Integrand size = 23, antiderivative size = 381 \[ \int \frac {(d+e x)^{3/2} \log (a+b x)}{a+b x} \, dx=-\frac {16 (b d-a e) \sqrt {d+e x}}{3 b^2}-\frac {4 (d+e x)^{3/2}}{9 b}+\frac {16 (b d-a e)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{3 b^{5/2}}+\frac {2 (b d-a e)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{b^{5/2}}+\frac {2 (b d-a e) \sqrt {d+e x} \log (a+b x)}{b^2}+\frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {2 (b d-a e)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{5/2}}-\frac {4 (b d-a e)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{5/2}}-\frac {2 (b d-a e)^{3/2} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{5/2}} \]
-4/9*(e*x+d)^(3/2)/b+16/3*(-a*e+b*d)^(3/2)*arctanh(b^(1/2)*(e*x+d)^(1/2)/( -a*e+b*d)^(1/2))/b^(5/2)+2*(-a*e+b*d)^(3/2)*arctanh(b^(1/2)*(e*x+d)^(1/2)/ (-a*e+b*d)^(1/2))^2/b^(5/2)+2/3*(e*x+d)^(3/2)*ln(b*x+a)/b-2*(-a*e+b*d)^(3/ 2)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))*ln(b*x+a)/b^(5/2)-4*(-a *e+b*d)^(3/2)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))*ln(2/(1-b^(1 /2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2)))/b^(5/2)-2*(-a*e+b*d)^(3/2)*polylog(2, 1-2/(1-b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2)))/b^(5/2)-16/3*(-a*e+b*d)*(e *x+d)^(1/2)/b^2+2*(-a*e+b*d)*ln(b*x+a)*(e*x+d)^(1/2)/b^2
Time = 0.59 (sec) , antiderivative size = 556, normalized size of antiderivative = 1.46 \[ \int \frac {(d+e x)^{3/2} \log (a+b x)}{a+b x} \, dx=\frac {-72 \sqrt {b} (b d-a e) \sqrt {d+e x}-8 \sqrt {b} \sqrt {d+e x} (4 b d-3 a e+b e x)+96 (b d-a e)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )+36 \sqrt {b} (b d-a e) \sqrt {d+e x} \log (a+b x)+12 b^{3/2} (d+e x)^{3/2} \log (a+b x)+18 (b d-a e)^{3/2} \log (a+b x) \log \left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right )-18 (b d-a e)^{3/2} \log (a+b x) \log \left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right )-9 (b d-a e)^{3/2} \left (\log \left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right ) \left (\log \left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {b} \sqrt {d+e x}}{2 \sqrt {b d-a e}}\right )\right )+9 (b d-a e)^{3/2} \left (\log \left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right ) \left (\log \left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {b} \sqrt {d+e x}}{2 \sqrt {b d-a e}}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )\right )\right )}{18 b^{5/2}} \]
(-72*Sqrt[b]*(b*d - a*e)*Sqrt[d + e*x] - 8*Sqrt[b]*Sqrt[d + e*x]*(4*b*d - 3*a*e + b*e*x) + 96*(b*d - a*e)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt [b*d - a*e]] + 36*Sqrt[b]*(b*d - a*e)*Sqrt[d + e*x]*Log[a + b*x] + 12*b^(3 /2)*(d + e*x)^(3/2)*Log[a + b*x] + 18*(b*d - a*e)^(3/2)*Log[a + b*x]*Log[S qrt[b*d - a*e] - Sqrt[b]*Sqrt[d + e*x]] - 18*(b*d - a*e)^(3/2)*Log[a + b*x ]*Log[Sqrt[b*d - a*e] + Sqrt[b]*Sqrt[d + e*x]] - 9*(b*d - a*e)^(3/2)*(Log[ Sqrt[b*d - a*e] - Sqrt[b]*Sqrt[d + e*x]]*(Log[Sqrt[b*d - a*e] - Sqrt[b]*Sq rt[d + e*x]] + 2*Log[(1 + (Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e])/2]) + 2 *PolyLog[2, 1/2 - (Sqrt[b]*Sqrt[d + e*x])/(2*Sqrt[b*d - a*e])]) + 9*(b*d - a*e)^(3/2)*(Log[Sqrt[b*d - a*e] + Sqrt[b]*Sqrt[d + e*x]]*(Log[Sqrt[b*d - a*e] + Sqrt[b]*Sqrt[d + e*x]] + 2*Log[1/2 - (Sqrt[b]*Sqrt[d + e*x])/(2*Sqr t[b*d - a*e])]) + 2*PolyLog[2, (1 + (Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e ])/2]))/(18*b^(5/2))
Time = 2.86 (sec) , antiderivative size = 695, normalized size of antiderivative = 1.82, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.870, Rules used = {2858, 2788, 2756, 60, 60, 73, 221, 2788, 2756, 60, 73, 221, 2790, 27, 7267, 2092, 6546, 6470, 2849, 2752}
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.
