Optimal. Leaf size=100 \[ \frac{1}{315} \left (189 d^2 e x^5+105 d^3 x^3+135 d e^2 x^7+35 e^3 x^9\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3}{25} b d^2 e n x^5-\frac{1}{9} b d^3 n x^3-\frac{3}{49} b d e^2 n x^7-\frac{1}{81} b e^3 n x^9 \]
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Rubi [A] time = 0.0865099, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {270, 2334} \[ \frac{1}{315} \left (189 d^2 e x^5+105 d^3 x^3+135 d e^2 x^7+35 e^3 x^9\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3}{25} b d^2 e n x^5-\frac{1}{9} b d^3 n x^3-\frac{3}{49} b d e^2 n x^7-\frac{1}{81} b e^3 n x^9 \]
Antiderivative was successfully verified.
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Rule 270
Rule 2334
Rubi steps
\begin{align*} \int x^2 \left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{315} \left (105 d^3 x^3+189 d^2 e x^5+135 d e^2 x^7+35 e^3 x^9\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (\frac{d^3 x^2}{3}+\frac{3}{5} d^2 e x^4+\frac{3}{7} d e^2 x^6+\frac{e^3 x^8}{9}\right ) \, dx\\ &=-\frac{1}{9} b d^3 n x^3-\frac{3}{25} b d^2 e n x^5-\frac{3}{49} b d e^2 n x^7-\frac{1}{81} b e^3 n x^9+\frac{1}{315} \left (105 d^3 x^3+189 d^2 e x^5+135 d e^2 x^7+35 e^3 x^9\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0469742, size = 133, normalized size = 1.33 \[ \frac{3}{5} d^2 e x^5 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{3} d^3 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{3}{7} d e^2 x^7 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{9} e^3 x^9 \left (a+b \log \left (c x^n\right )\right )-\frac{3}{25} b d^2 e n x^5-\frac{1}{9} b d^3 n x^3-\frac{3}{49} b d e^2 n x^7-\frac{1}{81} b e^3 n x^9 \]
Antiderivative was successfully verified.
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Maple [C] time = 0.213, size = 602, normalized size = 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 = 1.03093, size = 193, normalized size = 1.93 \begin{align*} -\frac{1}{81} \, b e^{3} n x^{9} + \frac{1}{9} \, b e^{3} x^{9} \log \left (c x^{n}\right ) + \frac{1}{9} \, a e^{3} x^{9} - \frac{3}{49} \, b d e^{2} n x^{7} + \frac{3}{7} \, b d e^{2} x^{7} \log \left (c x^{n}\right ) + \frac{3}{7} \, a d e^{2} x^{7} - \frac{3}{25} \, b d^{2} e n x^{5} + \frac{3}{5} \, b d^{2} e x^{5} \log \left (c x^{n}\right ) + \frac{3}{5} \, a d^{2} e x^{5} - \frac{1}{9} \, b d^{3} n x^{3} + \frac{1}{3} \, b d^{3} x^{3} \log \left (c x^{n}\right ) + \frac{1}{3} \, a d^{3} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34886, size = 409, normalized size = 4.09 \begin{align*} -\frac{1}{81} \,{\left (b e^{3} n - 9 \, a e^{3}\right )} x^{9} - \frac{3}{49} \,{\left (b d e^{2} n - 7 \, a d e^{2}\right )} x^{7} - \frac{3}{25} \,{\left (b d^{2} e n - 5 \, a d^{2} e\right )} x^{5} - \frac{1}{9} \,{\left (b d^{3} n - 3 \, a d^{3}\right )} x^{3} + \frac{1}{315} \,{\left (35 \, b e^{3} x^{9} + 135 \, b d e^{2} x^{7} + 189 \, b d^{2} e x^{5} + 105 \, b d^{3} x^{3}\right )} \log \left (c\right ) + \frac{1}{315} \,{\left (35 \, b e^{3} n x^{9} + 135 \, b d e^{2} n x^{7} + 189 \, b d^{2} e n x^{5} + 105 \, b d^{3} n x^{3}\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 20.5486, size = 230, normalized size = 2.3 \begin{align*} \frac{a d^{3} x^{3}}{3} + \frac{3 a d^{2} e x^{5}}{5} + \frac{3 a d e^{2} x^{7}}{7} + \frac{a e^{3} x^{9}}{9} + \frac{b d^{3} n x^{3} \log{\left (x \right )}}{3} - \frac{b d^{3} n x^{3}}{9} + \frac{b d^{3} x^{3} \log{\left (c \right )}}{3} + \frac{3 b d^{2} e n x^{5} \log{\left (x \right )}}{5} - \frac{3 b d^{2} e n x^{5}}{25} + \frac{3 b d^{2} e x^{5} \log{\left (c \right )}}{5} + \frac{3 b d e^{2} n x^{7} \log{\left (x \right )}}{7} - \frac{3 b d e^{2} n x^{7}}{49} + \frac{3 b d e^{2} x^{7} \log{\left (c \right )}}{7} + \frac{b e^{3} n x^{9} \log{\left (x \right )}}{9} - \frac{b e^{3} n x^{9}}{81} + \frac{b e^{3} x^{9} \log{\left (c \right )}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31562, size = 234, normalized size = 2.34 \begin{align*} \frac{1}{9} \, b n x^{9} e^{3} \log \left (x\right ) - \frac{1}{81} \, b n x^{9} e^{3} + \frac{1}{9} \, b x^{9} e^{3} \log \left (c\right ) + \frac{3}{7} \, b d n x^{7} e^{2} \log \left (x\right ) + \frac{1}{9} \, a x^{9} e^{3} - \frac{3}{49} \, b d n x^{7} e^{2} + \frac{3}{7} \, b d x^{7} e^{2} \log \left (c\right ) + \frac{3}{5} \, b d^{2} n x^{5} e \log \left (x\right ) + \frac{3}{7} \, a d x^{7} e^{2} - \frac{3}{25} \, b d^{2} n x^{5} e + \frac{3}{5} \, b d^{2} x^{5} e \log \left (c\right ) + \frac{3}{5} \, a d^{2} x^{5} e + \frac{1}{3} \, b d^{3} n x^{3} \log \left (x\right ) - \frac{1}{9} \, b d^{3} n x^{3} + \frac{1}{3} \, b d^{3} x^{3} \log \left (c\right ) + \frac{1}{3} \, a d^{3} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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