Integrand size = 23, antiderivative size = 418 \[ \int x^3 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{560 c^6 e}-\frac {b \left (13 c^2 d+25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 e}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}+\frac {b \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{560 c^7 e^{3/2}}+\frac {2 b d^{7/2} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{35 e^2} \]
-1/5*d*(e*x^2+d)^(5/2)*(a+b*arcsech(c*x))/e^2+1/7*(e*x^2+d)^(7/2)*(a+b*arc sech(c*x))/e^2+1/560*b*(35*c^6*d^3-35*c^4*d^2*e-63*c^2*d*e^2-25*e^3)*arcta n(e^(1/2)*(-c^2*x^2+1)^(1/2)/c/(e*x^2+d)^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^ (1/2)/c^7/e^(3/2)+2/35*b*d^(7/2)*arctanh((e*x^2+d)^(1/2)/d^(1/2)/(-c^2*x^2 +1)^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)/e^2-1/840*b*(13*c^2*d+25*e)*(e* x^2+d)^(3/2)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/c^4/e-1/42 *b*(e*x^2+d)^(5/2)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/c^2/ e+1/560*b*(3*c^4*d^2-38*c^2*d*e-25*e^2)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(- c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/2)/c^6/e
Time = 37.50 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.75 \[ \int x^3 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=-\frac {\sqrt {d+e x^2} \left (48 a c^6 \left (2 d-5 e x^2\right ) \left (d+e x^2\right )^2+b e \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (75 e^2+2 c^2 e \left (82 d+25 e x^2\right )+c^4 \left (57 d^2+106 d e x^2+40 e^2 x^4\right )\right )+48 b c^6 \left (2 d-5 e x^2\right ) \left (d+e x^2\right )^2 \text {sech}^{-1}(c x)\right )}{1680 c^6 e^2}-\frac {b \sqrt {\frac {1-c x}{1+c x}} \sqrt {-1+c^2 x^2} \left (-32 c^7 d^{7/2} \arctan \left (\frac {\sqrt {d} \sqrt {-1+c^2 x^2}}{\sqrt {d+e x^2}}\right )+\sqrt {e} \left (-35 c^6 d^3+35 c^4 d^2 e+63 c^2 d e^2+25 e^3\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )\right )}{560 c^7 e^2 (-1+c x)} \]
-1/1680*(Sqrt[d + e*x^2]*(48*a*c^6*(2*d - 5*e*x^2)*(d + e*x^2)^2 + b*e*Sqr t[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(75*e^2 + 2*c^2*e*(82*d + 25*e*x^2) + c^4 *(57*d^2 + 106*d*e*x^2 + 40*e^2*x^4)) + 48*b*c^6*(2*d - 5*e*x^2)*(d + e*x^ 2)^2*ArcSech[c*x]))/(c^6*e^2) - (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[-1 + c^2 *x^2]*(-32*c^7*d^(7/2)*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2] ] + Sqrt[e]*(-35*c^6*d^3 + 35*c^4*d^2*e + 63*c^2*d*e^2 + 25*e^3)*ArcTanh[( Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])]))/(560*c^7*e^2*(-1 + c*x) )
Time = 0.66 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.85, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {6855, 27, 435, 171, 27, 171, 27, 171, 27, 175, 66, 104, 218, 220}
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.
