Optimal. Leaf size=196 \[ \frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac{2 b e^3 n \sqrt{d+e x^2}}{35 d^2 x}+\frac{2 b e^2 n \left (d+e x^2\right )^{3/2}}{105 d^2 x^3}-\frac{2 b e^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{35 d^2}+\frac{2 b e n \left (d+e x^2\right )^{5/2}}{175 d^2 x^5}-\frac{b n \left (d+e x^2\right )^{7/2}}{49 d^2 x^7} \]
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Rubi [A] time = 0.171883, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {271, 264, 2350, 12, 451, 277, 217, 206} \[ \frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac{2 b e^3 n \sqrt{d+e x^2}}{35 d^2 x}+\frac{2 b e^2 n \left (d+e x^2\right )^{3/2}}{105 d^2 x^3}-\frac{2 b e^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{35 d^2}+\frac{2 b e n \left (d+e x^2\right )^{5/2}}{175 d^2 x^5}-\frac{b n \left (d+e x^2\right )^{7/2}}{49 d^2 x^7} \]
Antiderivative was successfully verified.
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Rule 271
Rule 264
Rule 2350
Rule 12
Rule 451
Rule 277
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx &=-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-(b n) \int \frac{\left (d+e x^2\right )^{5/2} \left (-5 d+2 e x^2\right )}{35 d^2 x^8} \, dx\\ &=-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac{(b n) \int \frac{\left (d+e x^2\right )^{5/2} \left (-5 d+2 e x^2\right )}{x^8} \, dx}{35 d^2}\\ &=-\frac{b n \left (d+e x^2\right )^{7/2}}{49 d^2 x^7}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac{(2 b e n) \int \frac{\left (d+e x^2\right )^{5/2}}{x^6} \, dx}{35 d^2}\\ &=\frac{2 b e n \left (d+e x^2\right )^{5/2}}{175 d^2 x^5}-\frac{b n \left (d+e x^2\right )^{7/2}}{49 d^2 x^7}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac{\left (2 b e^2 n\right ) \int \frac{\left (d+e x^2\right )^{3/2}}{x^4} \, dx}{35 d^2}\\ &=\frac{2 b e^2 n \left (d+e x^2\right )^{3/2}}{105 d^2 x^3}+\frac{2 b e n \left (d+e x^2\right )^{5/2}}{175 d^2 x^5}-\frac{b n \left (d+e x^2\right )^{7/2}}{49 d^2 x^7}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac{\left (2 b e^3 n\right ) \int \frac{\sqrt{d+e x^2}}{x^2} \, dx}{35 d^2}\\ &=\frac{2 b e^3 n \sqrt{d+e x^2}}{35 d^2 x}+\frac{2 b e^2 n \left (d+e x^2\right )^{3/2}}{105 d^2 x^3}+\frac{2 b e n \left (d+e x^2\right )^{5/2}}{175 d^2 x^5}-\frac{b n \left (d+e x^2\right )^{7/2}}{49 d^2 x^7}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac{\left (2 b e^4 n\right ) \int \frac{1}{\sqrt{d+e x^2}} \, dx}{35 d^2}\\ &=\frac{2 b e^3 n \sqrt{d+e x^2}}{35 d^2 x}+\frac{2 b e^2 n \left (d+e x^2\right )^{3/2}}{105 d^2 x^3}+\frac{2 b e n \left (d+e x^2\right )^{5/2}}{175 d^2 x^5}-\frac{b n \left (d+e x^2\right )^{7/2}}{49 d^2 x^7}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac{\left (2 b e^4 n\right ) \operatorname{Subst}\left (\int \frac{1}{1-e x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )}{35 d^2}\\ &=\frac{2 b e^3 n \sqrt{d+e x^2}}{35 d^2 x}+\frac{2 b e^2 n \left (d+e x^2\right )^{3/2}}{105 d^2 x^3}+\frac{2 b e n \left (d+e x^2\right )^{5/2}}{175 d^2 x^5}-\frac{b n \left (d+e x^2\right )^{7/2}}{49 d^2 x^7}-\frac{2 b e^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{35 d^2}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}\\ \end{align*}
Mathematica [A] time = 0.222432, size = 145, normalized size = 0.74 \[ -\frac{\sqrt{d+e x^2} \left (105 a \left (5 d-2 e x^2\right ) \left (d+e x^2\right )^2+b n \left (183 d^2 e x^2+75 d^3+71 d e^2 x^4-247 e^3 x^6\right )\right )+105 b \left (5 d-2 e x^2\right ) \left (d+e x^2\right )^{5/2} \log \left (c x^n\right )+210 b e^{7/2} n x^7 \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )}{3675 d^2 x^7} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.499, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{8}} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84193, size = 1013, normalized size = 5.17 \begin{align*} \left [\frac{105 \, b e^{\frac{7}{2}} n x^{7} \log \left (-2 \, e x^{2} + 2 \, \sqrt{e x^{2} + d} \sqrt{e} x - d\right ) +{\left ({\left (247 \, b e^{3} n + 210 \, a e^{3}\right )} x^{6} - 75 \, b d^{3} n -{\left (71 \, b d e^{2} n + 105 \, a d e^{2}\right )} x^{4} - 525 \, a d^{3} - 3 \,{\left (61 \, b d^{2} e n + 280 \, a d^{2} e\right )} x^{2} + 105 \,{\left (2 \, b e^{3} x^{6} - b d e^{2} x^{4} - 8 \, b d^{2} e x^{2} - 5 \, b d^{3}\right )} \log \left (c\right ) + 105 \,{\left (2 \, b e^{3} n x^{6} - b d e^{2} n x^{4} - 8 \, b d^{2} e n x^{2} - 5 \, b d^{3} n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{3675 \, d^{2} x^{7}}, \frac{210 \, b \sqrt{-e} e^{3} n x^{7} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) +{\left ({\left (247 \, b e^{3} n + 210 \, a e^{3}\right )} x^{6} - 75 \, b d^{3} n -{\left (71 \, b d e^{2} n + 105 \, a d e^{2}\right )} x^{4} - 525 \, a d^{3} - 3 \,{\left (61 \, b d^{2} e n + 280 \, a d^{2} e\right )} x^{2} + 105 \,{\left (2 \, b e^{3} x^{6} - b d e^{2} x^{4} - 8 \, b d^{2} e x^{2} - 5 \, b d^{3}\right )} \log \left (c\right ) + 105 \,{\left (2 \, b e^{3} n x^{6} - b d e^{2} n x^{4} - 8 \, b d^{2} e n x^{2} - 5 \, b d^{3} n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{3675 \, d^{2} x^{7}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}^{\frac{3}{2}}{\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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