Optimal. Leaf size=256 \[ -\frac{8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}+\frac{4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}-\frac{8 b e^4 n \sqrt{d+e x^2}}{315 d^3 x}-\frac{8 b e^3 n \left (d+e x^2\right )^{3/2}}{945 d^3 x^3}-\frac{8 b e^2 n \left (d+e x^2\right )^{5/2}}{1575 d^3 x^5}+\frac{8 b e^{9/2} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{315 d^3}+\frac{50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}-\frac{b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.220568, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {271, 264, 2350, 12, 1265, 451, 277, 217, 206} \[ -\frac{8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}+\frac{4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}-\frac{8 b e^4 n \sqrt{d+e x^2}}{315 d^3 x}-\frac{8 b e^3 n \left (d+e x^2\right )^{3/2}}{945 d^3 x^3}-\frac{8 b e^2 n \left (d+e x^2\right )^{5/2}}{1575 d^3 x^5}+\frac{8 b e^{9/2} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{315 d^3}+\frac{50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}-\frac{b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 271
Rule 264
Rule 2350
Rule 12
Rule 1265
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^{10}} \, dx &=-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac{4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac{8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}-(b n) \int \frac{\left (d+e x^2\right )^{5/2} \left (-35 d^2+20 d e x^2-8 e^2 x^4\right )}{315 d^3 x^{10}} \, dx\\ &=-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac{4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac{8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}-\frac{(b n) \int \frac{\left (d+e x^2\right )^{5/2} \left (-35 d^2+20 d e x^2-8 e^2 x^4\right )}{x^{10}} \, dx}{315 d^3}\\ &=-\frac{b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac{4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac{8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}+\frac{(b n) \int \frac{\left (d+e x^2\right )^{5/2} \left (-250 d^2 e+72 d e^2 x^2\right )}{x^8} \, dx}{2835 d^4}\\ &=-\frac{b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9}+\frac{50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac{4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac{8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}+\frac{\left (8 b e^2 n\right ) \int \frac{\left (d+e x^2\right )^{5/2}}{x^6} \, dx}{315 d^3}\\ &=-\frac{8 b e^2 n \left (d+e x^2\right )^{5/2}}{1575 d^3 x^5}-\frac{b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9}+\frac{50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac{4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac{8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}+\frac{\left (8 b e^3 n\right ) \int \frac{\left (d+e x^2\right )^{3/2}}{x^4} \, dx}{315 d^3}\\ &=-\frac{8 b e^3 n \left (d+e x^2\right )^{3/2}}{945 d^3 x^3}-\frac{8 b e^2 n \left (d+e x^2\right )^{5/2}}{1575 d^3 x^5}-\frac{b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9}+\frac{50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac{4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac{8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}+\frac{\left (8 b e^4 n\right ) \int \frac{\sqrt{d+e x^2}}{x^2} \, dx}{315 d^3}\\ &=-\frac{8 b e^4 n \sqrt{d+e x^2}}{315 d^3 x}-\frac{8 b e^3 n \left (d+e x^2\right )^{3/2}}{945 d^3 x^3}-\frac{8 b e^2 n \left (d+e x^2\right )^{5/2}}{1575 d^3 x^5}-\frac{b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9}+\frac{50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac{4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac{8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}+\frac{\left (8 b e^5 n\right ) \int \frac{1}{\sqrt{d+e x^2}} \, dx}{315 d^3}\\ &=-\frac{8 b e^4 n \sqrt{d+e x^2}}{315 d^3 x}-\frac{8 b e^3 n \left (d+e x^2\right )^{3/2}}{945 d^3 x^3}-\frac{8 b e^2 n \left (d+e x^2\right )^{5/2}}{1575 d^3 x^5}-\frac{b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9}+\frac{50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac{4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac{8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}+\frac{\left (8 b e^5 n\right ) \operatorname{Subst}\left (\int \frac{1}{1-e x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )}{315 d^3}\\ &=-\frac{8 b e^4 n \sqrt{d+e x^2}}{315 d^3 x}-\frac{8 b e^3 n \left (d+e x^2\right )^{3/2}}{945 d^3 x^3}-\frac{8 b e^2 n \left (d+e x^2\right )^{5/2}}{1575 d^3 x^5}-\frac{b n \left (d+e x^2\right )^{7/2}}{81 d^2 x^9}+\frac{50 b e n \left (d+e x^2\right )^{7/2}}{3969 d^3 x^7}+\frac{8 b e^{9/2} n \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{315 d^3}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{9 d x^9}+\frac{4 e \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{63 d^2 x^7}-\frac{8 e^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{315 d^3 x^5}\\ \end{align*}
Mathematica [A] time = 0.268336, size = 178, normalized size = 0.7 \[ -\frac{\sqrt{d+e x^2} \left (315 a \left (35 d^2-20 d e x^2+8 e^2 x^4\right ) \left (d+e x^2\right )^2+b n \left (429 d^2 e^2 x^4+2425 d^3 e x^2+1225 d^4-677 d e^3 x^6+2614 e^4 x^8\right )\right )+315 b \left (d+e x^2\right )^{5/2} \left (35 d^2-20 d e x^2+8 e^2 x^4\right ) \log \left (c x^n\right )-2520 b e^{9/2} n x^9 \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )}{99225 d^3 x^9} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.51, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{10}} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.18455, size = 1287, normalized size = 5.03 \begin{align*} \left [\frac{1260 \, b e^{\frac{9}{2}} n x^{9} \log \left (-2 \, e x^{2} - 2 \, \sqrt{e x^{2} + d} \sqrt{e} x - d\right ) -{\left (2 \,{\left (1307 \, b e^{4} n + 1260 \, a e^{4}\right )} x^{8} -{\left (677 \, b d e^{3} n + 1260 \, a d e^{3}\right )} x^{6} + 1225 \, b d^{4} n + 11025 \, a d^{4} + 3 \,{\left (143 \, b d^{2} e^{2} n + 315 \, a d^{2} e^{2}\right )} x^{4} + 25 \,{\left (97 \, b d^{3} e n + 630 \, a d^{3} e\right )} x^{2} + 315 \,{\left (8 \, b e^{4} x^{8} - 4 \, b d e^{3} x^{6} + 3 \, b d^{2} e^{2} x^{4} + 50 \, b d^{3} e x^{2} + 35 \, b d^{4}\right )} \log \left (c\right ) + 315 \,{\left (8 \, b e^{4} n x^{8} - 4 \, b d e^{3} n x^{6} + 3 \, b d^{2} e^{2} n x^{4} + 50 \, b d^{3} e n x^{2} + 35 \, b d^{4} n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{99225 \, d^{3} x^{9}}, -\frac{2520 \, b \sqrt{-e} e^{4} n x^{9} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) +{\left (2 \,{\left (1307 \, b e^{4} n + 1260 \, a e^{4}\right )} x^{8} -{\left (677 \, b d e^{3} n + 1260 \, a d e^{3}\right )} x^{6} + 1225 \, b d^{4} n + 11025 \, a d^{4} + 3 \,{\left (143 \, b d^{2} e^{2} n + 315 \, a d^{2} e^{2}\right )} x^{4} + 25 \,{\left (97 \, b d^{3} e n + 630 \, a d^{3} e\right )} x^{2} + 315 \,{\left (8 \, b e^{4} x^{8} - 4 \, b d e^{3} x^{6} + 3 \, b d^{2} e^{2} x^{4} + 50 \, b d^{3} e x^{2} + 35 \, b d^{4}\right )} \log \left (c\right ) + 315 \,{\left (8 \, b e^{4} n x^{8} - 4 \, b d e^{3} n x^{6} + 3 \, b d^{2} e^{2} n x^{4} + 50 \, b d^{3} e n x^{2} + 35 \, b d^{4} n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{99225 \, d^{3} x^{9}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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^{10}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]