Optimal. Leaf size=74 \[ \frac{1}{60} \left (10 d^2 x^6+15 d e x^8+6 e^2 x^{10}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{36} b d^2 n x^6-\frac{1}{32} b d e n x^8-\frac{1}{100} b e^2 n x^{10} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0865796, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {266, 43, 2334, 12, 14} \[ \frac{1}{60} \left (10 d^2 x^6+15 d e x^8+6 e^2 x^{10}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{36} b d^2 n x^6-\frac{1}{32} b d e n x^8-\frac{1}{100} b e^2 n x^{10} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rule 2334
Rule 12
Rule 14
Rubi steps
\begin{align*} \int x^5 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{60} \left (10 d^2 x^6+15 d e x^8+6 e^2 x^{10}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{1}{60} x^5 \left (10 d^2+15 d e x^2+6 e^2 x^4\right ) \, dx\\ &=\frac{1}{60} \left (10 d^2 x^6+15 d e x^8+6 e^2 x^{10}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{60} (b n) \int x^5 \left (10 d^2+15 d e x^2+6 e^2 x^4\right ) \, dx\\ &=\frac{1}{60} \left (10 d^2 x^6+15 d e x^8+6 e^2 x^{10}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{60} (b n) \int \left (10 d^2 x^5+15 d e x^7+6 e^2 x^9\right ) \, dx\\ &=-\frac{1}{36} b d^2 n x^6-\frac{1}{32} b d e n x^8-\frac{1}{100} b e^2 n x^{10}+\frac{1}{60} \left (10 d^2 x^6+15 d e x^8+6 e^2 x^{10}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0398215, size = 84, normalized size = 1.14 \[ \frac{x^6 \left (1200 d^2 \left (a+b \log \left (c x^n\right )\right )+1800 d e x^2 \left (a+b \log \left (c x^n\right )\right )+720 e^2 x^4 \left (a+b \log \left (c x^n\right )\right )-200 b d^2 n-225 b d e n x^2-72 b e^2 n x^4\right )}{7200} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.195, size = 434, normalized size = 5.9 \begin{align*}{\frac{b{x}^{6} \left ( 6\,{e}^{2}{x}^{4}+15\,de{x}^{2}+10\,{d}^{2} \right ) \ln \left ({x}^{n} \right ) }{60}}-{\frac{i}{8}}\pi \,bde{x}^{8} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{12}}\pi \,b{d}^{2}{x}^{6}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+{\frac{i}{8}}\pi \,bde{x}^{8} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{i}{8}}\pi \,bde{x}^{8}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+{\frac{\ln \left ( c \right ) b{e}^{2}{x}^{10}}{10}}-{\frac{b{e}^{2}n{x}^{10}}{100}}+{\frac{a{e}^{2}{x}^{10}}{10}}-{\frac{i}{8}}\pi \,bde{x}^{8}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{20}}\pi \,b{e}^{2}{x}^{10}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +{\frac{i}{20}}\pi \,b{e}^{2}{x}^{10} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{i}{20}}\pi \,b{e}^{2}{x}^{10}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+{\frac{\ln \left ( c \right ) bde{x}^{8}}{4}}-{\frac{bden{x}^{8}}{32}}+{\frac{ade{x}^{8}}{4}}+{\frac{i}{12}}\pi \,b{d}^{2}{x}^{6} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -{\frac{i}{12}}\pi \,b{d}^{2}{x}^{6} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-{\frac{i}{12}}\pi \,b{d}^{2}{x}^{6}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{20}}\pi \,b{e}^{2}{x}^{10} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{\ln \left ( c \right ) b{d}^{2}{x}^{6}}{6}}-{\frac{b{d}^{2}n{x}^{6}}{36}}+{\frac{a{d}^{2}{x}^{6}}{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.05436, size = 135, normalized size = 1.82 \begin{align*} -\frac{1}{100} \, b e^{2} n x^{10} + \frac{1}{10} \, b e^{2} x^{10} \log \left (c x^{n}\right ) + \frac{1}{10} \, a e^{2} x^{10} - \frac{1}{32} \, b d e n x^{8} + \frac{1}{4} \, b d e x^{8} \log \left (c x^{n}\right ) + \frac{1}{4} \, a d e x^{8} - \frac{1}{36} \, b d^{2} n x^{6} + \frac{1}{6} \, b d^{2} x^{6} \log \left (c x^{n}\right ) + \frac{1}{6} \, a d^{2} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.33028, size = 297, normalized size = 4.01 \begin{align*} -\frac{1}{100} \,{\left (b e^{2} n - 10 \, a e^{2}\right )} x^{10} - \frac{1}{32} \,{\left (b d e n - 8 \, a d e\right )} x^{8} - \frac{1}{36} \,{\left (b d^{2} n - 6 \, a d^{2}\right )} x^{6} + \frac{1}{60} \,{\left (6 \, b e^{2} x^{10} + 15 \, b d e x^{8} + 10 \, b d^{2} x^{6}\right )} \log \left (c\right ) + \frac{1}{60} \,{\left (6 \, b e^{2} n x^{10} + 15 \, b d e n x^{8} + 10 \, b d^{2} n x^{6}\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 34.1154, size = 151, normalized size = 2.04 \begin{align*} \frac{a d^{2} x^{6}}{6} + \frac{a d e x^{8}}{4} + \frac{a e^{2} x^{10}}{10} + \frac{b d^{2} n x^{6} \log{\left (x \right )}}{6} - \frac{b d^{2} n x^{6}}{36} + \frac{b d^{2} x^{6} \log{\left (c \right )}}{6} + \frac{b d e n x^{8} \log{\left (x \right )}}{4} - \frac{b d e n x^{8}}{32} + \frac{b d e x^{8} \log{\left (c \right )}}{4} + \frac{b e^{2} n x^{10} \log{\left (x \right )}}{10} - \frac{b e^{2} n x^{10}}{100} + \frac{b e^{2} x^{10} \log{\left (c \right )}}{10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.30353, size = 166, normalized size = 2.24 \begin{align*} \frac{1}{10} \, b n x^{10} e^{2} \log \left (x\right ) - \frac{1}{100} \, b n x^{10} e^{2} + \frac{1}{10} \, b x^{10} e^{2} \log \left (c\right ) + \frac{1}{4} \, b d n x^{8} e \log \left (x\right ) + \frac{1}{10} \, a x^{10} e^{2} - \frac{1}{32} \, b d n x^{8} e + \frac{1}{4} \, b d x^{8} e \log \left (c\right ) + \frac{1}{4} \, a d x^{8} e + \frac{1}{6} \, b d^{2} n x^{6} \log \left (x\right ) - \frac{1}{36} \, b d^{2} n x^{6} + \frac{1}{6} \, b d^{2} x^{6} \log \left (c\right ) + \frac{1}{6} \, a d^{2} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]