Integrand size = 23, antiderivative size = 554 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{x^8} \, dx=\frac {b c \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{3675 d^2 \sqrt {c^2 x^2}}+\frac {b c \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{3675 d x^2 \sqrt {c^2 x^2}}+\frac {b c \left (30 c^2 d+11 e\right ) \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac {b c^2 \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3675 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {2 b \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3675 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]
-1/7*(e*x^2+d)^(5/2)*(a+b*arcsec(c*x))/d/x^7+2/35*e*(e*x^2+d)^(5/2)*(a+b*a rcsec(c*x))/d^2/x^5+1/1225*b*c*(30*c^2*d+11*e)*(e*x^2+d)^(3/2)*(c^2*x^2-1) ^(1/2)/d/x^4/(c^2*x^2)^(1/2)+1/49*b*c*(e*x^2+d)^(5/2)*(c^2*x^2-1)^(1/2)/d/ x^6/(c^2*x^2)^(1/2)+1/3675*b*c*(240*c^6*d^3+528*c^4*d^2*e+193*c^2*d*e^2-24 7*e^3)*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d^2/(c^2*x^2)^(1/2)+1/3675*b*c*(1 20*c^4*d^2+159*c^2*d*e-37*e^2)*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d/x^2/(c^ 2*x^2)^(1/2)-1/3675*b*c^2*(240*c^6*d^3+528*c^4*d^2*e+193*c^2*d*e^2-247*e^3 )*x*EllipticE(c*x,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/2)/d^2 /(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(1+e*x^2/d)^(1/2)+2/3675*b*(c^2*d+e)*(1 20*c^6*d^3+204*c^4*d^2*e+17*c^2*d*e^2-105*e^3)*x*EllipticF(c*x,(-e/c^2/d)^ (1/2))*(-c^2*x^2+1)^(1/2)*(1+e*x^2/d)^(1/2)/d^2/(c^2*x^2)^(1/2)/(c^2*x^2-1 )^(1/2)/(e*x^2+d)^(1/2)
Result contains complex when optimal does not.
Time = 11.82 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.69 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{x^8} \, dx=\frac {\sqrt {d+e x^2} \left (-105 a \left (5 d-2 e x^2\right ) \left (d+e x^2\right )^2+b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (-247 e^3 x^6+d e^2 x^4 \left (71+193 c^2 x^2\right )+3 d^2 e x^2 \left (61+83 c^2 x^2+176 c^4 x^4\right )+15 d^3 \left (5+6 c^2 x^2+8 c^4 x^4+16 c^6 x^6\right )\right )-105 b \left (5 d-2 e x^2\right ) \left (d+e x^2\right )^2 \sec ^{-1}(c x)\right )}{3675 d^2 x^7}-\frac {i b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (c^2 d \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) E\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )-2 \left (120 c^8 d^4+324 c^6 d^3 e+221 c^4 d^2 e^2-88 c^2 d e^3-105 e^4\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),-\frac {e}{c^2 d}\right )\right )}{3675 \sqrt {-c^2} d^2 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]
(Sqrt[d + e*x^2]*(-105*a*(5*d - 2*e*x^2)*(d + e*x^2)^2 + b*c*Sqrt[1 - 1/(c ^2*x^2)]*x*(-247*e^3*x^6 + d*e^2*x^4*(71 + 193*c^2*x^2) + 3*d^2*e*x^2*(61 + 83*c^2*x^2 + 176*c^4*x^4) + 15*d^3*(5 + 6*c^2*x^2 + 8*c^4*x^4 + 16*c^6*x ^6)) - 105*b*(5*d - 2*e*x^2)*(d + e*x^2)^2*ArcSec[c*x]))/(3675*d^2*x^7) - ((I/3675)*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(240*c^6* d^3 + 528*c^4*d^2*e + 193*c^2*d*e^2 - 247*e^3)*EllipticE[I*ArcSinh[Sqrt[-c ^2]*x], -(e/(c^2*d))] - 2*(120*c^8*d^4 + 324*c^6*d^3*e + 221*c^4*d^2*e^2 - 88*c^2*d*e^3 - 105*e^4)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]) )/(Sqrt[-c^2]*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])
Time = 1.01 (sec) , antiderivative size = 504, normalized size of antiderivative = 0.91, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.739, Rules used = {5761, 27, 442, 25, 442, 25, 442, 25, 445, 27, 399, 323, 323, 321, 331, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{x^8} \, dx\) |
| \(\Big \downarrow \) 5761 |
| \(\displaystyle -\frac {b c x \int -\frac {\left (5 d-2 e x^2\right ) \left (e x^2+d\right )^{5/2}}{35 d^2 x^8 \sqrt {c^2 x^2-1}}dx}{\sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c x \int \frac {\left (5 d-2 e x^2\right ) \left (e x^2+d\right )^{5/2}}{x^8 \sqrt {c^2 x^2-1}}dx}{35 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 442 |
| \(\displaystyle \frac {b c x \left (\frac {5 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{7 x^7}-\frac {1}{7} \int -\frac {\left (e x^2+d\right )^{3/2} \left (\left (5 c^2 d-14 e\right ) e x^2+d \left (30 d c^2+11 e\right )\right )}{x^6 \sqrt {c^2 x^2-1}}dx\right )}{35 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b c x \left (\frac {1}{7} \int \frac {\left (e x^2+d\right )^{3/2} \left (\left (5 c^2 d-14 e\right ) e x^2+d \left (30 d c^2+11 e\right )\right )}{x^6 \sqrt {c^2 x^2-1}}dx+\frac {5 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 442 |
