Integrand size = 23, antiderivative size = 326 \[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^3 \, dx=\frac {16}{25} a b^2 e^4 x-\frac {298 b^3 e^4 \sqrt {1+(c+d x)^2}}{375 d}+\frac {76 b^3 e^4 \left (1+(c+d x)^2\right )^{3/2}}{1125 d}-\frac {6 b^3 e^4 \left (1+(c+d x)^2\right )^{5/2}}{625 d}+\frac {16 b^3 e^4 (c+d x) \text {arcsinh}(c+d x)}{25 d}-\frac {8 b^2 e^4 (c+d x)^3 (a+b \text {arcsinh}(c+d x))}{75 d}+\frac {6 b^2 e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))}{125 d}-\frac {8 b e^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{25 d}+\frac {4 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{25 d}-\frac {3 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{25 d}+\frac {e^4 (c+d x)^5 (a+b \text {arcsinh}(c+d x))^3}{5 d} \]
16/25*a*b^2*e^4*x+76/1125*b^3*e^4*(1+(d*x+c)^2)^(3/2)/d-6/625*b^3*e^4*(1+( d*x+c)^2)^(5/2)/d+16/25*b^3*e^4*(d*x+c)*arcsinh(d*x+c)/d-8/75*b^2*e^4*(d*x +c)^3*(a+b*arcsinh(d*x+c))/d+6/125*b^2*e^4*(d*x+c)^5*(a+b*arcsinh(d*x+c))/ d+1/5*e^4*(d*x+c)^5*(a+b*arcsinh(d*x+c))^3/d-298/375*b^3*e^4*(1+(d*x+c)^2) ^(1/2)/d-8/25*b*e^4*(a+b*arcsinh(d*x+c))^2*(1+(d*x+c)^2)^(1/2)/d+4/25*b*e^ 4*(d*x+c)^2*(a+b*arcsinh(d*x+c))^2*(1+(d*x+c)^2)^(1/2)/d-3/25*b*e^4*(d*x+c )^4*(a+b*arcsinh(d*x+c))^2*(1+(d*x+c)^2)^(1/2)/d
Time = 0.43 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.09 \[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^3 \, dx=\frac {e^4 \left (240 a b^2 (c+d x)-40 a b^2 (c+d x)^3+3 a \left (25 a^2+6 b^2\right ) (c+d x)^5+\frac {1}{15} b \sqrt {1+(c+d x)^2} \left (-8 \left (225 a^2+518 b^2\right )+4 \left (225 a^2+68 b^2\right ) (c+d x)^2-27 \left (25 a^2+2 b^2\right ) (c+d x)^4\right )-b \left (-240 b^2 (c+d x)+40 b^2 (c+d x)^3-225 a^2 (c+d x)^5-18 b^2 (c+d x)^5+240 a b \sqrt {1+(c+d x)^2}-120 a b (c+d x)^2 \sqrt {1+(c+d x)^2}+90 a b (c+d x)^4 \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)-15 b^2 \left (-15 a (c+d x)^5+8 b \sqrt {1+(c+d x)^2}-4 b (c+d x)^2 \sqrt {1+(c+d x)^2}+3 b (c+d x)^4 \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)^2+75 b^3 (c+d x)^5 \text {arcsinh}(c+d x)^3\right )}{375 d} \]
(e^4*(240*a*b^2*(c + d*x) - 40*a*b^2*(c + d*x)^3 + 3*a*(25*a^2 + 6*b^2)*(c + d*x)^5 + (b*Sqrt[1 + (c + d*x)^2]*(-8*(225*a^2 + 518*b^2) + 4*(225*a^2 + 68*b^2)*(c + d*x)^2 - 27*(25*a^2 + 2*b^2)*(c + d*x)^4))/15 - b*(-240*b^2 *(c + d*x) + 40*b^2*(c + d*x)^3 - 225*a^2*(c + d*x)^5 - 18*b^2*(c + d*x)^5 + 240*a*b*Sqrt[1 + (c + d*x)^2] - 120*a*b*(c + d*x)^2*Sqrt[1 + (c + d*x)^ 2] + 90*a*b*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] - 15*b^2*( -15*a*(c + d*x)^5 + 8*b*Sqrt[1 + (c + d*x)^2] - 4*b*(c + d*x)^2*Sqrt[1 + ( c + d*x)^2] + 3*b*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^2 + 75*b^3*(c + d*x)^5*ArcSinh[c + d*x]^3))/(375*d)
Time = 1.35 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.02, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {6274, 27, 6191, 6227, 6191, 243, 53, 2009, 6227, 6191, 243, 53, 2009, 6213, 2009}
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.
