Integrand size = 18, antiderivative size = 315 \[ \int \sqrt {d+e x} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}+\frac {4 b \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b d^2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
2/3*(e*x+d)^(3/2)*(a+b*arcsec(c*x))/e+4/3*b*EllipticE(1/2*(-c*x+1)^(1/2)*2 ^(1/2),2^(1/2)*(e/(c*d+e))^(1/2))*(e*x+d)^(1/2)*(-c^2*x^2+1)^(1/2)/c^2/x/( 1-1/c^2/x^2)^(1/2)/(c*(e*x+d)/(c*d+e))^(1/2)+4/3*b*d*EllipticF(1/2*(-c*x+1 )^(1/2)*2^(1/2),2^(1/2)*(e/(c*d+e))^(1/2))*(c*(e*x+d)/(c*d+e))^(1/2)*(-c^2 *x^2+1)^(1/2)/c^2/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)+4/3*b*d^2*EllipticPi (1/2*(-c*x+1)^(1/2)*2^(1/2),2,2^(1/2)*(e/(c*d+e))^(1/2))*(c*(e*x+d)/(c*d+e ))^(1/2)*(-c^2*x^2+1)^(1/2)/c/e/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)
Result contains complex when optimal does not.
Time = 33.34 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.88 \[ \int \sqrt {d+e x} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {2 \left (a (d+e x)^{3/2}+b (d+e x)^{3/2} \sec ^{-1}(c x)+\frac {2 i b \sqrt {\frac {e (1+c x)}{-c d+e}} \sqrt {\frac {e-c e x}{c d+e}} \left ((-c d+e) E\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right )|\frac {c d+e}{c d-e}\right )+(2 c d-e) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right ),\frac {c d+e}{c d-e}\right )-c d \operatorname {EllipticPi}\left (1+\frac {e}{c d},i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right ),\frac {c d+e}{c d-e}\right )\right )}{c^2 \sqrt {-\frac {c}{c d+e}} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{3 e} \]
(2*(a*(d + e*x)^(3/2) + b*(d + e*x)^(3/2)*ArcSec[c*x] + ((2*I)*b*Sqrt[(e*( 1 + c*x))/(-(c*d) + e)]*Sqrt[(e - c*e*x)/(c*d + e)]*((-(c*d) + e)*Elliptic E[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] + (2 *c*d - e)*EllipticF[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] - c*d*EllipticPi[1 + e/(c*d), I*ArcSinh[Sqrt[-(c/(c*d + e))] *Sqrt[d + e*x]], (c*d + e)/(c*d - e)]))/(c^2*Sqrt[-(c/(c*d + e))]*Sqrt[1 - 1/(c^2*x^2)]*x)))/(3*e)
Time = 0.85 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.11, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5749, 1898, 634, 600, 509, 508, 327, 512, 511, 321, 633, 632, 186, 413, 412}
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.
| \(\displaystyle \int \sqrt {d+e x} \left (a+b \sec ^{-1}(c x)\right ) \, dx\) |
| \(\Big \downarrow \) 5749 |
| \(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac {2 b \int \frac {(d+e x)^{3/2}}{\sqrt {1-\frac {1}{c^2 x^2}} x^2}dx}{3 c e}\) |
| \(\Big \downarrow \) 1898 |
| \(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \int \frac {(d+e x)^{3/2}}{x \sqrt {x^2-\frac {1}{c^2}}}dx}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
| \(\Big \downarrow \) 634 |
| \(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx-\int \frac {-x e^2-2 d e}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx+d e \int \frac {1}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx+e \int \frac {\sqrt {d+e x}}{\sqrt {x^2-\frac {1}{c^2}}}dx\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx+d e \int \frac {1}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx+\frac {e \sqrt {1-c^2 x^2} \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx}{\sqrt {x^2-\frac {1}{c^2}}}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx+d e \int \frac {1}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e (1-c x)}{c d+e}}}{\sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx+d e \int \frac {1}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx+\frac {d e \sqrt {1-c^2 x^2} \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{\sqrt {x^2-\frac {1}{c^2}}}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx-\frac {2 d e \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \int \frac {1}{\sqrt {1-\frac {e (1-c x)}{c d+e}} \sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx-\frac {2 d e \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
| \(\Big \downarrow \) 633 |
| \(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\frac {d^2 \sqrt {1-c^2 x^2} \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{\sqrt {x^2-\frac {1}{c^2}}}-\frac {2 d e \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
| \(\Big \downarrow \) 632 |
| \(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\frac {d^2 \sqrt {1-c^2 x^2} \int \frac {1}{x \sqrt {1-c x} \sqrt {c x+1} \sqrt {d+e x}}dx}{\sqrt {x^2-\frac {1}{c^2}}}-\frac {2 d e \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {2 d^2 \sqrt {1-c^2 x^2} \int \frac {1}{c x \sqrt {c x+1} \sqrt {d+\frac {e}{c}-\frac {e (1-c x)}{c}}}d\sqrt {1-c x}}{\sqrt {x^2-\frac {1}{c^2}}}-\frac {2 d e \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {2 d^2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \int \frac {1}{c x \sqrt {c x+1} \sqrt {1-\frac {e (1-c x)}{c d+e}}}d\sqrt {1-c x}}{\sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 d e \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {2 d^2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 d e \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