| \(\displaystyle \int \frac {(d+e x)^{3/2} \log (a+b x)}{a+b x} \, dx\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {\int \frac {\left (d-\frac {a e}{b}+\frac {e (a+b x)}{b}\right )^{3/2} \log (a+b x)}{a+b x}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 2788 |
| \(\displaystyle \frac {\frac {e \int \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}} \log (a+b x)d(a+b x)}{b}+\left (d-\frac {a e}{b}\right ) \int \frac {\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}} \log (a+b x)}{a+b x}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {\frac {e \left (\frac {2 b \log (a+b x) \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}}{3 e}-\frac {2 b \int \frac {\left (d-\frac {a e}{b}+\frac {e (a+b x)}{b}\right )^{3/2}}{a+b x}d(a+b x)}{3 e}\right )}{b}+\left (d-\frac {a e}{b}\right ) \int \frac {\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}} \log (a+b x)}{a+b x}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {e \left (\frac {2 b \log (a+b x) \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}}{3 e}-\frac {2 b \left (\left (d-\frac {a e}{b}\right ) \int \frac {\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{a+b x}d(a+b x)+\frac {2}{3} \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}\right )}{3 e}\right )}{b}+\left (d-\frac {a e}{b}\right ) \int \frac {\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}} \log (a+b x)}{a+b x}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {e \left (\frac {2 b \log (a+b x) \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}}{3 e}-\frac {2 b \left (\left (d-\frac {a e}{b}\right ) \left (\left (d-\frac {a e}{b}\right ) \int \frac {1}{(a+b x) \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}d(a+b x)+2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}\right )+\frac {2}{3} \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}\right )}{3 e}\right )}{b}+\left (d-\frac {a e}{b}\right ) \int \frac {\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}} \log (a+b x)}{a+b x}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {e \left (\frac {2 b \log (a+b x) \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}}{3 e}-\frac {2 b \left (\left (d-\frac {a e}{b}\right ) \left (\frac {2 b \left (d-\frac {a e}{b}\right ) \int \frac {1}{a+\frac {b \left (d-\frac {a e}{b}+\frac {e (a+b x)}{b}\right )}{e}-\frac {b d}{e}}d\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{e}+2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}\right )+\frac {2}{3} \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}\right )}{3 e}\right )}{b}+\left (d-\frac {a e}{b}\right ) \int \frac {\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}} \log (a+b x)}{a+b x}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (d-\frac {a e}{b}\right ) \int \frac {\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}} \log (a+b x)}{a+b x}d(a+b x)+\frac {e \left (\frac {2 b \log (a+b x) \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}}{3 e}-\frac {2 b \left (\left (d-\frac {a e}{b}\right ) \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {2}{3} \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}\right )}{3 e}\right )}{b}}{b}\) |
| \(\Big \downarrow \) 2788 |
| \(\displaystyle \frac {\left (d-\frac {a e}{b}\right ) \left (\frac {e \int \frac {\log (a+b x)}{\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}d(a+b x)}{b}+\left (d-\frac {a e}{b}\right ) \int \frac {\log (a+b x)}{(a+b x) \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}d(a+b x)\right )+\frac {e \left (\frac {2 b \log (a+b x) \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}}{3 e}-\frac {2 b \left (\left (d-\frac {a e}{b}\right ) \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {2}{3} \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}\right )}{3 e}\right )}{b}}{b}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {\left (d-\frac {a e}{b}\right ) \left (\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \int \frac {\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{a+b x}d(a+b x)}{e}\right )}{b}+\left (d-\frac {a