| \(\displaystyle \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int e^4 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^3d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^4 \int (c+d x)^4 (a+b \text {arcsinh}(c+d x))^3d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{5} b \int \frac {(c+d x)^5 (a+b \text {arcsinh}(c+d x))^2}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{5} b \left (-\frac {2}{5} b \int (c+d x)^4 (a+b \text {arcsinh}(c+d x))d(c+d x)-\frac {4}{5} \int \frac {(c+d x)^3 (a+b \text {arcsinh}(c+d x))^2}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{5} b \left (-\frac {2}{5} b \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))-\frac {1}{5} b \int \frac {(c+d x)^5}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )-\frac {4}{5} \int \frac {(c+d x)^3 (a+b \text {arcsinh}(c+d x))^2}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{5} b \left (-\frac {2}{5} b \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))-\frac {1}{10} b \int \frac {(c+d x)^4}{\sqrt {(c+d x)^2+1}}d(c+d x)^2\right )-\frac {4}{5} \int \frac {(c+d x)^3 (a+b \text {arcsinh}(c+d x))^2}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{5} b \left (-\frac {2}{5} b \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))-\frac {1}{10} b \int \left (\left ((c+d x)^2+1\right )^{3/2}-2 \sqrt {(c+d x)^2+1}+\frac {1}{\sqrt {(c+d x)^2+1}}\right )d(c+d x)^2\right )-\frac {4}{5} \int \frac {(c+d x)^3 (a+b \text {arcsinh}(c+d x))^2}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{5} b \left (-\frac {4}{5} \int \frac {(c+d x)^3 (a+b \text {arcsinh}(c+d x))^2}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2-\frac {2}{5} b \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))-\frac {1}{10} b \left (\frac {2}{5} \left ((c+d x)^2+1\right )^{5/2}-\frac {4}{3} \left ((c+d x)^2+1\right )^{3/2}+2 \sqrt {(c+d x)^2+1}\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{5} b \left (-\frac {4}{5} \left (-\frac {2}{3} b \int (c+d x)^2 (a+b \text {arcsinh}(c+d x))d(c+d x)-\frac {2}{3} \int \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^2}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{3} (c+d x)^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2\right )+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2-\frac {2}{5} b \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))-\frac {1}{10} b \left (\frac {2}{5} \left ((c+d x)^2+1\right )^{5/2}-\frac {4}{3} \left ((c+d x)^2+1\right )^{3/2}+2 \sqrt {(c+d x)^2+1}\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{5} b \left (-\frac {4}{5} \left (-\frac {2}{3} b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))-\frac {1}{3} b \int \frac {(c+d x)^3}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )-\frac {2}{3} \int \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^2}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{3} (c+d x)^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2\right )+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2-\frac {2}{5} b \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))-\frac {1}{10} b \left (\frac {2}{5} \left ((c+d x)^2+1\right )^{5/2}-\frac {4}{3} \left ((c+d x)^2+1\right )^{3/2}+2 \sqrt {(c+d x)^2+1}\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{5} b \left (-\frac {4}{5} \left (-\frac {2}{3} b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))-\frac {1}{6} b \int \frac {(c+d x)^2}{\sqrt {(c+d x)^2+1}}d(c+d x)^2\right )-\frac {2}{3} \int \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^2}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{3} (c+d x)^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2\right )+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2-\frac {2}{5} b \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))-\frac {1}{10} b \left (\frac {2}{5} \left ((c+d x)^2+1\right )^{5/2}-\frac {4}{3} \left ((c+d x)^2+1\right )^{3/2}+2 \sqrt {(c+d x)^2+1}\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{5} b \left (-\frac {4}{5} \left (-\frac {2}{3} b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))-\frac {1}{6} b \int \left (\sqrt {(c+d x)^2+1}-\frac {1}{\sqrt {(c+d x)^2+1}}\right )d(c+d x)^2\right )-\frac {2}{3} \int \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^2}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{3} (c+d x)^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2\right )+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2-\frac {2}{5} b \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))-\frac {1}{10} b \left (\frac {2}{5} \left ((c+d x)^2+1\right )^{5/2}-\frac {4}{3} \left ((c+d x)^2+1\right )^{3/2}+2 \sqrt {(c+d x)^2+1}\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{5} b \left (-\frac {4}{5} \left (-\frac {2}{3} \int \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^2}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{3} (c+d x)^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2-\frac {2}{3} b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))-\frac {1}{6} b \left (\frac {2}{3} \left ((c+d x)^2+1\right )^{3/2}-2 \sqrt {(c+d x)^2+1}\right )\right )\right )+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2-\frac {2}{5} b \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))-\frac {1}{10} b \left (\frac {2}{5} \left ((c+d x)^2+1\right )^{5/2}-\frac {4}{3} \left ((c+d x)^2+1\right )^{3/2}+2 \sqrt {(c+d x)^2+1}\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{5} b \left (-\frac {4}{5} \left (-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2-2 b \int (a+b \text {arcsinh}(c+d x))d(c+d x)\right )+\frac {1}{3} (c+d x)^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2-\frac {2}{3} b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))-\frac {1}{6} b \left (\frac {2}{3} \left ((c+d x)^2+1\right )^{3/2}-2 \sqrt {(c+d x)^2+1}\right )\right )\right )+\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2-\frac {2}{5} b \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))-\frac {1}{10} b \left (\frac {2}{5} \left ((c+d x)^2+1\right )^{5/2}-\frac {4}{3} \left ((c+d x)^2+1\right )^{3/2}+2 \sqrt {(c+d x)^2+1}\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^4 \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{5} b \left (\frac {1}{5} \sqrt {(c+d x)^2+1} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2-\frac {2}{5} b \left (\frac {1}{5} (c+d x)^5 (a+b \text {arcsinh}(c+d x))-\frac {1}{10} b \left (\frac {2}{5} \left ((c+d x)^2+1\right )^{5/2}-\frac {4}{3} \left ((c+d x)^2+1\right )^{3/2}+2 \sqrt {(c+d x)^2+1}\right )\right )-\frac {4}{5} \left (\frac {1}{3} (c+d x)^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2-\frac {2}{3} b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))-\frac {1}{6} b \left (\frac {2}{3} \left ((c+d x)^2+1\right )^{3/2}-2 \sqrt {(c+d x)^2+1}\right )\right )-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2-2 b \left (a (c+d x)+b (c+d x) \text {arcsinh}(c+d x)-b \sqrt {(c+d x)^2+1}\right )\right )\right )\right )\right )}{d}\) |
(e^4*(((c + d*x)^5*(a + b*ArcSinh[c + d*x])^3)/5 - (3*b*(((c + d*x)^4*Sqrt [1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^2)/5 - (2*b*(-1/10*(b*(2*Sqrt[1 + (c + d*x)^2] - (4*(1 + (c + d*x)^2)^(3/2))/3 + (2*(1 + (c + d*x)^2)^(5/ 2))/5)) + ((c + d*x)^5*(a + b*ArcSinh[c + d*x]))/5))/5 - (4*(((c + d*x)^2* Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^2)/3 - (2*b*(-1/6*(b*(-2*Sq rt[1 + (c + d*x)^2] + (2*(1 + (c + d*x)^2)^(3/2))/3)) + ((c + d*x)^3*(a + b*ArcSinh[c + d*x]))/3))/3 - (2*(Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^2 - 2*b*(a*(c + d*x) - b*Sqrt[1 + (c + d*x)^2] + b*(c + d*x)*ArcSinh [c + d*x])))/3))/5))/5))/d
3.2.37.3.1 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] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int [(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] ) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ m, 1] && NeQ[m + 2*p + 1, 0]
Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 0.42 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.29
| method | result | size |
| derivativedivides | \(\frac {\frac {e^{4} a^{3} \left (d x +c \right )^{5}}{5}+e^{4} b^{3} \left (\frac {\left (d x +c \right )^{5} \operatorname {arcsinh}\left (d x +c \right )^{3}}{5}-\frac {8 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{25}-\frac {3 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{25}+\frac {4 \left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{25}+\frac {16 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{25}-\frac {4144 \sqrt {1+\left (d x +c \right )^{2}}}{5625}+\frac {6 \left (d x +c \right )^{5} \operatorname {arcsinh}\left (d x +c \right )}{125}-\frac {6 \left (d x +c \right )^{4} \sqrt {1+\left (d x +c \right )^{2}}}{625}+\frac {272 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{5625}-\frac {8 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )}{75}\right )+3 e^{4} a \,b^{2} \left (\frac {\left (d x +c \right )^{5} \operatorname {arcsinh}\left (d x +c \right )^{2}}{5}-\frac {16 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{75}-\frac {2 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{25}+\frac {8 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )^{2}}{75}+\frac {16 d x}{75}+\frac {16 c}{75}+\frac {2 \left (d x +c \right )^{5}}{125}-\frac {8 \left (d x +c \right )^{3}}{225}\right )+3 e^{4} a^{2} b \left (\frac {\left (d x +c \right )^{5} \operatorname {arcsinh}\left (d x +c \right )}{5}-\frac {\left (d x +c \right )^{4} \sqrt {1+\left (d x +c \right )^{2}}}{25}+\frac {4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{75}-\frac {8 \sqrt {1+\left (d x +c \right )^{2}}}{75}\right )}{d}\) | \(420\) |
