Integrand size = 23, antiderivative size = 409 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{x^6} \, dx=\frac {4 b \left (c^2 d+2 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{75 x^3}+\frac {b \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{75 d x}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 d x^5}+\frac {b c \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{75 d \sqrt {1+\frac {e x^2}{d}}}-\frac {b \left (c^2 d+e\right ) \left (8 c^4 d^2+19 c^2 d e+15 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{75 c d \sqrt {d+e x^2}} \]
-1/5*(e*x^2+d)^(5/2)*(a+b*arcsech(c*x))/d/x^5+1/25*b*(e*x^2+d)^(3/2)*(1/(c *x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/x^5+4/75*b*(c^2*d+2*e)*(1/(c *x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/2)/x^3+1/75*b*( 8*c^4*d^2+23*c^2*d*e+23*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+1/75*b*c*(8*c^4*d^2+23*c^2*d*e+23*e^2)*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/( 1+e*x^2/d)^(1/2)-1/75*b*(c^2*d+e)*(8*c^4*d^2+19*c^2*d*e+15*e^2)*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/(e*x^2+d)^(1/2)
Result contains complex when optimal does not.
Time = 27.98 (sec) , antiderivative size = 620, normalized size of antiderivative = 1.52 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{x^6} \, dx=\frac {-\frac {15 a \left (d+e x^2\right )^3}{x^5}+\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (d+e x^2\right ) \left (23 e^2 x^4+d e x^2 \left (11+23 c^2 x^2\right )+d^2 \left (3+4 c^2 x^2+8 c^4 x^4\right )\right )}{x^5}-\frac {15 b \left (d+e x^2\right )^3 \text {sech}^{-1}(c x)}{x^5}+\frac {b \sqrt {\frac {1-c x}{1+c x}} \left (-c^2 \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \left (d+e x^2\right )-\frac {i \left (c \sqrt {d}-i \sqrt {e}\right )^2 (1+c x) \sqrt {\frac {c \left (\sqrt {d}-i \sqrt {e} x\right )}{\left (c \sqrt {d}-i \sqrt {e}\right ) (1+c x)}} \sqrt {\frac {c \left (\sqrt {d}+i \sqrt {e} x\right )}{\left (c \sqrt {d}+i \sqrt {e}\right ) (1+c x)}} \left (\left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {\left (c^2 d+e\right ) (1-c x)}{\left (c \sqrt {d}+i \sqrt {e}\right )^2 (1+c x)}}\right )|\frac {\left (c \sqrt {d}+i \sqrt {e}\right )^2}{\left (c \sqrt {d}-i \sqrt {e}\right )^2}\right )+2 \sqrt {e} \left (8 i c^3 d^{3/2}-12 c^2 d \sqrt {e}+7 i c \sqrt {d} e-15 e^{3/2}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {\left (c^2 d+e\right ) (1-c x)}{\left (c \sqrt {d}+i \sqrt {e}\right )^2 (1+c x)}}\right ),\frac {\left (c \sqrt {d}+i \sqrt {e}\right )^2}{\left (c \sqrt {d}-i \sqrt {e}\right )^2}\right )\right )}{\sqrt {-\frac {\left (c \sqrt {d}-i \sqrt {e}\right ) (-1+c x)}{\left (c \sqrt {d}+i \sqrt {e}\right ) (1+c x)}}}\right )}{c}}{75 d \sqrt {d+e x^2}} \]
((-15*a*(d + e*x^2)^3)/x^5 + (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(d + e *x^2)*(23*e^2*x^4 + d*e*x^2*(11 + 23*c^2*x^2) + d^2*(3 + 4*c^2*x^2 + 8*c^4 *x^4)))/x^5 - (15*b*(d + e*x^2)^3*ArcSech[c*x])/x^5 + (b*Sqrt[(1 - c*x)/(1 + c*x)]*(-(c^2*(8*c^4*d^2 + 23*c^2*d*e + 23*e^2)*(d + e*x^2)) - (I*(c*Sqr t[d] - I*Sqrt[e])^2*(1 + c*x)*Sqrt[(c*(Sqrt[d] - I*Sqrt[e]*x))/((c*Sqrt[d] - I*Sqrt[e])*(1 + c*x))]*Sqrt[(c*(Sqrt[d] + I*Sqrt[e]*x))/((c*Sqrt[d] + I *Sqrt[e])*(1 + c*x))]*((8*c^4*d^2 + 23*c^2*d*e + 23*e^2)*EllipticE[I*ArcSi nh[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] + 2*Sqrt[e]*((8*I)*c^3 *d^(3/2) - 12*c^2*d*Sqrt[e] + (7*I)*c*Sqrt[d]*e - 15*e^(3/2))*EllipticF[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*Sqrt [d] - I*Sqrt[e])*(-1 + c*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x)))]))/c)/(7 5*d*Sqrt[d + e*x^2])
Time = 0.72 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.83, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {6855, 27, 376, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{x^6} \, dx\) |
| \(\Big \downarrow \) 6855 |
