Optimal. Leaf size=212 \[ \frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \left (d+e x^2\right )^{3/2}}-\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt{d+e x^2}}-\frac{3 d \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}-\frac{b d^2 n}{3 e^4 \sqrt{d+e x^2}}-\frac{16 b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{3 e^4}+\frac{8 b d n \sqrt{d+e x^2}}{3 e^4}-\frac{b n \left (d+e x^2\right )^{3/2}}{9 e^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.323787, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {266, 43, 2350, 12, 1799, 1619, 63, 208} \[ \frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \left (d+e x^2\right )^{3/2}}-\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt{d+e x^2}}-\frac{3 d \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}-\frac{b d^2 n}{3 e^4 \sqrt{d+e x^2}}-\frac{16 b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{3 e^4}+\frac{8 b d n \sqrt{d+e x^2}}{3 e^4}-\frac{b n \left (d+e x^2\right )^{3/2}}{9 e^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rule 2350
Rule 12
Rule 1799
Rule 1619
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^7 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \left (d+e x^2\right )^{3/2}}-\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt{d+e x^2}}-\frac{3 d \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}-(b n) \int \frac{-16 d^3-24 d^2 e x^2-6 d e^2 x^4+e^3 x^6}{3 e^4 x \left (d+e x^2\right )^{3/2}} \, dx\\ &=\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \left (d+e x^2\right )^{3/2}}-\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt{d+e x^2}}-\frac{3 d \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}-\frac{(b n) \int \frac{-16 d^3-24 d^2 e x^2-6 d e^2 x^4+e^3 x^6}{x \left (d+e x^2\right )^{3/2}} \, dx}{3 e^4}\\ &=\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \left (d+e x^2\right )^{3/2}}-\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt{d+e x^2}}-\frac{3 d \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}-\frac{(b n) \operatorname{Subst}\left (\int \frac{-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3}{x (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 e^4}\\ &=\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \left (d+e x^2\right )^{3/2}}-\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt{d+e x^2}}-\frac{3 d \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}-\frac{(b n) \operatorname{Subst}\left (\int \left (-\frac{d^2 e}{(d+e x)^{3/2}}-\frac{7 d e}{\sqrt{d+e x}}-\frac{16 d^2}{x \sqrt{d+e x}}+\frac{e^2 x}{\sqrt{d+e x}}\right ) \, dx,x,x^2\right )}{6 e^4}\\ &=-\frac{b d^2 n}{3 e^4 \sqrt{d+e x^2}}+\frac{7 b d n \sqrt{d+e x^2}}{3 e^4}+\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \left (d+e x^2\right )^{3/2}}-\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt{d+e x^2}}-\frac{3 d \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac{\left (8 b d^2 n\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d+e x}} \, dx,x,x^2\right )}{3 e^4}-\frac{(b n) \operatorname{Subst}\left (\int \frac{x}{\sqrt{d+e x}} \, dx,x,x^2\right )}{6 e^2}\\ &=-\frac{b d^2 n}{3 e^4 \sqrt{d+e x^2}}+\frac{7 b d n \sqrt{d+e x^2}}{3 e^4}+\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \left (d+e x^2\right )^{3/2}}-\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt{d+e x^2}}-\frac{3 d \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac{\left (16 b d^2 n\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )}{3 e^5}-\frac{(b n) \operatorname{Subst}\left (\int \left (-\frac{d}{e \sqrt{d+e x}}+\frac{\sqrt{d+e x}}{e}\right ) \, dx,x,x^2\right )}{6 e^2}\\ &=-\frac{b d^2 n}{3 e^4 \sqrt{d+e x^2}}+\frac{8 b d n \sqrt{d+e x^2}}{3 e^4}-\frac{b n \left (d+e x^2\right )^{3/2}}{9 e^4}-\frac{16 b d^{3/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{3 e^4}+\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \left (d+e x^2\right )^{3/2}}-\frac{3 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt{d+e x^2}}-\frac{3 d \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}\\ \end{align*}
Mathematica [A] time = 0.249205, size = 240, normalized size = 1.13 \[ \frac{-72 a d^2 e x^2-48 a d^3-18 a d e^2 x^4+3 a e^3 x^6-3 b \left (24 d^2 e x^2+16 d^3+6 d e^2 x^4-e^3 x^6\right ) \log \left (c x^n\right )+42 b d^2 e n x^2-48 b d^{3/2} e n x^2 \sqrt{d+e x^2} \log \left (\sqrt{d} \sqrt{d+e x^2}+d\right )+48 b d^{3/2} n \log (x) \left (d+e x^2\right )^{3/2}-48 b d^{5/2} n \sqrt{d+e x^2} \log \left (\sqrt{d} \sqrt{d+e x^2}+d\right )+20 b d^3 n+21 b d e^2 n x^4-b e^3 n x^6}{9 e^4 \left (d+e x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.406, size = 0, normalized size = 0. \begin{align*} \int{{x}^{7} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \left ( e{x}^{2}+d \right ) ^{-{\frac{5}{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 = 1.95926, size = 1134, normalized size = 5.35 \begin{align*} \left [\frac{24 \,{\left (b d e^{2} n x^{4} + 2 \, b d^{2} e n x^{2} + b d^{3} n\right )} \sqrt{d} \log \left (-\frac{e x^{2} - 2 \, \sqrt{e x^{2} + d} \sqrt{d} + 2 \, d}{x^{2}}\right ) -{\left ({\left (b e^{3} n - 3 \, a e^{3}\right )} x^{6} - 20 \, b d^{3} n - 3 \,{\left (7 \, b d e^{2} n - 6 \, a d e^{2}\right )} x^{4} + 48 \, a d^{3} - 6 \,{\left (7 \, b d^{2} e n - 12 \, a d^{2} e\right )} x^{2} - 3 \,{\left (b e^{3} x^{6} - 6 \, b d e^{2} x^{4} - 24 \, b d^{2} e x^{2} - 16 \, b d^{3}\right )} \log \left (c\right ) - 3 \,{\left (b e^{3} n x^{6} - 6 \, b d e^{2} n x^{4} - 24 \, b d^{2} e n x^{2} - 16 \, b d^{3} n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{9 \,{\left (e^{6} x^{4} + 2 \, d e^{5} x^{2} + d^{2} e^{4}\right )}}, \frac{48 \,{\left (b d e^{2} n x^{4} + 2 \, b d^{2} e n x^{2} + b d^{3} n\right )} \sqrt{-d} \arctan \left (\frac{\sqrt{-d}}{\sqrt{e x^{2} + d}}\right ) -{\left ({\left (b e^{3} n - 3 \, a e^{3}\right )} x^{6} - 20 \, b d^{3} n - 3 \,{\left (7 \, b d e^{2} n - 6 \, a d e^{2}\right )} x^{4} + 48 \, a d^{3} - 6 \,{\left (7 \, b d^{2} e n - 12 \, a d^{2} e\right )} x^{2} - 3 \,{\left (b e^{3} x^{6} - 6 \, b d e^{2} x^{4} - 24 \, b d^{2} e x^{2} - 16 \, b d^{3}\right )} \log \left (c\right ) - 3 \,{\left (b e^{3} n x^{6} - 6 \, b d e^{2} n x^{4} - 24 \, b d^{2} e n x^{2} - 16 \, b d^{3} n\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{9 \,{\left (e^{6} x^{4} + 2 \, d e^{5} x^{2} + d^{2} e^{4}\right )}}\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 (b \log \left (c x^{n}\right ) + a\right )} x^{7}}{{\left (e x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]