Integrand size = 23, antiderivative size = 246 \[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {b x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {d+e x^2}}+\frac {x^3 \left (a+b \text {sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b c \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3 d e \left (c^2 d+e\right ) \sqrt {1+\frac {e x^2}{d}}}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3 c d e \sqrt {d+e x^2}} \]
1/3*x^3*(a+b*arcsech(c*x))/d/(e*x^2+d)^(3/2)-1/3*b*x*(1/(c*x+1))^(1/2)*(c* x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/d/(c^2*d+e)/(e*x^2+d)^(1/2)-1/3*b*c*Elliptic E(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/ e/(c^2*d+e)/(1+e*x^2/d)^(1/2)+1/3*b*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/(e*x^2+d)^(1/2)
Result contains complex when optimal does not.
Time = 3.14 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.98 \[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {a x^3-\frac {b \sqrt {\frac {1-c x}{1+c x}} (-c d+e x) \left (d+e x^2\right )}{e \left (c^2 d+e\right )}+b x^3 \text {sech}^{-1}(c x)+\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \sqrt {\frac {c \left (\sqrt {d}+i \sqrt {e} x\right )}{\left (c \sqrt {d}+i \sqrt {e}\right ) (1+c x)}} \sqrt {\frac {c \left (i \sqrt {d}+\sqrt {e} x\right )}{\left (i c \sqrt {d}+\sqrt {e}\right ) (1+c x)}} \left (d+e x^2\right ) \left (\left (i c \sqrt {d}+\sqrt {e}\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {\left (c^2 d+e\right ) (1-c x)}{\left (c \sqrt {d}+i \sqrt {e}\right )^2 (1+c x)}}\right )|\frac {\left (c \sqrt {d}+i \sqrt {e}\right )^2}{\left (c \sqrt {d}-i \sqrt {e}\right )^2}\right )-2 \sqrt {e} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {\left (c^2 d+e\right ) (1-c x)}{\left (c \sqrt {d}+i \sqrt {e}\right )^2 (1+c x)}}\right ),\frac {\left (c \sqrt {d}+i \sqrt {e}\right )^2}{\left (c \sqrt {d}-i \sqrt {e}\right )^2}\right )\right )}{c \left (c \sqrt {d}+i \sqrt {e}\right ) e \sqrt {\frac {\left (i c \sqrt {d}+\sqrt {e}\right ) (-1+c x)}{\left (-i c \sqrt {d}+\sqrt {e}\right ) (1+c x)}}}}{3 d \left (d+e x^2\right )^{3/2}} \]
(a*x^3 - (b*Sqrt[(1 - c*x)/(1 + c*x)]*(-(c*d) + e*x)*(d + e*x^2))/(e*(c^2* d + e)) + b*x^3*ArcSech[c*x] + (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*Sqrt [(c*(Sqrt[d] + I*Sqrt[e]*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x))]*Sqrt[(c* (I*Sqrt[d] + Sqrt[e]*x))/((I*c*Sqrt[d] + Sqrt[e])*(1 + c*x))]*(d + e*x^2)* ((I*c*Sqrt[d] + Sqrt[e])*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] - 2*Sqrt[e]*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]))/(c*(c*Sqrt[d] + I*Sqrt[e])*e*Sqrt[((I*c*Sqrt[ d] + Sqrt[e])*(-1 + c*x))/(((-I)*c*Sqrt[d] + Sqrt[e])*(1 + c*x))]))/(3*d*( d + e*x^2)^(3/2))
Time = 0.49 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.83, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6855, 27, 373, 326, 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.
