x^3\right ) \, dx,x,x^2\right )}{16 e}\\ &=-\frac{1}{4} b d^3 n x^2-\frac{3}{16} b d^2 e n x^4-\frac{1}{12} b d e^2 n x^6-\frac{1}{64} b e^3 n x^8-\frac{b d^4 n \log (x)}{8 e}+\frac{\left (d+e x^2\right )^4 \left (a+b \log \left (c x^n\right )\right )}{8 e}\\ \end{align*}
Mathematica [A] time = 0.0510205, size = 118, normalized size = 1.3 \[ \frac{1}{192} x^2 \left (24 a \left (6 d^2 e x^2+4 d^3+4 d e^2 x^4+e^3 x^6\right )+24 b \left (6 d^2 e x^2+4 d^3+4 d e^2 x^4+e^3 x^6\right ) \log \left (c x^n\right )-b n \left (36 d^2 e x^2+48 d^3+16 d e^2 x^4+3 e^3 x^6\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.209, size = 601, normalized size = 6.6 \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 = 1.20182, size = 193, normalized size = 2.12 \begin{align*} -\frac{1}{64} \, b e^{3} n x^{8} + \frac{1}{8} \, b e^{3} x^{8} \log \left (c x^{n}\right ) + \frac{1}{8} \, a e^{3} x^{8} - \frac{1}{12} \, b d e^{2} n x^{6} + \frac{1}{2} \, b d e^{2} x^{6} \log \left (c x^{n}\right ) + \frac{1}{2} \, a d e^{2} x^{6} - \frac{3}{16} \, b d^{2} e n x^{4} + \frac{3}{4} \, b d^{2} e x^{4} \log \left (c x^{n}\right ) + \frac{3}{4} \, a d^{2} e x^{4} - \frac{1}{4} \, b d^{3} n x^{2} + \frac{1}{2} \, b d^{3} x^{2} \log \left (c x^{n}\right ) + \frac{1}{2} \, a d^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.40845, size = 379, normalized size = 4.16 \begin{align*} -\frac{1}{64} \,{\left (b e^{3} n - 8 \, a e^{3}\right )} x^{8} - \frac{1}{12} \,{\left (b d e^{2} n - 6 \, a d e^{2}\right )} x^{6} - \frac{3}{16} \,{\left (b d^{2} e n - 4 \, a d^{2} e\right )} x^{4} - \frac{1}{4} \,{\left (b d^{3} n - 2 \, a d^{3}\right )} x^{2} + \frac{1}{8} \,{\left (b e^{3} x^{8} + 4 \, b d e^{2} x^{6} + 6 \, b d^{2} e x^{4} + 4 \, b d^{3} x^{2}\right )} \log \left (c\right ) + \frac{1}{8} \,{\left (b e^{3} n x^{8} + 4 \, b d e^{2} n x^{6} + 6 \, b d^{2} e n x^{4} + 4 \, b d^{3} n x^{2}\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 13.6113, size = 223, normalized size = 2.45 \begin{align*} \frac{a d^{3} x^{2}}{2} + \frac{3 a d^{2} e x^{4}}{4} + \frac{a d e^{2} x^{6}}{2} + \frac{a e^{3} x^{8}}{8} + \frac{b d^{3} n x^{2} \log{\left (x \right )}}{2} - \frac{b d^{3} n x^{2}}{4} + \frac{b d^{3} x^{2} \log{\left (c \right )}}{2} + \frac{3 b d^{2} e n x^{4} \log{\left (x \right )}}{4} - \frac{3 b d^{2} e n x^{4}}{16} + \frac{3 b d^{2} e x^{4} \log{\left (c \right )}}{4} + \frac{b d e^{2} n x^{6} \log{\left (x \right )}}{2} - \frac{b d e^{2} n x^{6}}{12} + \frac{b d e^{2} x^{6} \log{\left (c \right )}}{2} + \frac{b e^{3} n x^{8} \log{\left (x \right )}}{8} - \frac{b e^{3} n x^{8}}{64} + \frac{b e^{3} x^{8} \log{\left (c \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.32775, size = 234, normalized size = 2.57 \begin{align*} \frac{1}{8} \, b n x^{8} e^{3} \log \left (x\right ) - \frac{1}{64} \, b n x^{8} e^{3} + \frac{1}{8} \, b x^{8} e^{3} \log \left (c\right ) + \frac{1}{2} \, b d n x^{6} e^{2} \log \left (x\right ) + \frac{1}{8} \, a x^{8} e^{3} - \frac{1}{12} \, b d n x^{6} e^{2} + \frac{1}{2} \, b d x^{6} e^{2} \log \left (c\right ) + \frac{3}{4} \, b d^{2} n x^{4} e \log \left (x\right ) + \frac{1}{2} \, a d x^{6} e^{2} - \frac{3}{16} \, b d^{2} n x^{4} e + \frac{3}{4} \, b d^{2} x^{4} e \log \left (c\right ) + \frac{3}{4} \, a d^{2} x^{4} e + \frac{1}{2} \, b d^{3} n x^{2} \log \left (x\right ) - \frac{1}{4} \, b d^{3} n x^{2} + \frac{1}{2} \, b d^{3} x^{2} \log \left (c\right ) + \frac{1}{2} \, a d^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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