| \(\displaystyle \frac {b c x \left (\frac {1}{7} \left (\frac {d \sqrt {c^2 x^2-1} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{5 x^5}-\frac {1}{5} \int -\frac {\sqrt {e x^2+d} \left (2 e \left (15 d^2 c^4+18 d e c^2-35 e^2\right ) x^2+d \left (120 d^2 c^4+159 d e c^2-37 e^2\right )\right )}{x^4 \sqrt {c^2 x^2-1}}dx\right )+\frac {5 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b c x \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {\sqrt {e x^2+d} \left (2 e \left (15 d^2 c^4+18 d e c^2-35 e^2\right ) x^2+d \left (120 d^2 c^4+159 d e c^2-37 e^2\right )\right )}{x^4 \sqrt {c^2 x^2-1}}dx+\frac {d \sqrt {c^2 x^2-1} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{5 x^5}\right )+\frac {5 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 442 |
| \(\displaystyle \frac {b c x \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {d \sqrt {c^2 x^2-1} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3 x^3}-\frac {1}{3} \int -\frac {e \left (120 d^3 c^6+249 d^2 e c^4+71 d e^2 c^2-210 e^3\right ) x^2+d \left (240 d^3 c^6+528 d^2 e c^4+193 d e^2 c^2-247 e^3\right )}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx\right )+\frac {d \sqrt {c^2 x^2-1} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{5 x^5}\right )+\frac {5 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b c x \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {e \left (120 d^3 c^6+249 d^2 e c^4+71 d e^2 c^2-210 e^3\right ) x^2+d \left (240 d^3 c^6+528 d^2 e c^4+193 d e^2 c^2-247 e^3\right )}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx+\frac {d \sqrt {c^2 x^2-1} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {d \sqrt {c^2 x^2-1} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{5 x^5}\right )+\frac {5 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {b c x \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\int \frac {d e \left (120 d^3 c^6+249 d^2 e c^4+71 d e^2 c^2-\left (240 d^3 c^6+528 d^2 e c^4+193 d e^2 c^2-247 e^3\right ) x^2 c^2-210 e^3\right )}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx}{d}+\frac {\sqrt {c^2 x^2-1} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {d \sqrt {c^2 x^2-1} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{5 x^5}\right )+\frac {5 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c x \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (e \int \frac {120 d^3 c^6+249 d^2 e c^4+71 d e^2 c^2-\left (240 d^3 c^6+528 d^2 e c^4+193 d e^2 c^2-247 e^3\right ) x^2 c^2-210 e^3}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx+\frac {\sqrt {c^2 x^2-1} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {d \sqrt {c^2 x^2-1} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{5 x^5}\right )+\frac {5 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {b c x \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {2 \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) \int \frac {1}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx}{e}-\frac {c^2 \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx}{e}\right )+\frac {\sqrt {c^2 x^2-1} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {d \sqrt {c^2 x^2-1} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{5 x^5}\right )+\frac {5 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {b c x \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {2 \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) \sqrt {\frac {e x^2}{d}+1} \int \frac {1}{\sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}dx}{e \sqrt {d+e x^2}}-\frac {c^2 \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx}{e}\right )+\frac {\sqrt {c^2 x^2-1} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {d \sqrt {c^2 x^2-1} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{5 x^5}\right )+\frac {5 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {b c x \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {2 \sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) \sqrt {\frac {e x^2}{d}+1} \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1}}dx}{e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {c^2 \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx}{e}\right )+\frac {\sqrt {c^2 x^2-1} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {d \sqrt {c^2 x^2-1} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{5 x^5}\right )+\frac {5 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {b c x \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {2 \sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {c^2 \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx}{e}\right )+\frac {\sqrt {c^2 x^2-1} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {d \sqrt {c^2 x^2-1} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{5 x^5}\right )+\frac {5 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 331 |