(2*(d + e*x)^(3/2)*(a + b*ArcSec[c*x]))/(3*e) - (2*b*Sqrt[-c^(-2) + x^2]*( (-2*e*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[ 2]], (2*e)/(c*d + e)])/(c*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[-c^(-2) + x^2 ]) - (2*d*e*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcS in[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*Sqrt[d + e*x]*Sqrt[-c^(-2) + x^2]) - (2*d^2*Sqrt[1 - c^2*x^2]*Sqrt[1 - (e*(1 - c*x))/(c*d + e)]*Elli pticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[-c^(-2) + x^2]*Sqrt[d + e/c - (e*(1 - c*x))/c])))/(3*c*e*Sqrt[1 - 1/(c^2*x^2)]*x)
3.1.64.3.1 Defintions of rubi rules used
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ )]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2 Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c*f)/d, 0]
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[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[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > With[{q = Rt[-b/a, 2]}, Simp[1/Sqrt[a] Int[1/(x*Sqrt[c + d*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > Simp[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(x*Sqrt[c + d*x]*Sqrt[1 + b*(x^2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[((c_) + (d_.)*(x_))^(n_)/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp[c^(n + 1/2) Int[1/(x*Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] - Int[( 1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]))*ExpandToSum[(c^(n + 1/2) - (c + d*x)^(n + 1/2))/x, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n - 1/2, 0]
Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^ (q_.), x_Symbol] :> Simp[x^(2*n*FracPart[p])*((a + c/x^(2*n))^FracPart[p]/( c + a*x^(2*n))^FracPart[p]) Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + a*x^(2*n ))^p, x], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[mn2, -2*n] && !I ntegerQ[p] && !IntegerQ[q] && PosQ[n]
Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol ] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSec[c*x])/(e*(m + 1))), x] - Simp[b/ (c*e*(m + 1)) Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x] / ; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
Time = 7.74 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.23
| method | result | size |
| derivativedivides | \(\frac {\frac {2 \left (e x +d \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arcsec}\left (c x \right )}{3}-\frac {2 \left (2 d \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c -\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d -d \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c +\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e -\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e \right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}}{3 c^{2} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e}\) | \(386\) |
| default | \(\frac {\frac {2 \left (e x +d \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arcsec}\left (c x \right )}{3}-\frac {2 \left (2 d \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c -\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d -d \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c +\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e -\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e \right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}}{3 c^{2} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e}\) | \(386\) |
| parts | \(\frac {2 a \left (e x +d \right )^{\frac {3}{2}}}{3 e}+\frac {2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arcsec}\left (c x \right )}{3}-\frac {2 \left (2 d \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c -\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d -d \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c +\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e -\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e \right ) \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}}{3 c^{2} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e}\) | \(390\) |
2/e*(1/3*(e*x+d)^(3/2)*a+b*(1/3*(e*x+d)^(3/2)*arcsec(c*x)-2/3/c^2*(2*d*Ell ipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c-Elliptic E((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c*d-d*EllipticP i((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),1/c*(c*d-e)/d,(c/(c*d+e))^(1/2)/(c/(c*d- e))^(1/2))*c+EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^( 1/2))*e-EllipticE((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2)) *e)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)/ (c/(c*d-e))^(1/2)/x/((c^2*(e*x+d)^2-2*c^2*d*(e*x+d)+c^2*d^2-e^2)/c^2/e^2/x ^2)^(1/2)))
\[ \int \sqrt {d+e x} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int { \sqrt {e x + d} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} \,d x } \]
\[ \int \sqrt {d+e x} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int \left (a + b \operatorname {asec}{\left (c x \right )}\right ) \sqrt {d + e x}\, dx \]
Exception generated. \[ \int \sqrt {d+e x} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\text {Exception raised: ValueError} \]
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+c*d>0)', see `assume?` for mor e details)
\[ \int \sqrt {d+e x} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int { \sqrt {e x + d} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} \,d x } \]
Timed out. \[ \int \sqrt {d+e x} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int \left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )\,\sqrt {d+e\,x} \,d x \]