Integrand size = 22, antiderivative size = 378 \[ \int \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {9}{32} b d n x \sqrt {d+e x^2}-\frac {1}{16} b n x \left (d+e x^2\right )^{3/2}+\frac {3 b d^{5/2} n \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{16 \sqrt {e} \sqrt {d+e x^2}}-\frac {9 b d^2 n \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{32 \sqrt {e}}-\frac {3 b d^{5/2} n \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{8 \sqrt {e} \sqrt {d+e x^2}}+\frac {3}{8} d x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^{5/2} \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 \sqrt {e} \sqrt {d+e x^2}}-\frac {3 b d^{5/2} n \sqrt {1+\frac {e x^2}{d}} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{16 \sqrt {e} \sqrt {d+e x^2}} \]
-1/16*b*n*x*(e*x^2+d)^(3/2)+1/4*x*(e*x^2+d)^(3/2)*(a+b*ln(c*x^n))-9/32*b*d ^2*n*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/e^(1/2)-9/32*b*d*n*x*(e*x^2+d)^(1/ 2)+3/8*d*x*(a+b*ln(c*x^n))*(e*x^2+d)^(1/2)+3/16*b*d^(5/2)*n*arcsinh(x*e^(1 /2)/d^(1/2))^2*(1+e*x^2/d)^(1/2)/e^(1/2)/(e*x^2+d)^(1/2)-3/8*b*d^(5/2)*n*a rcsinh(x*e^(1/2)/d^(1/2))*ln(1-(x*e^(1/2)/d^(1/2)+(1+e*x^2/d)^(1/2))^2)*(1 +e*x^2/d)^(1/2)/e^(1/2)/(e*x^2+d)^(1/2)+3/8*d^(5/2)*arcsinh(x*e^(1/2)/d^(1 /2))*(a+b*ln(c*x^n))*(1+e*x^2/d)^(1/2)/e^(1/2)/(e*x^2+d)^(1/2)-3/16*b*d^(5 /2)*n*polylog(2,(x*e^(1/2)/d^(1/2)+(1+e*x^2/d)^(1/2))^2)*(1+e*x^2/d)^(1/2) /e^(1/2)/(e*x^2+d)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.55 (sec) , antiderivative size = 314, normalized size of antiderivative = 0.83 \[ \int \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {-8 b e^{3/2} n x^3 \sqrt {d+e x^2} \, _3F_2\left (-\frac {1}{2},\frac {3}{2},\frac {3}{2};\frac {5}{2},\frac {5}{2};-\frac {e x^2}{d}\right )+9 \left (-4 b d \sqrt {e} n x \sqrt {d+e x^2} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-\frac {e x^2}{d}\right )+b d^{3/2} n \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) (-2+3 \log (x))+\sqrt {1+\frac {e x^2}{d}} \left (\sqrt {e} x \sqrt {d+e x^2} \left (5 a d-2 b d n+2 a e x^2\right )+3 d^2 (a-b n \log (x)) \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )+b \log \left (c x^n\right ) \left (\sqrt {e} x \sqrt {d+e x^2} \left (5 d+2 e x^2\right )+3 d^2 \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )\right )\right )\right )}{72 \sqrt {e} \sqrt {1+\frac {e x^2}{d}}} \]
(-8*b*e^(3/2)*n*x^3*Sqrt[d + e*x^2]*HypergeometricPFQ[{-1/2, 3/2, 3/2}, {5 /2, 5/2}, -((e*x^2)/d)] + 9*(-4*b*d*Sqrt[e]*n*x*Sqrt[d + e*x^2]*Hypergeome tricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, -((e*x^2)/d)] + b*d^(3/2)*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(-2 + 3*Log[x]) + Sqrt[1 + (e*x^2)/d]* (Sqrt[e]*x*Sqrt[d + e*x^2]*(5*a*d - 2*b*d*n + 2*a*e*x^2) + 3*d^2*(a - b*n* Log[x])*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]] + b*Log[c*x^n]*(Sqrt[e]*x*Sqrt[ d + e*x^2]*(5*d + 2*e*x^2) + 3*d^2*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]]))))/ (72*Sqrt[e]*Sqrt[1 + (e*x^2)/d])
Result contains complex when optimal does not.