e}{b}\right ) \int \frac {\log (a+b x)}{(a+b x) \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}d(a+b x)\right )+\frac {e \left (\frac {2 b \log (a+b x) \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}}{3 e}-\frac {2 b \left (\left (d-\frac {a e}{b}\right ) \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {2}{3} \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}\right )}{3 e}\right )}{b}}{b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\left (d-\frac {a e}{b}\right ) \left (\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \left (\left (d-\frac {a e}{b}\right ) \int \frac {1}{(a+b x) \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}d(a+b x)+2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}\right )}{e}\right )}{b}+\left (d-\frac {a e}{b}\right ) \int \frac {\log (a+b x)}{(a+b x) \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}d(a+b x)\right )+\frac {e \left (\frac {2 b \log (a+b x) \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}}{3 e}-\frac {2 b \left (\left (d-\frac {a e}{b}\right ) \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {2}{3} \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}\right )}{3 e}\right )}{b}}{b}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\left (d-\frac {a e}{b}\right ) \left (\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \left (\frac {2 b \left (d-\frac {a e}{b}\right ) \int \frac {1}{a+\frac {b \left (d-\frac {a e}{b}+\frac {e (a+b x)}{b}\right )}{e}-\frac {b d}{e}}d\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{e}+2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}\right )}{e}\right )}{b}+\left (d-\frac {a e}{b}\right ) \int \frac {\log (a+b x)}{(a+b x) \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}d(a+b x)\right )+\frac {e \left (\frac {2 b \log (a+b x) \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}}{3 e}-\frac {2 b \left (\left (d-\frac {a e}{b}\right ) \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {2}{3} \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}\right )}{3 e}\right )}{b}}{b}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (d-\frac {a e}{b}\right ) \left (\left (d-\frac {a e}{b}\right ) \int \frac {\log (a+b x)}{(a+b x) \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}d(a+b x)+\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )}{e}\right )}{b}\right )+\frac {e \left (\frac {2 b \log (a+b x) \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}}{3 e}-\frac {2 b \left (\left (d-\frac {a e}{b}\right ) \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {2}{3} \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}\right )}{3 e}\right )}{b}}{b}\) |
| \(\Big \downarrow \) 2790 |
| \(\displaystyle \frac {\left (d-\frac {a e}{b}\right ) \left (\left (d-\frac {a e}{b}\right ) \left (-\int -\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e} (a+b x)}d(a+b x)-\frac {2 \sqrt {b} \log (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )}{e}\right )}{b}\right )+\frac {e \left (\frac {2 b \log (a+b x) \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}}{3 e}-\frac {2 b \left (\left (d-\frac {a e}{b}\right ) \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {2}{3} \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}\right )}{3 e}\right )}{b}}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (d-\frac {a e}{b}\right ) \left (\left (d-\frac {a e}{b}\right ) \left (\frac {2 \sqrt {b} \int \frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}\right )}{a+b x}d(a+b x)}{\sqrt {b d-a e}}-\frac {2 \sqrt {b} \log (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )}{e}\right )}{b}\right )+\frac {e \left (\frac {2 b \log (a+b x) \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}}{3 e}-\frac {2 b \left (\left (d-\frac {a e}{b}\right ) \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {2}{3} \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}\right )}{3 e}\right )}{b}}{b}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {\left (d-\frac {a e}{b}\right ) \left (\left (d-\frac {a e}{b}\right ) \left (\frac {4 b^{3/2} \int \frac {\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}\right )}{a e-b \left (\frac {a e}{b}-\frac {e (a+b x)}{b}\right )}d\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}-\frac {2 \sqrt {b} \log (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )}{e}\right )}{b}\right )+\frac {e \left (\frac {2 b \log (a+b x) \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}}{3 e}-\frac {2 b \left (\left (d-\frac {a e}{b}\right ) \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {2}{3} \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}\right )}{3 e}\right )}{b}}{b}\) |
