Integrand size = 23, antiderivative size = 556 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx=\frac {b \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{3675 d x^3}+\frac {b \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{3675 d^2 x}+\frac {b \left (30 c^2 d+11 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^5}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^7}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}+\frac {b c \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3675 d^2 \sqrt {1+\frac {e x^2}{d}}}-\frac {2 b \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3675 c d^2 \sqrt {d+e x^2}} \]
-1/7*(e*x^2+d)^(5/2)*(a+b*arcsech(c*x))/d/x^7+2/35*e*(e*x^2+d)^(5/2)*(a+b* arcsech(c*x))/d^2/x^5+1/1225*b*(30*c^2*d+11*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)/d/x^5+1/49*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)/d/x^7+1/3675*b*(120*c^4*d^2+ 159*c^2*d*e-37*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)/d/x^3+1/3675*b*(240*c^6*d^3+528*c^4*d^2*e+193*c^2*d*e^2-247*e ^3)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/2)/d^2 /x+1/3675*b*c*(240*c^6*d^3+528*c^4*d^2*e+193*c^2*d*e^2-247*e^3)*EllipticE( c*x,(-e/c^2/d)^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(e*x^2+d)^(1/2)/d^2/ (1+e*x^2/d)^(1/2)-2/3675*b*(c^2*d+e)*(120*c^6*d^3+204*c^4*d^2*e+17*c^2*d*e ^2-105*e^3)*EllipticF(c*x,(-e/c^2/d)^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2 )*(1+e*x^2/d)^(1/2)/c/d^2/(e*x^2+d)^(1/2)
Result contains complex when optimal does not.
Time = 29.47 (sec) , antiderivative size = 1187, normalized size of antiderivative = 2.13 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx=\left (-\frac {a d}{7 x^7}-\frac {8 a e}{35 x^5}-\frac {a e^2}{35 d x^3}+\frac {2 a e^3}{35 d^2 x}\right ) \sqrt {d+e x^2}+\left (\frac {b c \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right )}{3675 d^2}+\frac {b d}{49 x^7}+\frac {b c d}{49 x^6}+\frac {b \left (30 c^2 d+61 e\right )}{1225 x^5}+\frac {b c \left (30 c^2 d+61 e\right )}{1225 x^4}+\frac {b \left (120 c^4 d^2+249 c^2 d e+71 e^2\right )}{3675 d x^3}+\frac {b c \left (120 c^4 d^2+249 c^2 d e+71 e^2\right )}{3675 d x^2}+\frac {b \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right )}{3675 d^2 x}\right ) \sqrt {\frac {1-c x}{1+c x}} \sqrt {d+e x^2}-\frac {b \left (5 d-2 e x^2\right ) \left (d+e x^2\right )^{5/2} \text {sech}^{-1}(c x)}{35 d^2 x^7}+\frac {-b c \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {\frac {1-c x}{1+c x}} \sqrt {\frac {e \left (-1+\frac {1-c x}{1+c x}\right )^2+c^2 d \left (1+\frac {1-c x}{1+c x}\right )^2}{c^2 \left (1+\frac {1-c x}{1+c x}\right )^2}}-\frac {i b \left (c \sqrt {d}-i \sqrt {e}\right )^2 \sqrt {1+\frac {\left (c^2 d+e\right ) (1-c x)}{\left (c \sqrt {d}-i \sqrt {e}\right )^2 (1+c x)}} \sqrt {1+\frac {\left (c^2 d+e\right ) (1-c x)}{\left (c \sqrt {d}+i \sqrt {e}\right )^2 (1+c x)}} \left (\frac {\sqrt {\frac {c^2 d+e}{\left (c \sqrt {d}+i \sqrt {e}\right )^2}} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {\frac {1-c x}{1+c x}} E\left (i \text {arcsinh}\left (\sqrt {\frac {\left (c^2 d+e\right ) (1-c x)}{\left (c \sqrt {d}+i \sqrt {e}\right )^2 (1+c x)}}\right )|\frac {\left (c \sqrt {d}+i \sqrt {e}\right )^2}{\left (c \sqrt {d}-i \sqrt {e}\right )^2}\right )}{\sqrt {\frac {\left (c^2 d+e\right ) (1-c x)}{\left (c \sqrt {d}+i \sqrt {e}\right )^2 (1+c x)}}}+\frac {2 \sqrt {e} \sqrt {\frac {c^2 d+e}{\left (c \sqrt {d}+i \sqrt {e}\right )^2}} \left (240 i c^5 d^{5/2}-360 c^4 d^2 \sqrt {e}+48 i c^3 d^{3/2} e-207 c^2 d e^{3/2}-173 i c \sqrt {d} e^2+210 e^{5/2}\right ) \sqrt {\frac {1-c x}{1+c x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {\left (c^2 d+e\right ) (1-c x)}{\left (c \sqrt {d}+i \sqrt {e}\right )^2 (1+c x)}}\right ),\frac {\left (c \sqrt {d}+i \sqrt {e}\right )^2}{\left (c \sqrt {d}-i \sqrt {e}\right )^2}\right )}{\sqrt {\frac {\left (c^2 d+e\right ) (1-c x)}{\left (c \sqrt {d}+i \sqrt {e}\right )^2 (1+c x)}}}\right )}{c \sqrt {\frac {c \sqrt {d}-i \sqrt {e}}{c \sqrt {d}+i \sqrt {e}}} \left (1+\frac {1-c x}{1+c x}\right ) \sqrt {\frac {e \left (-1+\frac {1-c x}{1+c x}\right )^2+c^2 d \left (1+\frac {1-c x}{1+c x}\right )^2}{c^2 \left (1+\frac {1-c x}{1+c x}\right )^2}}}}{3675 d^2} \]
(-1/7*(a*d)/x^7 - (8*a*e)/(35*x^5) - (a*e^2)/(35*d*x^3) + (2*a*e^3)/(35*d^ 2*x))*Sqrt[d + e*x^2] + ((b*c*(240*c^6*d^3 + 528*c^4*d^2*e + 193*c^2*d*e^2 - 247*e^3))/(3675*d^2) + (b*d)/(49*x^7) + (b*c*d)/(49*x^6) + (b*(30*c^2*d + 61*e))/(1225*x^5) + (b*c*(30*c^2*d + 61*e))/(1225*x^4) + (b*(120*c^4*d^ 2 + 249*c^2*d*e + 71*e^2))/(3675*d*x^3) + (b*c*(120*c^4*d^2 + 249*c^2*d*e + 71*e^2))/(3675*d*x^2) + (b*(240*c^6*d^3 + 528*c^4*d^2*e + 193*c^2*d*e^2 - 247*e^3))/(3675*d^2*x))*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[d + e*x^2] - (b*( 5*d - 2*e*x^2)*(d + e*x^2)^(5/2)*ArcSech[c*x])/(35*d^2*x^7) + (-(b*c*(240* c^6*d^3 + 528*c^4*d^2*e + 193*c^2*d*e^2 - 247*e^3)*Sqrt[(1 - c*x)/(1 + c*x )]*Sqrt[(e*(-1 + (1 - c*x)/(1 + c*x))^2 + c^2*d*(1 + (1 - c*x)/(1 + c*x))^ 2)/(c^2*(1 + (1 - c*x)/(1 + c*x))^2)]) - (I*b*(c*Sqrt[d] - I*Sqrt[e])^2*Sq rt[1 + ((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] - I*Sqrt[e])^2*(1 + c*x))]*Sqrt [1 + ((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]*((Sqrt [(c^2*d + e)/(c*Sqrt[d] + I*Sqrt[e])^2]*(240*c^6*d^3 + 528*c^4*d^2*e + 193 *c^2*d*e^2 - 247*e^3)*Sqrt[(1 - c*x)/(1 + c*x)]*EllipticE[I*ArcSinh[Sqrt[( (c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2])/Sqrt[((c^2*d + e)*(1 - c*x))/( (c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))] + (2*Sqrt[e]*Sqrt[(c^2*d + e)/(c*Sqr t[d] + I*Sqrt[e])^2]*((240*I)*c^5*d^(5/2) - 360*c^4*d^2*Sqrt[e] + (48*I)*c ^3*d^(3/2)*e - 207*c^2*d*e^(3/2) - (173*I)*c*Sqrt[d]*e^2 + 210*e^(5/2))...