Time = 1.12 (sec) , antiderivative size = 355, normalized size of antiderivative = 0.94, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.864, Rules used = {2758, 211, 211, 224, 219, 2758, 211, 224, 219, 2764, 2762, 6190, 3042, 26, 4199, 25, 2620, 2715, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
| \(\Big \downarrow \) 2758 |
| \(\displaystyle \frac {3}{4} d \int \sqrt {e x^2+d} \left (a+b \log \left (c x^n\right )\right )dx-\frac {1}{4} b n \int \left (e x^2+d\right )^{3/2}dx+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {3}{4} d \int \sqrt {e x^2+d} \left (a+b \log \left (c x^n\right )\right )dx-\frac {1}{4} b n \left (\frac {3}{4} d \int \sqrt {e x^2+d}dx+\frac {1}{4} x \left (d+e x^2\right )^{3/2}\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {3}{4} d \int \sqrt {e x^2+d} \left (a+b \log \left (c x^n\right )\right )dx-\frac {1}{4} b n \left (\frac {3}{4} d \left (\frac {1}{2} d \int \frac {1}{\sqrt {e x^2+d}}dx+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2}\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {3}{4} d \int \sqrt {e x^2+d} \left (a+b \log \left (c x^n\right )\right )dx-\frac {1}{4} b n \left (\frac {3}{4} d \left (\frac {1}{2} d \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2}\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{4} d \int \sqrt {e x^2+d} \left (a+b \log \left (c x^n\right )\right )dx+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \left (\frac {3}{4} d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 2758 |
| \(\displaystyle \frac {3}{4} d \left (\frac {1}{2} d \int \frac {a+b \log \left (c x^n\right )}{\sqrt {e x^2+d}}dx-\frac {1}{2} b n \int \sqrt {e x^2+d}dx+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \left (\frac {3}{4} d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {3}{4} d \left (\frac {1}{2} d \int \frac {a+b \log \left (c x^n\right )}{\sqrt {e x^2+d}}dx-\frac {1}{2} b n \left (\frac {1}{2} d \int \frac {1}{\sqrt {e x^2+d}}dx+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \left (\frac {3}{4} d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {3}{4} d \left (\frac {1}{2} d \int \frac {a+b \log \left (c x^n\right )}{\sqrt {e x^2+d}}dx-\frac {1}{2} b n \left (\frac {1}{2} d \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \left (\frac {3}{4} d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{4} d \left (\frac {1}{2} d \int \frac {a+b \log \left (c x^n\right )}{\sqrt {e x^2+d}}dx+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \left (\frac {3}{4} d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 2764 |
| \(\displaystyle \frac {3}{4} d \left (\frac {d \sqrt {\frac {e x^2}{d}+1} \int \frac {a+b \log \left (c x^n\right )}{\sqrt {\frac {e x^2}{d}+1}}dx}{2 \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \left (\frac {3}{4} d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 2762 |
| \(\displaystyle \frac {3}{4} d \left (\frac {d \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}-\frac {b \sqrt {d} n \int \frac {\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x}dx}{\sqrt {e}}\right )}{2 \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \left (\frac {3}{4} d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 6190 |
| \(\displaystyle \frac {3}{4} d \left (\frac {d \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}-\frac {b \sqrt {d} n \int \frac {\sqrt {d} \sqrt {\frac {e x^2}{d}+1} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e} x}d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}\right )}{2 \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \left (\frac {3}{4} d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} d \left (\frac {d \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}-\frac {b \sqrt {d} n \int -i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \tan \left (i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+\frac {\pi }{2}\right )d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}\right )}{2 \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \left (\frac {3}{4} d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {3}{4} d \left (\frac {d \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {i b \sqrt {d} n \int \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \tan \left (i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+\frac {\pi }{2}\right )d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}\right )}{2 \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \left (\frac {3}{4} d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 4199 |