| \(\Big \downarrow \) 2092 |
| \(\displaystyle \frac {\left (d-\frac {a e}{b}\right ) \left (\left (d-\frac {a e}{b}\right ) \left (\frac {4 b^{3/2} \int \frac {\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}\right )}{-b d+a e+b \left (d-\frac {a e}{b}+\frac {e (a+b x)}{b}\right )}d\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}-\frac {2 \sqrt {b} \log (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )}{e}\right )}{b}\right )+\frac {e \left (\frac {2 b \log (a+b x) \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}}{3 e}-\frac {2 b \left (\left (d-\frac {a e}{b}\right ) \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {2}{3} \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}\right )}{3 e}\right )}{b}}{b}\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle \frac {\left (d-\frac {a e}{b}\right ) \left (\left (d-\frac {a e}{b}\right ) \left (\frac {4 b^{3/2} \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )^2}{2 b}-\frac {\int \frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}\right )}{1-\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}}d\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b} \sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}-\frac {2 \sqrt {b} \log (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )}{e}\right )}{b}\right )+\frac {e \left (\frac {2 b \log (a+b x) \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}}{3 e}-\frac {2 b \left (\left (d-\frac {a e}{b}\right ) \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {2}{3} \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}\right )}{3 e}\right )}{b}}{b}\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {\left (d-\frac {a e}{b}\right ) \left (\left (d-\frac {a e}{b}\right ) \left (\frac {4 b^{3/2} \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )^2}{2 b}-\frac {\frac {\sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}}\right )}{\sqrt {b}}-\int \frac {\log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}}\right )}{1-\frac {b \left (d-\frac {a e}{b}+\frac {e (a+b x)}{b}\right )}{b d-a e}}d\sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b} \sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}-\frac {2 \sqrt {b} \log (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )}{e}\right )}{b}\right )+\frac {e \left (\frac {2 b \log (a+b x) \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}}{3 e}-\frac {2 b \left (\left (d-\frac {a e}{b}\right ) \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {2}{3} \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}\right )}{3 e}\right )}{b}}{b}\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {\left (d-\frac {a e}{b}\right ) \left (\left (d-\frac {a e}{b}\right ) \left (\frac {4 b^{3/2} \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )^2}{2 b}-\frac {\frac {\sqrt {b d-a e} \int \frac {\log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}}\right )}{1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}}}d\frac {1}{1-\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}}}{\sqrt {b}}+\frac {\sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}}\right )}{\sqrt {b}}}{\sqrt {b} \sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}-\frac {2 \sqrt {b} \log (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )}{e}\right )}{b}\right )+\frac {e \left (\frac {2 b \log (a+b x) \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}}{3 e}-\frac {2 b \left (\left (d-\frac {a