| \(\displaystyle \int x^3 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx\) |
| \(\Big \downarrow \) 6855 |
| \(\displaystyle b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int -\frac {\left (2 d-5 e x^2\right ) \left (e x^2+d\right )^{5/2}}{35 e^2 x \sqrt {1-c^2 x^2}}dx+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {\left (2 d-5 e x^2\right ) \left (e x^2+d\right )^{5/2}}{x \sqrt {1-c^2 x^2}}dx}{35 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}\) |
| \(\Big \downarrow \) 435 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {\left (2 d-5 e x^2\right ) \left (e x^2+d\right )^{5/2}}{x^2 \sqrt {1-c^2 x^2}}dx^2}{70 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {5 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{3 c^2}-\frac {\int -\frac {\left (e x^2+d\right )^{3/2} \left (12 c^2 d^2-e \left (13 d c^2+25 e\right ) x^2\right )}{2 x^2 \sqrt {1-c^2 x^2}}dx^2}{3 c^2}\right )}{70 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\int \frac {\left (e x^2+d\right )^{3/2} \left (12 c^2 d^2-e \left (13 d c^2+25 e\right ) x^2\right )}{x^2 \sqrt {1-c^2 x^2}}dx^2}{6 c^2}+\frac {5 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{3 c^2}\right )}{70 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {e \sqrt {1-c^2 x^2} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}-\frac {\int -\frac {3 \sqrt {e x^2+d} \left (16 d^3 c^4+e \left (3 d^2 c^4-38 d e c^2-25 e^2\right ) x^2\right )}{2 x^2 \sqrt {1-c^2 x^2}}dx^2}{2 c^2}}{6 c^2}+\frac {5 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{3 c^2}\right )}{70 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {3 \int \frac {\sqrt {e x^2+d} \left (16 d^3 c^4+e \left (3 d^2 c^4-38 d e c^2-25 e^2\right ) x^2\right )}{x^2 \sqrt {1-c^2 x^2}}dx^2}{4 c^2}+\frac {e \sqrt {1-c^2 x^2} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{6 c^2}+\frac {5 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{3 c^2}\right )}{70 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {3 \left (-\frac {\int -\frac {32 d^4 c^6+e \left (35 d^3 c^6-35 d^2 e c^4-63 d e^2 c^2-25 e^3\right ) x^2}{2 x^2 \sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx^2}{c^2}-\frac {e \sqrt {1-c^2 x^2} \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {d+e x^2}}{c^2}\right )}{4 c^2}+\frac {e \sqrt {1-c^2 x^2} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{6 c^2}+\frac {5 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{3 c^2}\right )}{70 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {3 \left (\frac {\int \frac {32 d^4 c^6+e \left (35 d^3 c^6-35 d^2 e c^4-63 d e^2 c^2-25 e^3\right ) x^2}{x^2 \sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx^2}{2 c^2}-\frac {e \sqrt {1-c^2 x^2} \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {d+e x^2}}{c^2}\right )}{4 c^2}+\frac {e \sqrt {1-c^2 x^2} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{6 c^2}+\frac {5 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{3 c^2}\right )}{70 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {3 \left (\frac {32 c^6 d^4 \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx^2+e \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx^2}{2 c^2}-\frac {e \sqrt {1-c^2 x^2} \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {d+e x^2}}{c^2}\right )}{4 c^2}+\frac {e \sqrt {1-c^2 x^2} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{6 c^2}+\frac {5 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{3 c^2}\right )}{70 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {3 \left (\frac {32 c^6 d^4 \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx^2+2 e \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) \int \frac {1}{-e x^4-c^2}d\frac {\sqrt {1-c^2 x^2}}{\sqrt {e x^2+d}}}{2 c^2}-\frac {e \sqrt {1-c^2 x^2} \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {d+e x^2}}{c^2}\right )}{4 c^2}+\frac {e \sqrt {1-c^2 x^2} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{6 c^2}+\frac {5 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{3 c^2}\right )}{70 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {3 \left (\frac {64 c^6 d^4 \int \frac {1}{x^4-d}d\frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}+2 e \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) \int \frac {1}{-e x^4-c^2}d\frac {\sqrt {1-c^2 x^2}}{\sqrt {e x^2+d}}}{2 c^2}-\frac {e \sqrt {1-c^2 x^2} \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {d+e x^2}}{c^2}\right )}{4 c^2}+\frac {e \sqrt {1-c^2 x^2} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{6 c^2}+\frac {5 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{3 c^2}\right )}{70 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {3 \left (\frac {64 c^6 d^4 \int \frac {1}{x^4-d}d\frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}-\frac {2 \sqrt {e} \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{c}}{2 c^2}-\frac {e \sqrt {1-c^2 x^2} \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {d+e x^2}}{c^2}\right )}{4 c^2}+\frac {e \sqrt {1-c^2 x^2} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{6 c^2}+\frac {5 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{3 c^2}\right )}{70 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle \frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {3 \left (\frac {-\frac {2 \sqrt {e} \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{c}-64 c^6 d^{7/2} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 c^2}-\frac {e \sqrt {1-c^2 x^2} \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {d+e x^2}}{c^2}\right )}{4 c^2}+\frac {e \sqrt {1-c^2 x^2} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{6 c^2}+\frac {5 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{3 c^2}\right )}{70 e^2}\) |
-1/5*(d*(d + e*x^2)^(5/2)*(a + b*ArcSech[c*x]))/e^2 + ((d + e*x^2)^(7/2)*( a + b*ArcSech[c*x]))/(7*e^2) - (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*((5*e *Sqrt[1 - c^2*x^2]*(d + e*x^2)^(5/2))/(3*c^2) + ((e*(13*c^2*d + 25*e)*Sqrt [1 - c^2*x^2]*(d + e*x^2)^(3/2))/(2*c^2) + (3*(-((e*(3*c^4*d^2 - 38*c^2*d* e - 25*e^2)*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/c^2) + ((-2*Sqrt[e]*(35*c^6 *d^3 - 35*c^4*d^2*e - 63*c^2*d*e^2 - 25*e^3)*ArcTan[(Sqrt[e]*Sqrt[1 - c^2* x^2])/(c*Sqrt[d + e*x^2])])/c - 64*c^6*d^(7/2)*ArcTanh[Sqrt[d + e*x^2]/(Sq rt[d]*Sqrt[1 - c^2*x^2])])/(2*c^2)))/(4*c^2))/(6*c^2)))/(70*e^2)
3.2.40.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^(q_.)*(( e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2) *(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && IntegerQ[(m - 1)/2]
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*( x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Si mp[(a + b*ArcSech[c*x]) u, x] + Simp[b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)] Int[SimplifyIntegrand[u/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), x], x], x]] /; Fre eQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] && !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] && !(ILtQ[p, 0] && GtQ[m + 2 *p + 3, 0])) || (ILtQ[(m + 2*p + 1)/2, 0] && !ILtQ[(m - 1)/2, 0]))
\[\int x^{3} \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )d x\]
Time = 1.47 (sec) , antiderivative size = 1989, normalized size of antiderivative = 4.76 \[ \int x^3 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\text {Too large to display} \]
[1/6720*(96*b*c^7*d^(7/2)*log(((c^4*d^2 - 6*c^2*d*e + e^2)*x^4 - 8*(c^2*d^ 2 - d*e)*x^2 - 4*((c^3*d - c*e)*x^3 - 2*c*d*x)*sqrt(e*x^2 + d)*sqrt(d)*sqr t(-(c^2*x^2 - 1)/(c^2*x^2)) + 8*d^2)/x^4) + 3*(35*b*c^6*d^3 - 35*b*c^4*d^2 *e - 63*b*c^2*d*e^2 - 25*b*e^3)*sqrt(-e)*log(8*c^4*e^2*x^4 + c^4*d^2 - 6*c ^2*d*e + 8*(c^4*d*e - c^2*e^2)*x^2 - 4*(2*c^4*e*x^3 + (c^4*d - c^2*e)*x)*s qrt(e*x^2 + d)*sqrt(-e)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + e^2) + 192*(5*b*c ^7*e^3*x^6 + 8*b*c^7*d*e^2*x^4 + b*c^7*d^2*e*x^2 - 2*b*c^7*d^3)*sqrt(e*x^2 + d)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) + 4*(240*a*c^7*e ^3*x^6 + 384*a*c^7*d*e^2*x^4 + 48*a*c^7*d^2*e*x^2 - 96*a*c^7*d^3 - (40*b*c ^6*e^3*x^5 + 2*(53*b*c^6*d*e^2 + 25*b*c^4*e^3)*x^3 + (57*b*c^6*d^2*e + 164 *b*c^4*d*e^2 + 75*b*c^2*e^3)*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))*sqrt(e*x^2 + d))/(c^7*e^2), 1/3360*(48*b*c^7*d^(7/2)*log(((c^4*d^2 - 6*c^2*d*e + e^2 )*x^4 - 8*(c^2*d^2 - d*e)*x^2 - 4*((c^3*d - c*e)*x^3 - 2*c*d*x)*sqrt(e*x^2 + d)*sqrt(d)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 8*d^2)/x^4) + 3*(35*b*c^6*d ^3 - 35*b*c^4*d^2*e - 63*b*c^2*d*e^2 - 25*b*e^3)*sqrt(e)*arctan(1/2*(2*c^2 *e*x^3 + (c^2*d - e)*x)*sqrt(e*x^2 + d)*sqrt(e)*sqrt(-(c^2*x^2 - 1)/(c^2*x ^2))/(c^2*e^2*x^4 + (c^2*d*e - e^2)*x^2 - d*e)) + 96*(5*b*c^7*e^3*x^6 + 8* b*c^7*d*e^2*x^4 + b*c^7*d^2*e*x^2 - 2*b*c^7*d^3)*sqrt(e*x^2 + d)*log((c*x* sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) + 2*(240*a*c^7*e^3*x^6 + 384*a* c^7*d*e^2*x^4 + 48*a*c^7*d^2*e*x^2 - 96*a*c^7*d^3 - (40*b*c^6*e^3*x^5 +...
\[ \int x^3 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int x^{3} \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}\, dx \]
Exception generated. \[ \int x^3 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\text {Exception raised: ValueError} \]
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
\[ \int x^3 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{3} \,d x } \]
Timed out. \[ \int x^3 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int x^3\,{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]