| default | \(\frac {\frac {e^{4} a^{3} \left (d x +c \right )^{5}}{5}+e^{4} b^{3} \left (\frac {\left (d x +c \right )^{5} \operatorname {arcsinh}\left (d x +c \right )^{3}}{5}-\frac {8 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{25}-\frac {3 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{25}+\frac {4 \left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{25}+\frac {16 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{25}-\frac {4144 \sqrt {1+\left (d x +c \right )^{2}}}{5625}+\frac {6 \left (d x +c \right )^{5} \operatorname {arcsinh}\left (d x +c \right )}{125}-\frac {6 \left (d x +c \right )^{4} \sqrt {1+\left (d x +c \right )^{2}}}{625}+\frac {272 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{5625}-\frac {8 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )}{75}\right )+3 e^{4} a \,b^{2} \left (\frac {\left (d x +c \right )^{5} \operatorname {arcsinh}\left (d x +c \right )^{2}}{5}-\frac {16 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{75}-\frac {2 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{25}+\frac {8 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )^{2}}{75}+\frac {16 d x}{75}+\frac {16 c}{75}+\frac {2 \left (d x +c \right )^{5}}{125}-\frac {8 \left (d x +c \right )^{3}}{225}\right )+3 e^{4} a^{2} b \left (\frac {\left (d x +c \right )^{5} \operatorname {arcsinh}\left (d x +c \right )}{5}-\frac {\left (d x +c \right )^{4} \sqrt {1+\left (d x +c \right )^{2}}}{25}+\frac {4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{75}-\frac {8 \sqrt {1+\left (d x +c \right )^{2}}}{75}\right )}{d}\) | \(420\) |
| parts | \(\frac {e^{4} a^{3} \left (d x +c \right )^{5}}{5 d}+\frac {e^{4} b^{3} \left (\frac {\left (d x +c \right )^{5} \operatorname {arcsinh}\left (d x +c \right )^{3}}{5}-\frac {8 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{25}-\frac {3 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{25}+\frac {4 \left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{25}+\frac {16 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{25}-\frac {4144 \sqrt {1+\left (d x +c \right )^{2}}}{5625}+\frac {6 \left (d x +c \right )^{5} \operatorname {arcsinh}\left (d x +c \right )}{125}-\frac {6 \left (d x +c \right )^{4} \sqrt {1+\left (d x +c \right )^{2}}}{625}+\frac {272 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{5625}-\frac {8 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )}{75}\right )}{d}+\frac {3 e^{4} a \,b^{2} \left (\frac {\left (d x +c \right )^{5} \operatorname {arcsinh}\left (d x +c \right )^{2}}{5}-\frac {16 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{75}-\frac {2 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{25}+\frac {8 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )^{2}}{75}+\frac {16 d x}{75}+\frac {16 c}{75}+\frac {2 \left (d x +c \right )^{5}}{125}-\frac {8 \left (d x +c \right )^{3}}{225}\right )}{d}+\frac {3 e^{4} a^{2} b \left (\frac {\left (d x +c \right )^{5} \operatorname {arcsinh}\left (d x +c \right )}{5}-\frac {\left (d x +c \right )^{4} \sqrt {1+\left (d x +c \right )^{2}}}{25}+\frac {4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{75}-\frac {8 \sqrt {1+\left (d x +c \right )^{2}}}{75}\right )}{d}\) | \(428\) |
1/d*(1/5*e^4*a^3*(d*x+c)^5+e^4*b^3*(1/5*(d*x+c)^5*arcsinh(d*x+c)^3-8/25*ar csinh(d*x+c)^2*(1+(d*x+c)^2)^(1/2)-3/25*(d*x+c)^4*arcsinh(d*x+c)^2*(1+(d*x +c)^2)^(1/2)+4/25*(d*x+c)^2*arcsinh(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+16/25*(d* x+c)*arcsinh(d*x+c)-4144/5625*(1+(d*x+c)^2)^(1/2)+6/125*(d*x+c)^5*arcsinh( d*x+c)-6/625*(d*x+c)^4*(1+(d*x+c)^2)^(1/2)+272/5625*(d*x+c)^2*(1+(d*x+c)^2 )^(1/2)-8/75*(d*x+c)^3*arcsinh(d*x+c))+3*e^4*a*b^2*(1/5*(d*x+c)^5*arcsinh( d*x+c)^2-16/75*arcsinh(d*x+c)*(1+(d*x+c)^2)^(1/2)-2/25*(d*x+c)^4*arcsinh(d *x+c)*(1+(d*x+c)^2)^(1/2)+8/75*arcsinh(d*x+c)*(1+(d*x+c)^2)^(1/2)*(d*x+c)^ 2+16/75*d*x+16/75*c+2/125*(d*x+c)^5-8/225*(d*x+c)^3)+3*e^4*a^2*b*(1/5*(d*x +c)^5*arcsinh(d*x+c)-1/25*(d*x+c)^4*(1+(d*x+c)^2)^(1/2)+4/75*(d*x+c)^2*(1+ (d*x+c)^2)^(1/2)-8/75*(1+(d*x+c)^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 1077 vs. \(2 (292) = 584\).