e}{b}\right ) \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {2}{3} \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}\right )}{3 e}\right )}{b}}{b}\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {\left (d-\frac {a e}{b}\right ) \left (\left (d-\frac {a e}{b}\right ) \left (\frac {4 b^{3/2} \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )^2}{2 b}-\frac {\frac {\sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}}\right )}{\sqrt {b}}+\frac {\sqrt {b d-a e} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e (a+b x)}{b}}}{\sqrt {b d-a e}}}\right )}{2 \sqrt {b}}}{\sqrt {b} \sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}-\frac {2 \sqrt {b} \log (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {e \left (\frac {2 b \log (a+b x) \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{e}-\frac {2 b \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )}{e}\right )}{b}\right )+\frac {e \left (\frac {2 b \log (a+b x) \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}}{3 e}-\frac {2 b \left (\left (d-\frac {a e}{b}\right ) \left (2 \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}-\frac {2 \sqrt {b} \left (d-\frac {a e}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\frac {e (a+b x)}{b}-\frac {a e}{b}+d}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e}}\right )+\frac {2}{3} \left (\frac {e (a+b x)}{b}-\frac {a e}{b}+d\right )^{3/2}\right )}{3 e}\right )}{b}}{b}\) |
((e*((-2*b*((2*(d - (a*e)/b + (e*(a + b*x))/b)^(3/2))/3 + (d - (a*e)/b)*(2 *Sqrt[d - (a*e)/b + (e*(a + b*x))/b] - (2*Sqrt[b]*(d - (a*e)/b)*ArcTanh[(S qrt[b]*Sqrt[d - (a*e)/b + (e*(a + b*x))/b])/Sqrt[b*d - a*e]])/Sqrt[b*d - a *e])))/(3*e) + (2*b*(d - (a*e)/b + (e*(a + b*x))/b)^(3/2)*Log[a + b*x])/(3 *e)))/b + (d - (a*e)/b)*((e*((-2*b*(2*Sqrt[d - (a*e)/b + (e*(a + b*x))/b] - (2*Sqrt[b]*(d - (a*e)/b)*ArcTanh[(Sqrt[b]*Sqrt[d - (a*e)/b + (e*(a + b*x ))/b])/Sqrt[b*d - a*e]])/Sqrt[b*d - a*e]))/e + (2*b*Sqrt[d - (a*e)/b + (e* (a + b*x))/b]*Log[a + b*x])/e))/b + (d - (a*e)/b)*((-2*Sqrt[b]*ArcTanh[(Sq rt[b]*Sqrt[d - (a*e)/b + (e*(a + b*x))/b])/Sqrt[b*d - a*e]]*Log[a + b*x])/ Sqrt[b*d - a*e] + (4*b^(3/2)*(ArcTanh[(Sqrt[b]*Sqrt[d - (a*e)/b + (e*(a + b*x))/b])/Sqrt[b*d - a*e]]^2/(2*b) - ((Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sq rt[d - (a*e)/b + (e*(a + b*x))/b])/Sqrt[b*d - a*e]]*Log[2/(1 - (Sqrt[b]*Sq rt[d - (a*e)/b + (e*(a + b*x))/b])/Sqrt[b*d - a*e])])/Sqrt[b] + (Sqrt[b*d - a*e]*PolyLog[2, 1 - 2/(1 - (Sqrt[b]*Sqrt[d - (a*e)/b + (e*(a + b*x))/b]) /Sqrt[b*d - a*e])])/(2*Sqrt[b]))/(Sqrt[b]*Sqrt[b*d - a*e])))/Sqrt[b*d - a* e])))/b
3.3.4.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(Px_)*(u_)^(p_.)*(z_)^(q_.), x_Symbol] :> Int[Px*ExpandToSum[z, x]^q*Ex pandToSum[u, x]^p, x] /; FreeQ[{p, q}, x] && BinomialQ[z, x] && BinomialQ[u , x] && !(BinomialMatchQ[z, x] && BinomialMatchQ[u, x])
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Simp[b*n*(p/(e*(q + 1))) Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] & & NeQ[q, 1]))
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.)) /(x_), x_Symbol] :> Simp[d Int[(d + e*x)^(q - 1)*((a + b*Log[c*x^n])^p/x) , x], x] + Simp[e Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /; F reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.)) /(x_), x_Symbol] :> With[{u = IntHide[(d + e*x^r)^q/x, x]}, Simp[u*(a + b*L og[c*x^n]), x] - Simp[b*n Int[1/x u, x], x]] /; FreeQ[{a, b, c, d, e, n , r}, x] && IntegerQ[q - 1/2]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-e/g Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ {c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ .)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e Subst[In t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol ] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c *(p/e) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 - c^2*x^ 2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2 , 0]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ (c*d) Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.05 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.09