Time = 0.96 (sec) , antiderivative size = 462, normalized size of antiderivative = 0.83, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {6855, 27, 442, 442, 442, 445, 25, 27, 399, 323, 321, 330, 327}
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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx\) |
| \(\Big \downarrow \) 6855 |
| \(\displaystyle b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int -\frac {\left (5 d-2 e x^2\right ) \left (e x^2+d\right )^{5/2}}{35 d^2 x^8 \sqrt {1-c^2 x^2}}dx+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {\left (5 d-2 e x^2\right ) \left (e x^2+d\right )^{5/2}}{x^8 \sqrt {1-c^2 x^2}}dx}{35 d^2}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 442 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{7} \int \frac {\left (e x^2+d\right )^{3/2} \left (\left (5 c^2 d-14 e\right ) e x^2+d \left (30 d c^2+11 e\right )\right )}{x^6 \sqrt {1-c^2 x^2}}dx-\frac {5 d \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 442 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {\sqrt {e x^2+d} \left (2 e \left (15 d^2 c^4+18 d e c^2-35 e^2\right ) x^2+d \left (120 d^2 c^4+159 d e c^2-37 e^2\right )\right )}{x^4 \sqrt {1-c^2 x^2}}dx-\frac {d \sqrt {1-c^2 x^2} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{5 x^5}\right )-\frac {5 d \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 442 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {e \left (120 d^3 c^6+249 d^2 e c^4+71 d e^2 c^2-210 e^3\right ) x^2+d \left (240 d^3 c^6+528 d^2 e c^4+193 d e^2 c^2-247 e^3\right )}{x^2 \sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx-\frac {d \sqrt {1-c^2 x^2} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3 x^3}\right )-\frac {d \sqrt {1-c^2 x^2} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{5 x^5}\right )-\frac {5 d \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (-\frac {\int -\frac {d e \left (120 d^3 c^6+249 d^2 e c^4+71 d e^2 c^2-\left (240 d^3 c^6+528 d^2 e c^4+193 d e^2 c^2-247 e^3\right ) x^2 c^2-210 e^3\right )}{\sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx}{d}-\frac {\sqrt {1-c^2 x^2} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2}}{x}\right )-\frac {d \sqrt {1-c^2 x^2} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3 x^3}\right )-\frac {d \sqrt {1-c^2 x^2} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{5 x^5}\right )-\frac {5 d \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\int \frac {d e \left (120 d^3 c^6+249 d^2 e c^4+71 d e^2 c^2-\left (240 d^3 c^6+528 d^2 e c^4+193 d e^2 c^2-247 e^3\right ) x^2 c^2-210 e^3\right )}{\sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx}{d}-\frac {\sqrt {1-c^2 x^2} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2}}{x}\right )-\frac {d \sqrt {1-c^2 x^2} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3 x^3}\right )-\frac {d \sqrt {1-c^2 x^2} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{5 x^5}\right )-\frac {5 d \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (e \int \frac {120 d^3 c^6+249 d^2 e c^4+71 d e^2 c^2-\left (240 d^3 c^6+528 d^2 e c^4+193 d e^2 c^2-247 e^3\right ) x^2 c^2-210 e^3}{\sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx-\frac {\sqrt {1-c^2 x^2} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2}}{x}\right )-\frac {d \sqrt {1-c^2 x^2} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3 x^3}\right )-\frac {d \sqrt {1-c^2 x^2} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{5 x^5}\right )-\frac {5 d \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {2 \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx}{e}-\frac {c^2 \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \int \frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}dx}{e}\right )-\frac {\sqrt {1-c^2 x^2} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2}}{x}\right )-\frac {d \sqrt {1-c^2 x^2} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3 x^3}\right )-\frac {d \sqrt {1-c^2 x^2} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{5 x^5}\right )-\frac {5 d \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {2 \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) \sqrt {\frac {e x^2}{d}+1} \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1}}dx}{e \sqrt {d+e x^2}}-\frac {c^2 \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \int \frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}dx}{e}\right )-\frac {\sqrt {1-c^2 x^2} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2}}{x}\right )-\frac {d \sqrt {1-c^2 x^2} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3 x^3}\right )-\frac {d \sqrt {1-c^2 x^2} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{5 x^5}\right )-\frac {5 d \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {2 \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {d+e x^2}}-\frac {c^2 \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \int \frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}dx}{e}\right )-\frac {\sqrt {1-c^2 x^2} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2}}{x}\right )-\frac {d \sqrt {1-c^2 x^2} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3 x^3}\right )-\frac {d \sqrt {1-c^2 x^2} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{5 x^5}\right )-\frac {5 d \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {2 \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {d+e x^2}}-\frac {c^2 \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-c^2 x^2}}dx}{e \sqrt {\frac {e x^2}{d}+1}}\right )-\frac {\sqrt {1-c^2 x^2} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2}}{x}\right )-\frac {d \sqrt {1-c^2 x^2} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3 x^3}\right )-\frac {d \sqrt {1-c^2 x^2} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{5 x^5}\right )-\frac {5 d \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 d x^7}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {2 \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {d+e x^2}}-\frac {c \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{e \sqrt {\frac {e x^2}{d}+1}}\right )-\frac {\sqrt {1-c^2 x^2} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2}}{x}\right )-\frac {d \sqrt {1-c^2 x^2} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3 x^3}\right )-\frac {d \sqrt {1-c^2 x^2} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{5 x^5}\right )-\frac {5 d \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2}\) |
-1/7*((d + e*x^2)^(5/2)*(a + b*ArcSech[c*x]))/(d*x^7) + (2*e*(d + e*x^2)^( 5/2)*(a + b*ArcSech[c*x]))/(35*d^2*x^5) - (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*((-5*d*Sqrt[1 - c^2*x^2]*(d + e*x^2)^(5/2))/(7*x^7) + (-1/5*(d*(30*c ^2*d + 11*e)*Sqrt[1 - c^2*x^2]*(d + e*x^2)^(3/2))/x^5 + (-1/3*(d*(120*c^4* d^2 + 159*c^2*d*e - 37*e^2)*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/x^3 + (-((( 240*c^6*d^3 + 528*c^4*d^2*e + 193*c^2*d*e^2 - 247*e^3)*Sqrt[1 - c^2*x^2]*S qrt[d + e*x^2])/x) + e*(-((c*(240*c^6*d^3 + 528*c^4*d^2*e + 193*c^2*d*e^2 - 247*e^3)*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(e*Sqrt[1 + (e*x^2)/d])) + (2*(c^2*d + e)*(120*c^6*d^3 + 204*c^4*d^2*e + 17*c^2*d*e ^2 - 105*e^3)*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(c *e*Sqrt[d + e*x^2])))/3)/5)/7))/(35*d^2)
3.2.49.3.1 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_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && !GtQ[c, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2] Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && !GtQ[a, 0]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(a*g*(m + 1))), x] - Simp[1/(a*g^2*(m + 1)) Int[(g*x) ^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*2 *(b*c*(p + 1) + a*d*q) + d*((b*e - a*f)*(m + 1) + b*e*2*(p + q + 1))*x^2, x ], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && GtQ[q, 0] && LtQ[m, -1] && !(EqQ[q, 1] && SimplerQ[e + f*x^2, c + d*x^2])
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
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]))
\[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}{x^{8}}d x\]
Time = 0.11 (sec) , antiderivative size = 447, normalized size of antiderivative = 0.80 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx=\frac {105 \, {\left (2 \, b c d e^{3} x^{6} - b c d^{2} e^{2} x^{4} - 8 \, b c d^{3} e x^{2} - 5 \, b c d^{4}\right )} \sqrt {e x^{2} + d} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (210 \, a c d e^{3} x^{6} - 105 \, a c d^{2} e^{2} x^{4} - 840 \, a c d^{3} e x^{2} - 525 \, a c d^{4} + {\left (75 \, b c^{2} d^{4} x + {\left (240 \, b c^{8} d^{4} + 528 \, b c^{6} d^{3} e + 193 \, b c^{4} d^{2} e^{2} - 247 \, b c^{2} d e^{3}\right )} x^{7} + {\left (120 \, b c^{6} d^{4} + 249 \, b c^{4} d^{3} e + 71 \, b c^{2} d^{2} e^{2}\right )} x^{5} + 3 \, {\left (30 \, b c^{4} d^{4} + 61 \, b c^{2} d^{3} e\right )} x^{3}\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \sqrt {e x^{2} + d} + {\left ({\left (240 \, b c^{10} d^{4} + 528 \, b c^{8} d^{3} e + 193 \, b c^{6} d^{2} e^{2} - 247 \, b c^{4} d e^{3}\right )} x^{7} E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left (240 \, b c^{10} d^{4} + 24 \, {\left (22 \, b c^{8} + 5 \, b c^{6}\right )} d^{3} e + {\left (193 \, b c^{6} + 249 \, b c^{4}\right )} d^{2} e^{2} - {\left (247 \, b c^{4} - 71 \, b c^{2}\right )} d e^{3} - 210 \, b e^{4}\right )} x^{7} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {d}}{3675 \, c d^{3} x^{7}} \]
1/3675*(105*(2*b*c*d*e^3*x^6 - b*c*d^2*e^2*x^4 - 8*b*c*d^3*e*x^2 - 5*b*c*d ^4)*sqrt(e*x^2 + d)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) + (210*a*c*d*e^3*x^6 - 105*a*c*d^2*e^2*x^4 - 840*a*c*d^3*e*x^2 - 525*a*c*d^4 + (75*b*c^2*d^4*x + (240*b*c^8*d^4 + 528*b*c^6*d^3*e + 193*b*c^4*d^2*e^2 - 247*b*c^2*d*e^3)*x^7 + (120*b*c^6*d^4 + 249*b*c^4*d^3*e + 71*b*c^2*d^2*e ^2)*x^5 + 3*(30*b*c^4*d^4 + 61*b*c^2*d^3*e)*x^3)*sqrt(-(c^2*x^2 - 1)/(c^2* x^2)))*sqrt(e*x^2 + d) + ((240*b*c^10*d^4 + 528*b*c^8*d^3*e + 193*b*c^6*d^ 2*e^2 - 247*b*c^4*d*e^3)*x^7*elliptic_e(arcsin(c*x), -e/(c^2*d)) - (240*b* c^10*d^4 + 24*(22*b*c^8 + 5*b*c^6)*d^3*e + (193*b*c^6 + 249*b*c^4)*d^2*e^2 - (247*b*c^4 - 71*b*c^2)*d*e^3 - 210*b*e^4)*x^7*elliptic_f(arcsin(c*x), - e/(c^2*d)))*sqrt(d))/(c*d^3*x^7)
Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x^{8}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{x^8} \,d x \]