Integrand size = 23, antiderivative size = 276 \[ \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {b c x^2 \sqrt {-1+c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3 d e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3 d e \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]
1/3*x^3*(a+b*arcsec(c*x))/d/(e*x^2+d)^(3/2)-1/3*b*c*x^2*(c^2*x^2-1)^(1/2)/ d/(c^2*d+e)/(c^2*x^2)^(1/2)/(e*x^2+d)^(1/2)+1/3*b*c^2*x*EllipticE(c*x,(-e/ c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/2)/d/e/(c^2*d+e)/(c^2*x^2)^( 1/2)/(c^2*x^2-1)^(1/2)/(1+e*x^2/d)^(1/2)-1/3*b*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/e/(c^2*x^2)^(1/2)/(c^2*x^2-1 )^(1/2)/(e*x^2+d)^(1/2)
Time = 0.31 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.67 \[ \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {x^2 \left (a \left (c^2 d+e\right ) x-b c \sqrt {1-\frac {1}{c^2 x^2}} \left (d+e x^2\right )+b \left (c^2 d+e\right ) x \sec ^{-1}(c x)\right )}{3 d \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} E\left (\arcsin \left (\sqrt {-\frac {e}{d}} x\right )|-\frac {c^2 d}{e}\right )}{3 d \sqrt {-\frac {e}{d}} \left (c^2 d+e\right ) \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]
(x^2*(a*(c^2*d + e)*x - b*c*Sqrt[1 - 1/(c^2*x^2)]*(d + e*x^2) + b*(c^2*d + e)*x*ArcSec[c*x]))/(3*d*(c^2*d + e)*(d + e*x^2)^(3/2)) + (b*c*Sqrt[1 - 1/ (c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*EllipticE[ArcSin[Sqrt[-(e/d)]*x], -((c^2* d)/e)])/(3*d*Sqrt[-(e/d)]*(c^2*d + e)*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])
Time = 0.53 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.91, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {5761, 27, 373, 326, 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.
| \(\displaystyle \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5761 |
| \(\displaystyle \frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b c x \int \frac {x^2}{3 d \sqrt {c^2 x^2-1} \left (e x^2+d\right )^{3/2}}dx}{\sqrt {c^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b c x \int \frac {x^2}{\sqrt {c^2 x^2-1} \left (e x^2+d\right )^{3/2}}dx}{3 d \sqrt {c^2 x^2}}\) |
| \(\Big \downarrow \) 373 |
| \(\displaystyle \frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b c x \left (\frac {x \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}-\frac {\int \frac {\sqrt {c^2 x^2-1}}{\sqrt {e x^2+d}}dx}{c^2 d+e}\right )}{3 d \sqrt {c^2 x^2}}\) |
| \(\Big \downarrow \) 326 |
| \(\displaystyle \frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b c x \left (\frac {x \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}-\frac {\frac {c^2 \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx}{e}-\frac {\left (c^2 d+e\right ) \int \frac {1}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx}{e}}{c^2 d+e}\right )}{3 d \sqrt {c^2 x^2}}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b c x \left (\frac {x \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}-\frac {\frac {c^2 \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx}{e}-\frac {\left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \int \frac {1}{\sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}dx}{e \sqrt {d+e x^2}}}{c^2 d+e}\right )}{3 d \sqrt {c^2 x^2}}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b c x \left (\frac {x \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}-\frac {\frac {c^2 \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx}{e}-\frac {\sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1}}dx}{e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}}{c^2 d+e}\right )}{3 d \sqrt {c^2 x^2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b c x \left (\frac {x \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}-\frac {\frac {c^2 \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx}{e}-\frac {\sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}}{c^2 d+e}\right )}{3 d \sqrt {c^2 x^2}}\) |
| \(\Big \downarrow \) 331 |
| \(\displaystyle \frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b c x \left (\frac {x \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}-\frac {\frac {c^2 \sqrt {1-c^2 x^2} \int \frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}dx}{e \sqrt {c^2 x^2-1}}-\frac {\sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}}{c^2 d+e}\right )}{3 d \sqrt {c^2 x^2}}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b c x \left (\frac {x \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}-\frac {\frac {c^2 \sqrt {1-c^2 x^2} \sqrt {d+e x^2} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-c^2 x^2}}dx}{e \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}-\frac {\sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}}{c^2 d+e}\right )}{3 d \sqrt {c^2 x^2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b c x \left (\frac {x \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}-\frac {\frac {c \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{e \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}-\frac {\sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}}{c^2 d+e}\right )}{3 d \sqrt {c^2 x^2}}\) |
(x^3*(a + b*ArcSec[c*x]))/(3*d*(d + e*x^2)^(3/2)) - (b*c*x*((x*Sqrt[-1 + c ^2*x^2])/((c^2*d + e)*Sqrt[d + e*x^2]) - ((c*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]) - ((c^2*d + e)*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]))/(c^ 2*d + e)))/(3*d*Sqrt[c^2*x^2])
3.2.58.3.1 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[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_.)*(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_.) + 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]))
\[\int \frac {x^{2} \left (a +b \,\operatorname {arcsec}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
Time = 0.12 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.04 \[ \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {{\left ({\left (b c^{3} d^{2} e + b c d e^{2}\right )} x^{3} \operatorname {arcsec}\left (c x\right ) + {\left (a c^{3} d^{2} e + a c d e^{2}\right )} x^{3} - {\left (b c d e^{2} x^{3} + b c d^{2} e x\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d} - {\left ({\left (b c^{4} d e^{2} x^{4} + 2 \, b c^{4} d^{2} e x^{2} + b c^{4} d^{3}\right )} E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left (b c^{4} d^{3} + {\left (b c^{4} d e^{2} + b e^{3}\right )} x^{4} + b d^{2} e + 2 \, {\left (b c^{4} d^{2} e + b d e^{2}\right )} x^{2}\right )} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {-d}}{3 \, {\left (c^{3} d^{5} e + c d^{4} e^{2} + {\left (c^{3} d^{3} e^{3} + c d^{2} e^{4}\right )} x^{4} + 2 \, {\left (c^{3} d^{4} e^{2} + c d^{3} e^{3}\right )} x^{2}\right )}} \]
1/3*(((b*c^3*d^2*e + b*c*d*e^2)*x^3*arcsec(c*x) + (a*c^3*d^2*e + a*c*d*e^2 )*x^3 - (b*c*d*e^2*x^3 + b*c*d^2*e*x)*sqrt(c^2*x^2 - 1))*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. \[ \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
-1/3*a*(x/((e*x^2 + d)^(3/2)*e) - x/(sqrt(e*x^2 + d)*d*e)) + b*integrate(x ^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))/((e^2*x^4 + 2*d*e*x^2 + d^2)*sqrt(e *x^2 + d)), x)
\[ \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]