Time = 0.30 (sec) , antiderivative size = 1077, normalized size of antiderivative = 3.30 \[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^3 \, dx=\text {Too large to display} \]
1/5625*(45*(25*a^3 + 6*a*b^2)*d^5*e^4*x^5 + 225*(25*a^3 + 6*a*b^2)*c*d^4*e ^4*x^4 - 150*(4*a*b^2 - 3*(25*a^3 + 6*a*b^2)*c^2)*d^3*e^4*x^3 - 450*(4*a*b ^2*c - (25*a^3 + 6*a*b^2)*c^3)*d^2*e^4*x^2 - 225*(8*a*b^2*c^2 - (25*a^3 + 6*a*b^2)*c^4 - 16*a*b^2)*d*e^4*x + 1125*(b^3*d^5*e^4*x^5 + 5*b^3*c*d^4*e^4 *x^4 + 10*b^3*c^2*d^3*e^4*x^3 + 10*b^3*c^3*d^2*e^4*x^2 + 5*b^3*c^4*d*e^4*x + b^3*c^5*e^4)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^3 + 225*( 15*a*b^2*d^5*e^4*x^5 + 75*a*b^2*c*d^4*e^4*x^4 + 150*a*b^2*c^2*d^3*e^4*x^3 + 150*a*b^2*c^3*d^2*e^4*x^2 + 75*a*b^2*c^4*d*e^4*x + 15*a*b^2*c^5*e^4 - (3 *b^3*d^4*e^4*x^4 + 12*b^3*c*d^3*e^4*x^3 + 2*(9*b^3*c^2 - 2*b^3)*d^2*e^4*x^ 2 + 4*(3*b^3*c^3 - 2*b^3*c)*d*e^4*x + (3*b^3*c^4 - 4*b^3*c^2 + 8*b^3)*e^4) *sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 + 15*(9*(25*a^2*b + 2*b^3)*d^5*e^4*x^5 + 45*(25*a^2*b + 2*b^3 )*c*d^4*e^4*x^4 - 10*(4*b^3 - 9*(25*a^2*b + 2*b^3)*c^2)*d^3*e^4*x^3 - 30*( 4*b^3*c - 3*(25*a^2*b + 2*b^3)*c^3)*d^2*e^4*x^2 - 15*(8*b^3*c^2 - 3*(25*a^ 2*b + 2*b^3)*c^4 - 16*b^3)*d*e^4*x - (40*b^3*c^3 - 9*(25*a^2*b + 2*b^3)*c^ 5 - 240*b^3*c)*e^4 - 30*(3*a*b^2*d^4*e^4*x^4 + 12*a*b^2*c*d^3*e^4*x^3 + 2* (9*a*b^2*c^2 - 2*a*b^2)*d^2*e^4*x^2 + 4*(3*a*b^2*c^3 - 2*a*b^2*c)*d*e^4*x + (3*a*b^2*c^4 - 4*a*b^2*c^2 + 8*a*b^2)*e^4)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) - (27*(25*a^2*b + 2 *b^3)*d^4*e^4*x^4 + 108*(25*a^2*b + 2*b^3)*c*d^3*e^4*x^3 - 2*(450*a^2*b...