| \(\displaystyle \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6855 |
| \(\displaystyle b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {x^2}{3 d \sqrt {1-c^2 x^2} \left (e x^2+d\right )^{3/2}}dx+\frac {x^3 \left (a+b \text {sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {x^2}{\sqrt {1-c^2 x^2} \left (e x^2+d\right )^{3/2}}dx}{3 d}+\frac {x^3 \left (a+b \text {sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 373 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\int \frac {\sqrt {1-c^2 x^2}}{\sqrt {e x^2+d}}dx}{c^2 d+e}-\frac {x \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{3 d}+\frac {x^3 \left (a+b \text {sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 326 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {\left (c^2 d+e\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx}{e}-\frac {c^2 \int \frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}dx}{e}}{c^2 d+e}-\frac {x \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{3 d}+\frac {x^3 \left (a+b \text {sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {\left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1}}dx}{e \sqrt {d+e x^2}}-\frac {c^2 \int \frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}dx}{e}}{c^2 d+e}-\frac {x \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{3 d}+\frac {x^3 \left (a+b \text {sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {\left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {d+e x^2}}-\frac {c^2 \int \frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}dx}{e}}{c^2 d+e}-\frac {x \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{3 d}+\frac {x^3 \left (a+b \text {sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {\left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {d+e x^2}}-\frac {c^2 \sqrt {d+e x^2} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-c^2 x^2}}dx}{e \sqrt {\frac {e x^2}{d}+1}}}{c^2 d+e}-\frac {x \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{3 d}+\frac {x^3 \left (a+b \text {sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {x^3 \left (a+b \text {sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {\left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {d+e x^2}}-\frac {c \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{e \sqrt {\frac {e x^2}{d}+1}}}{c^2 d+e}-\frac {x \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{3 d}\) |
(x^3*(a + b*ArcSech[c*x]))/(3*d*(d + e*x^2)^(3/2)) + (b*Sqrt[(1 + c*x)^(-1 )]*Sqrt[1 + c*x]*(-((x*Sqrt[1 - c^2*x^2])/((c^2*d + e)*Sqrt[d + e*x^2])) + (-((c*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(e*Sqrt[1 + ( e*x^2)/d])) + ((c^2*d + e)*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/ (c^2*d))])/(c*e*Sqrt[d + e*x^2]))/(c^2*d + e)))/(3*d)
3.2.75.3.1 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[ b/d Int[Sqrt[c + d*x^2]/Sqrt[a + b*x^2], x], x] - Simp[(b*c - a*d)/d In t[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && NegQ[b/a]
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[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*(b*c - a*d)*(p + 1))), x] - Simp[e^2/(2*(b*c - a*d)*(p + 1)) Int[(e *x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(m - 1) + d*(m + 2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 1] && LeQ[m, 3] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
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]))
\[\int \frac {x^{2} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (166) = 332\).
Time = 0.10 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.36 \[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {{\left (b c^{3} d^{2} e + b c d e^{2}\right )} \sqrt {e x^{2} + d} x^{3} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left ({\left (a c^{3} d^{2} e + a c d e^{2}\right )} x^{3} - {\left (b c^{2} d e^{2} x^{4} + b c^{2} d^{2} e x^{2}\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \sqrt {e x^{2} + d} - {\left ({\left (b c^{4} d e^{2} x^{4} + 2 \, b c^{4} d^{2} e x^{2} + b c^{4} d^{3}\right )} E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left (b c^{4} d^{3} + {\left (b c^{4} d e^{2} + b e^{3}\right )} x^{4} + b d^{2} e + 2 \, {\left (b c^{4} d^{2} e + b d e^{2}\right )} x^{2}\right )} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {d}}{3 \, {\left (c^{3} d^{5} e + c d^{4} e^{2} + {\left (c^{3} d^{3} e^{3} + c d^{2} e^{4}\right )} x^{4} + 2 \, {\left (c^{3} d^{4} e^{2} + c d^{3} e^{3}\right )} x^{2}\right )}} \]
1/3*((b*c^3*d^2*e + b*c*d*e^2)*sqrt(e*x^2 + d)*x^3*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) + ((a*c^3*d^2*e + a*c*d*e^2)*x^3 - (b*c^2*d*e ^2*x^4 + b*c^2*d^2*e*x^2)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))*sqrt(e*x^2 + d) - ((b*c^4*d*e^2*x^4 + 2*b*c^4*d^2*e*x^2 + b*c^4*d^3)*elliptic_e(arcsin(c*x ), -e/(c^2*d)) - (b*c^4*d^3 + (b*c^4*d*e^2 + b*e^3)*x^4 + b*d^2*e + 2*(b*c ^4*d^2*e + b*d*e^2)*x^2)*elliptic_f(arcsin(c*x), -e/(c^2*d)))*sqrt(d))/(c^ 3*d^5*e + c*d^4*e^2 + (c^3*d^3*e^3 + c*d^2*e^4)*x^4 + 2*(c^3*d^4*e^2 + c*d ^3*e^3)*x^2)
Timed out. \[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
-1/3*a*(x/((e*x^2 + d)^(3/2)*e) - x/(sqrt(e*x^2 + d)*d*e)) + b*integrate(x ^2*log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))/(e*x^2 + d)^(5/2), x )
\[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]