| \(\displaystyle b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int -\frac {\left (e x^2+d\right )^{5/2}}{5 d x^6 \sqrt {1-c^2 x^2}}dx-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {\left (e x^2+d\right )^{5/2}}{x^6 \sqrt {1-c^2 x^2}}dx}{5 d}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 376 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{5} \int \frac {\sqrt {e x^2+d} \left (e \left (d c^2+5 e\right ) x^2+4 d \left (d c^2+2 e\right )\right )}{x^4 \sqrt {1-c^2 x^2}}dx-\frac {d \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{5 d}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 442 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {e \left (4 d^2 c^4+11 d e c^2+15 e^2\right ) x^2+d \left (8 d^2 c^4+23 d e c^2+23 e^2\right )}{x^2 \sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx-\frac {4 d \sqrt {1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt {d+e x^2}}{3 x^3}\right )-\frac {d \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{5 d}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{5} \left (\frac {1}{3} \left (-\frac {\int -\frac {d e \left (4 d^2 c^4-\left (8 d^2 c^4+23 d e c^2+23 e^2\right ) x^2 c^2+11 d e c^2+15 e^2\right )}{\sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx}{d}-\frac {\sqrt {1-c^2 x^2} \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {d+e x^2}}{x}\right )-\frac {4 d \sqrt {1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt {d+e x^2}}{3 x^3}\right )-\frac {d \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{5 d}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\int \frac {d e \left (4 d^2 c^4-\left (8 d^2 c^4+23 d e c^2+23 e^2\right ) x^2 c^2+11 d e c^2+15 e^2\right )}{\sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx}{d}-\frac {\sqrt {1-c^2 x^2} \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {d+e x^2}}{x}\right )-\frac {4 d \sqrt {1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt {d+e x^2}}{3 x^3}\right )-\frac {d \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{5 d}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{5} \left (\frac {1}{3} \left (e \int \frac {4 d^2 c^4-\left (8 d^2 c^4+23 d e c^2+23 e^2\right ) x^2 c^2+11 d e c^2+15 e^2}{\sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx-\frac {\sqrt {1-c^2 x^2} \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {d+e x^2}}{x}\right )-\frac {4 d \sqrt {1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt {d+e x^2}}{3 x^3}\right )-\frac {d \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{5 d}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {\left (c^2 d+e\right ) \left (8 c^4 d^2+19 c^2 d e+15 e^2\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx}{e}-\frac {c^2 \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \int \frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}dx}{e}\right )-\frac {\sqrt {1-c^2 x^2} \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {d+e x^2}}{x}\right )-\frac {4 d \sqrt {1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt {d+e x^2}}{3 x^3}\right )-\frac {d \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{5 d}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {\left (c^2 d+e\right ) \left (8 c^4 d^2+19 c^2 d e+15 e^2\right ) \sqrt {\frac {e x^2}{d}+1} \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1}}dx}{e \sqrt {d+e x^2}}-\frac {c^2 \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \int \frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}dx}{e}\right )-\frac {\sqrt {1-c^2 x^2} \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {d+e x^2}}{x}\right )-\frac {4 d \sqrt {1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt {d+e x^2}}{3 x^3}\right )-\frac {d \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{5 d}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {\left (c^2 d+e\right ) \left (8 c^4 d^2+19 c^2 d e+15 e^2\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {d+e x^2}}-\frac {c^2 \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \int \frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}dx}{e}\right )-\frac {\sqrt {1-c^2 x^2} \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {d+e x^2}}{x}\right )-\frac {4 d \sqrt {1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt {d+e x^2}}{3 x^3}\right )-\frac {d \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{5 d}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {\left (c^2 d+e\right ) \left (8 c^4 d^2+19 c^2 d e+15 e^2\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {d+e x^2}}-\frac {c^2 \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {d+e