Leaf count of result is larger than twice the leaf count of optimal. 2518 vs. \(2 (306) = 612\).
Time = 1.02 (sec) , antiderivative size = 2518, normalized size of antiderivative = 7.72 \[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^3 \, dx=\text {Too large to display} \]
Piecewise((a**3*c**4*e**4*x + 2*a**3*c**3*d*e**4*x**2 + 2*a**3*c**2*d**2*e **4*x**3 + a**3*c*d**3*e**4*x**4 + a**3*d**4*e**4*x**5/5 + 3*a**2*b*c**5*e **4*asinh(c + d*x)/(5*d) + 3*a**2*b*c**4*e**4*x*asinh(c + d*x) - 3*a**2*b* c**4*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(25*d) + 6*a**2*b*c**3*d*e* *4*x**2*asinh(c + d*x) - 12*a**2*b*c**3*e**4*x*sqrt(c**2 + 2*c*d*x + d**2* x**2 + 1)/25 + 6*a**2*b*c**2*d**2*e**4*x**3*asinh(c + d*x) - 18*a**2*b*c** 2*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/25 + 4*a**2*b*c**2*e**4 *sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(25*d) + 3*a**2*b*c*d**3*e**4*x**4*a sinh(c + d*x) - 12*a**2*b*c*d**2*e**4*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/25 + 8*a**2*b*c*e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/25 + 3*a **2*b*d**4*e**4*x**5*asinh(c + d*x)/5 - 3*a**2*b*d**3*e**4*x**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/25 + 4*a**2*b*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/25 - 8*a**2*b*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(2 5*d) + 3*a*b**2*c**5*e**4*asinh(c + d*x)**2/(5*d) + 3*a*b**2*c**4*e**4*x*a sinh(c + d*x)**2 + 6*a*b**2*c**4*e**4*x/25 - 6*a*b**2*c**4*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/(25*d) + 6*a*b**2*c**3*d*e**4*x* *2*asinh(c + d*x)**2 + 12*a*b**2*c**3*d*e**4*x**2/25 - 24*a*b**2*c**3*e**4 *x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/25 + 6*a*b**2*c**2* d**2*e**4*x**3*asinh(c + d*x)**2 + 12*a*b**2*c**2*d**2*e**4*x**3/25 - 36*a *b**2*c**2*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d...
\[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^3 \, dx=\int { {\left (d e x + c e\right )}^{4} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
1/5*a^3*d^4*e^4*x^5 + a^3*c*d^3*e^4*x^4 + 2*a^3*c^2*d^2*e^4*x^3 + 2*a^3*c^ 3*d*e^4*x^2 + 3*(2*x^2*arcsinh(d*x + c) - d*(3*c^2*arcsinh(2*(d^2*x + c*d) /sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^3 + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*x/d^2 - (c^2 + 1)*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1) *d^2))/d^3 - 3*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c/d^3))*a^2*b*c^3*d*e^4 + (6*x^3*arcsinh(d*x + c) - d*(2*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*x^2/d^2 - 15*c^3*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^4 - 5*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c*x/d^3 + 9*(c^2 + 1)*c*arcsinh(2*(d^ 2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^4 + 15*sqrt(d^2*x^2 + 2*c *d*x + c^2 + 1)*c^2/d^4 - 4*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^2 + 1)/d^ 4))*a^2*b*c^2*d^2*e^4 + 1/8*(24*x^4*arcsinh(d*x + c) - (6*sqrt(d^2*x^2 + 2 *c*d*x + c^2 + 1)*x^3/d^2 - 14*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c*x^2/d^3 + 105*c^4*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^5 + 35*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c^2*x/d^4 - 90*(c^2 + 1)*c^2*arcsi nh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^5 - 105*sqrt(d^2* x^2 + 2*c*d*x + c^2 + 1)*c^3/d^5 - 9*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^ 2 + 1)*x/d^4 + 9*(c^2 + 1)^2*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*( c^2 + 1)*d^2))/d^5 + 55*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^2 + 1)*c/d^5) *d)*a^2*b*c*d^3*e^4 + 1/200*(120*x^5*arcsinh(d*x + c) - (24*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*x^4/d^2 - 54*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c*x^...
\[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^3 \, dx=\int { {\left (d e x + c e\right )}^{4} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int (c e+d e x)^4 (a+b \text {arcsinh}(c+d x))^3 \, dx=\int {\left (c\,e+d\,e\,x\right )}^4\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3 \,d x \]