| \(\displaystyle \frac {3}{4} d \left (\frac {d \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {i b \sqrt {d} n \left (2 i \int -\frac {e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}}d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2\right )}{\sqrt {e}}\right )}{2 \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \left (\frac {3}{4} d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{4} d \left (\frac {d \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {i b \sqrt {d} n \left (-2 i \int \frac {e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}}d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2\right )}{\sqrt {e}}\right )}{2 \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \left (\frac {3}{4} d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {3}{4} d \left (\frac {d \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {i b \sqrt {d} n \left (-2 i \left (\frac {1}{2} \int \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )d\text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-\frac {1}{2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2\right )}{\sqrt {e}}\right )}{2 \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \left (\frac {3}{4} d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {3}{4} d \left (\frac {d \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {i b \sqrt {d} n \left (-2 i \left (\frac {1}{4} \int e^{-2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )} \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )de^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}-\frac {1}{2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2\right )}{\sqrt {e}}\right )}{2 \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \left (\frac {3}{4} d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {3}{4} d \left (\frac {d \sqrt {\frac {e x^2}{d}+1} \left (\frac {\sqrt {d} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {i b \sqrt {d} n \left (-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )-\frac {1}{2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2\right )}{\sqrt {e}}\right )}{2 \sqrt {d+e x^2}}+\frac {1}{2} x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \left (\frac {3}{4} d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2}\right )\) |
-1/4*(b*n*((x*(d + e*x^2)^(3/2))/4 + (3*d*((x*Sqrt[d + e*x^2])/2 + (d*ArcT anh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*Sqrt[e])))/4)) + (x*(d + e*x^2)^(3/2) *(a + b*Log[c*x^n]))/4 + (3*d*(-1/2*(b*n*((x*Sqrt[d + e*x^2])/2 + (d*ArcTa nh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*Sqrt[e]))) + (x*Sqrt[d + e*x^2]*(a + b *Log[c*x^n]))/2 + (d*Sqrt[1 + (e*x^2)/d]*((Sqrt[d]*ArcSinh[(Sqrt[e]*x)/Sqr t[d]]*(a + b*Log[c*x^n]))/Sqrt[e] + (I*b*Sqrt[d]*n*((-1/2*I)*ArcSinh[(Sqrt [e]*x)/Sqrt[d]]^2 - (2*I)*(-1/2*(ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2 *ArcSinh[(Sqrt[e]*x)/Sqrt[d]])]) - PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqr t[d]])]/4)))/Sqrt[e]))/(2*Sqrt[d + e*x^2])))/4
3.3.69.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_.), x_Sy mbol] :> Simp[x*(d + e*x^2)^q*((a + b*Log[c*x^n])/(2*q + 1)), x] + (-Simp[b *(n/(2*q + 1)) Int[(d + e*x^2)^q, x], x] + Simp[2*d*(q/(2*q + 1)) Int[( d + e*x^2)^(q - 1)*(a + b*Log[c*x^n]), x], x]) /; FreeQ[{a, b, c, d, e, n}, x] && GtQ[q, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symb ol] :> Simp[ArcSinh[Rt[e, 2]*(x/Sqrt[d])]*((a + b*Log[c*x^n])/Rt[e, 2]), x] - Simp[b*(n/Rt[e, 2]) Int[ArcSinh[Rt[e, 2]*(x/Sqrt[d])]/x, x], x] /; Fre eQ[{a, b, c, d, e, n}, x] && GtQ[d, 0] && PosQ[e]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symb ol] :> Simp[Sqrt[1 + (e/d)*x^2]/Sqrt[d + e*x^2] Int[(a + b*Log[c*x^n])/Sq rt[1 + (e/d)*x^2], x], x] /; FreeQ[{a, b, c, d, e, n}, x] && !GtQ[d, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_ .)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp [2*I Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x ))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In tegerQ[4*k] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b Subst[Int[x^n*Coth[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a , b, c}, x] && IGtQ[n, 0]
\[\int \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )d x\]
\[ \int \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )} \,d x } \]
Timed out. \[ \int \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Timed out} \]
Exception generated. \[ \int \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )} \,d x } \]
Timed out. \[ \int \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\int {\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]