x^2} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-c^2 x^2}}dx}{e \sqrt {\frac {e x^2}{d}+1}}\right )-\frac {\sqrt {1-c^2 x^2} \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {d+e x^2}}{x}\right )-\frac {4 d \sqrt {1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt {d+e x^2}}{3 x^3}\right )-\frac {d \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{5 d}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 d x^5}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {\left (c^2 d+e\right ) \left (8 c^4 d^2+19 c^2 d e+15 e^2\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {d+e x^2}}-\frac {c \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{e \sqrt {\frac {e x^2}{d}+1}}\right )-\frac {\sqrt {1-c^2 x^2} \left (8 c^4 d^2+23 c^2 d e+23 e^2\right ) \sqrt {d+e x^2}}{x}\right )-\frac {4 d \sqrt {1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt {d+e x^2}}{3 x^3}\right )-\frac {d \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{5 d}\) |
-1/5*((d + e*x^2)^(5/2)*(a + b*ArcSech[c*x]))/(d*x^5) - (b*Sqrt[(1 + c*x)^ (-1)]*Sqrt[1 + c*x]*(-1/5*(d*Sqrt[1 - c^2*x^2]*(d + e*x^2)^(3/2))/x^5 + (( -4*d*(c^2*d + 2*e)*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(3*x^3) + (-(((8*c^4 *d^2 + 23*c^2*d*e + 23*e^2)*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/x) + e*(-(( c*(8*c^4*d^2 + 23*c^2*d*e + 23*e^2)*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(e*Sqrt[1 + (e*x^2)/d])) + ((c^2*d + e)*(8*c^4*d^2 + 19*c^ 2*d*e + 15*e^2)*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/ (c*e*Sqrt[d + e*x^2])))/3)/5))/(5*d)
3.2.48.3.1 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_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> Simp[c*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1 )/(a*e*(m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^ 2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b*c - a*d)*(m + 1) + 2*c*(b*c*(p + 1) + a* d*(q - 1)) + d*((b*c - a*d)*(m + 1) + 2*b*c*(p + q))*x^2, x], x], x] /; Fre eQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && LtQ[m, -1] & & IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
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]))
\[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}{x^{6}}d x\]
Time = 0.10 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.83 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{x^6} \, dx=-\frac {15 \, {\left (b c d e^{2} x^{4} + 2 \, b c d^{2} e x^{2} + b c d^{3}\right )} \sqrt {e x^{2} + d} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (15 \, a c d e^{2} x^{4} + 30 \, a c d^{2} e x^{2} + 15 \, a c d^{3} - {\left (3 \, b c^{2} d^{3} x + {\left (8 \, b c^{6} d^{3} + 23 \, b c^{4} d^{2} e + 23 \, b c^{2} d e^{2}\right )} x^{5} + {\left (4 \, b c^{4} d^{3} + 11 \, b c^{2} d^{2} e\right )} x^{3}\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \sqrt {e x^{2} + d} - {\left ({\left (8 \, b c^{8} d^{3} + 23 \, b c^{6} d^{2} e + 23 \, b c^{4} d e^{2}\right )} x^{5} E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left (8 \, b c^{8} d^{3} + {\left (23 \, b c^{6} + 4 \, b c^{4}\right )} d^{2} e + {\left (23 \, b c^{4} + 11 \, b c^{2}\right )} d e^{2} + 15 \, b e^{3}\right )} x^{5} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {d}}{75 \, c d^{2} x^{5}} \]
-1/75*(15*(b*c*d*e^2*x^4 + 2*b*c*d^2*e*x^2 + b*c*d^3)*sqrt(e*x^2 + d)*log( (c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) + (15*a*c*d*e^2*x^4 + 30*a *c*d^2*e*x^2 + 15*a*c*d^3 - (3*b*c^2*d^3*x + (8*b*c^6*d^3 + 23*b*c^4*d^2*e + 23*b*c^2*d*e^2)*x^5 + (4*b*c^4*d^3 + 11*b*c^2*d^2*e)*x^3)*sqrt(-(c^2*x^ 2 - 1)/(c^2*x^2)))*sqrt(e*x^2 + d) - ((8*b*c^8*d^3 + 23*b*c^6*d^2*e + 23*b *c^4*d*e^2)*x^5*elliptic_e(arcsin(c*x), -e/(c^2*d)) - (8*b*c^8*d^3 + (23*b *c^6 + 4*b*c^4)*d^2*e + (23*b*c^4 + 11*b*c^2)*d*e^2 + 15*b*e^3)*x^5*ellipt ic_f(arcsin(c*x), -e/(c^2*d)))*sqrt(d))/(c*d^2*x^5)
\[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{x^6} \, dx=\int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{x^{6}}\, dx \]
Exception generated. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{x^6} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{x^6} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{x^6} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{x^6} \,d x \]