| \(\displaystyle \frac {b c x \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {2 \sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {c^2 \sqrt {1-c^2 x^2} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \int \frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}dx}{e \sqrt {c^2 x^2-1}}\right )+\frac {\sqrt {c^2 x^2-1} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {d \sqrt {c^2 x^2-1} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{5 x^5}\right )+\frac {5 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {b c x \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {2 \sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {c^2 \sqrt {1-c^2 x^2} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-c^2 x^2}}dx}{e \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}\right )+\frac {\sqrt {c^2 x^2-1} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {d \sqrt {c^2 x^2-1} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{5 x^5}\right )+\frac {5 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{7 d x^7}+\frac {b c x \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {2 \sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {c \sqrt {1-c^2 x^2} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{e \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}\right )+\frac {\sqrt {c^2 x^2-1} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {d \sqrt {c^2 x^2-1} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{5 x^5}\right )+\frac {5 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{7 x^7}\right )}{35 d^2 \sqrt {c^2 x^2}}\) |
-1/7*((d + e*x^2)^(5/2)*(a + b*ArcSec[c*x]))/(d*x^7) + (2*e*(d + e*x^2)^(5 /2)*(a + b*ArcSec[c*x]))/(35*d^2*x^5) + (b*c*x*((5*d*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(5/2))/(7*x^7) + ((d*(30*c^2*d + 11*e)*Sqrt[-1 + c^2*x^2]*(d + e *x^2)^(3/2))/(5*x^5) + ((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])/(3*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]*Sqrt[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[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(e*Sqrt[-1 + c^2*x^2]*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 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x ], -(e/(c^2*d))])/(c*e*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])))/3)/5)/7))/(35 *d^2*Sqrt[c^2*x^2])
3.2.30.3.1 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[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && !GtQ[c, 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_.) + ArcSec[(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]}, Sim p[(a + b*ArcSec[c*x]) u, x] - Simp[b*c*(x/Sqrt[c^2*x^2]) Int[SimplifyIn tegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{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]))
\[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsec}\left (c x \right )\right )}{x^{8}}d x\]
Time = 0.15 (sec) , antiderivative size = 395, normalized size of antiderivative = 0.71 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{x^8} \, dx=\frac {{\left (210 \, a c d e^{3} x^{6} - 105 \, a c d^{2} e^{2} x^{4} - 840 \, a c d^{3} e x^{2} - 525 \, a c d^{4} + 105 \, {\left (2 \, b c d e^{3} x^{6} - b c d^{2} e^{2} x^{4} - 8 \, b c d^{3} e x^{2} - 5 \, b c d^{4}\right )} \operatorname {arcsec}\left (c x\right ) + {\left ({\left (240 \, b c^{7} d^{4} + 528 \, b c^{5} d^{3} e + 193 \, b c^{3} d^{2} e^{2} - 247 \, b c d e^{3}\right )} x^{6} + 75 \, b c d^{4} + {\left (120 \, b c^{5} d^{4} + 249 \, b c^{3} d^{3} e + 71 \, b c d^{2} e^{2}\right )} x^{4} + 3 \, {\left (30 \, b c^{3} d^{4} + 61 \, b c d^{3} e\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d} + {\left ({\left (240 \, b c^{10} d^{4} + 528 \, b c^{8} d^{3} e + 193 \, b c^{6} d^{2} e^{2} - 247 \, b c^{4} d e^{3}\right )} x^{7} E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left (240 \, b c^{10} d^{4} + 24 \, {\left (22 \, b c^{8} + 5 \, b c^{6}\right )} d^{3} e + {\left (193 \, b c^{6} + 249 \, b c^{4}\right )} d^{2} e^{2} - {\left (247 \, b c^{4} - 71 \, b c^{2}\right )} d e^{3} - 210 \, b e^{4}\right )} x^{7} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {-d}}{3675 \, c d^{3} x^{7}} \]
1/3675*((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 + 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)*arcsec(c*x) + ((240*b*c^7*d^4 + 528*b*c^5*d^3*e + 193*b*c^3*d^2*e^2 - 247*b*c*d*e^3)*x^6 + 75*b*c*d^4 + (120*b*c^5*d^4 + 249*b*c^3*d^3*e + 71 *b*c*d^2*e^2)*x^4 + 3*(30*b*c^3*d^4 + 61*b*c*d^3*e)*x^2)*sqrt(c^2*x^2 - 1) )*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 - (2 47*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. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{x^8} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{x^8} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{x^8} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}}{x^{8}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{x^8} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{x^8} \,d x \]