| method | result | size |
| derivativedivides | \(\frac {\frac {2 \left (e x +d \right )^{\frac {3}{2}} \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{e}\right )}{3}-\frac {4 b \left (-\frac {-\frac {b \left (e x +d \right )^{\frac {3}{2}}}{3}+\sqrt {e x +d}\, a e -\sqrt {e x +d}\, b d}{b^{2}}+\frac {\left (a^{2} e^{2}-2 a d e b +b^{2} d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b^{2} \sqrt {\left (a e -b d \right ) b}}\right )}{3}}{b}-\frac {2 \left (\sqrt {e x +d}\, \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{e}\right )-2 b \left (\frac {\sqrt {e x +d}}{b}+\frac {\left (-a e +b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b \sqrt {\left (a e -b d \right ) b}}\right )\right ) \left (a e -b d \right )}{b^{2}}+2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{2}+a e -b d \right )}{\sum }\frac {\left (\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{e}\right )-2 b \left (\frac {\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{4 \underline {\hspace {1.25 ex}}\alpha b}+\frac {\underline {\hspace {1.25 ex}}\alpha \ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 a e -2 b d}+\frac {\underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 a e -2 b d}\right )\right ) \left (a^{2} e^{2}-2 a d e b +b^{2} d^{2}\right )}{2 b^{3} \underline {\hspace {1.25 ex}}\alpha }\right )\) | \(416\) |
| default | \(\frac {\frac {2 \left (e x +d \right )^{\frac {3}{2}} \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{e}\right )}{3}-\frac {4 b \left (-\frac {-\frac {b \left (e x +d \right )^{\frac {3}{2}}}{3}+\sqrt {e x +d}\, a e -\sqrt {e x +d}\, b d}{b^{2}}+\frac {\left (a^{2} e^{2}-2 a d e b +b^{2} d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b^{2} \sqrt {\left (a e -b d \right ) b}}\right )}{3}}{b}-\frac {2 \left (\sqrt {e x +d}\, \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{e}\right )-2 b \left (\frac {\sqrt {e x +d}}{b}+\frac {\left (-a e +b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b \sqrt {\left (a e -b d \right ) b}}\right )\right ) \left (a e -b d \right )}{b^{2}}+2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{2}+a e -b d \right )}{\sum }\frac {\left (\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{e}\right )-2 b \left (\frac {\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{4 \underline {\hspace {1.25 ex}}\alpha b}+\frac {\underline {\hspace {1.25 ex}}\alpha \ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 a e -2 b d}+\frac {\underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 a e -2 b d}\right )\right ) \left (a^{2} e^{2}-2 a d e b +b^{2} d^{2}\right )}{2 b^{3} \underline {\hspace {1.25 ex}}\alpha }\right )\) | \(416\) |
2*(1/3*(e*x+d)^(3/2)*ln(((e*x+d)*b+a*e-b*d)/e)-2/3*b*(-1/b^2*(-1/3*b*(e*x+ d)^(3/2)+(e*x+d)^(1/2)*a*e-(e*x+d)^(1/2)*b*d)+(a^2*e^2-2*a*b*d*e+b^2*d^2)/ b^2/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))))/b-2* ((e*x+d)^(1/2)*ln(((e*x+d)*b+a*e-b*d)/e)-2*b*((e*x+d)^(1/2)/b+(-a*e+b*d)/b /((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))))*(a*e-b* d)/b^2+2*Sum(1/2*(ln((e*x+d)^(1/2)-_alpha)*ln(((e*x+d)*b+a*e-b*d)/e)-2*b*( 1/4/_alpha/b*ln((e*x+d)^(1/2)-_alpha)^2+1/2*_alpha/(a*e-b*d)*ln((e*x+d)^(1 /2)-_alpha)*ln(1/2*((e*x+d)^(1/2)+_alpha)/_alpha)+1/2*_alpha/(a*e-b*d)*dil og(1/2*((e*x+d)^(1/2)+_alpha)/_alpha)))*(a^2*e^2-2*a*b*d*e+b^2*d^2)/b^3/_a lpha,_alpha=RootOf(_Z^2*b+a*e-b*d))
\[ \int \frac {(d+e x)^{3/2} \log (a+b x)}{a+b x} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}} \log \left (b x + a\right )}{b x + a} \,d x } \]
Timed out. \[ \int \frac {(d+e x)^{3/2} \log (a+b x)}{a+b x} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(d+e x)^{3/2} \log (a+b x)}{a+b x} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m ore detail
\[ \int \frac {(d+e x)^{3/2} \log (a+b x)}{a+b x} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}} \log \left (b x + a\right )}{b x + a} \,d x } \]
Timed out. \[ \int \frac {(d+e x)^{3/2} \log (a+b x)}{a+b x} \, dx=\int \frac {\ln \left (a+b\,x\right )\,{\left (d+e\,x\right )}^{3/2}}{a+b\,x} \,d x \]