Integrand size = 21, antiderivative size = 362 \[ \int \sqrt {f+g x} \sqrt {a+c x^2} \, dx=-\frac {4 f \sqrt {f+g x} \sqrt {a+c x^2}}{15 g}+\frac {2 (f+g x)^{3/2} \sqrt {a+c x^2}}{5 g}+\frac {4 \sqrt {-a} \left (c f^2-3 a g^2\right ) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{15 \sqrt {c} g^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {4 \sqrt {-a} f \left (c f^2+a g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{15 \sqrt {c} g^2 \sqrt {f+g x} \sqrt {a+c x^2}} \]
2/5*(g*x+f)^(3/2)*(c*x^2+a)^(1/2)/g-4/15*f*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/g +4/15*(-3*a*g^2+c*f^2)*EllipticE(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2 ),(-2*a*g/(-a*g+f*(-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*(g*x+f)^(1/2)*(1+ c*x^2/a)^(1/2)/g^2/c^(1/2)/(c*x^2+a)^(1/2)/((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+ f*c^(1/2)))^(1/2)-4/15*f*(a*g^2+c*f^2)*EllipticF(1/2*(1-x*c^(1/2)/(-a)^(1/ 2))^(1/2)*2^(1/2),(-2*a*g/(-a*g+f*(-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*( 1+c*x^2/a)^(1/2)*((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)/g^2/c^(1 /2)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 23.16 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.44 \[ \int \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\frac {\sqrt {f+g x} \left (\frac {2 (f+3 g x) \left (a+c x^2\right )}{g}-\frac {4 \left (g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \left (c f^2-3 a g^2\right ) \left (a+c x^2\right )+\sqrt {c} \left (-i c^{3/2} f^3+\sqrt {a} c f^2 g+3 i a \sqrt {c} f g^2-3 a^{3/2} g^3\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )-\sqrt {a} \sqrt {c} g \left (c f^2+4 i \sqrt {a} \sqrt {c} f g-3 a g^2\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{c g^3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)}\right )}{15 \sqrt {a+c x^2}} \]
(Sqrt[f + g*x]*((2*(f + 3*g*x)*(a + c*x^2))/g - (4*(g^2*Sqrt[-f - (I*Sqrt[ a]*g)/Sqrt[c]]*(c*f^2 - 3*a*g^2)*(a + c*x^2) + Sqrt[c]*((-I)*c^(3/2)*f^3 + Sqrt[a]*c*f^2*g + (3*I)*a*Sqrt[c]*f*g^2 - 3*a^(3/2)*g^3)*Sqrt[(g*((I*Sqrt [a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g* x))]*(f + g*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/ Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] - Sqr t[a]*Sqrt[c]*g*(c*f^2 + (4*I)*Sqrt[a]*Sqrt[c]*f*g - 3*a*g^2)*Sqrt[(g*((I*S qrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c ]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)]))/ (c*g^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x))))/(15*Sqrt[a + c*x^2])
Time = 0.81 (sec) , antiderivative size = 693, normalized size of antiderivative = 1.91, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {493, 687, 27, 599, 25, 1511, 1416, 1509}
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.
| \(\displaystyle \int \sqrt {a+c x^2} \sqrt {f+g x} \, dx\) |
| \(\Big \downarrow \) 493 |
| \(\displaystyle \frac {2 \int \frac {(a g-c f x) \sqrt {f+g x}}{\sqrt {c x^2+a}}dx}{5 g}+\frac {2 \sqrt {a+c x^2} (f+g x)^{3/2}}{5 g}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {2 \left (\frac {2 \int \frac {c \left (4 a f g-\left (c f^2-3 a g^2\right ) x\right )}{2 \sqrt {f+g x} \sqrt {c x^2+a}}dx}{3 c}-\frac {2}{3} f \sqrt {a+c x^2} \sqrt {f+g x}\right )}{5 g}+\frac {2 \sqrt {a+c x^2} (f+g x)^{3/2}}{5 g}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {1}{3} \int \frac {4 a f g-\left (c f^2-3 a g^2\right ) x}{\sqrt {f+g x} \sqrt {c x^2+a}}dx-\frac {2}{3} f \sqrt {a+c x^2} \sqrt {f+g x}\right )}{5 g}+\frac {2 \sqrt {a+c x^2} (f+g x)^{3/2}}{5 g}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {2 \left (-\frac {2 \int -\frac {f \left (c f^2+a g^2\right )-\left (c f^2-3 a g^2\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{3 g^2}-\frac {2}{3} f \sqrt {a+c x^2} \sqrt {f+g x}\right )}{5 g}+\frac {2 \sqrt {a+c x^2} (f+g x)^{3/2}}{5 g}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (\frac {2 \int \frac {f \left (c f^2+a g^2\right )-\left (c f^2-3 a g^2\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{3 g^2}-\frac {2}{3} f \sqrt {a+c x^2} \sqrt {f+g x}\right )}{5 g}+\frac {2 \sqrt {a+c x^2} (f+g x)^{3/2}}{5 g}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {2 \left (-\frac {2 \left (\frac {\sqrt {a g^2+c f^2} \left (-\sqrt {c} f \sqrt {a g^2+c f^2}-3 a g^2+c f^2\right ) \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}-\frac {\left (c f^2-3 a g^2\right ) \sqrt {a g^2+c f^2} \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}\right )}{3 g^2}-\frac {2}{3} f \sqrt {a+c x^2} \sqrt {f+g x}\right )}{5 g}+\frac {2 \sqrt {a+c x^2} (f+g x)^{3/2}}{5 g}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {2 \left (-\frac {2 \left (\frac {\left (a g^2+c f^2\right )^{3/4} \left (-\sqrt {c} f \sqrt {a g^2+c f^2}-3 a g^2+c f^2\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}-\frac {\left (c f^2-3 a g^2\right ) \sqrt {a g^2+c f^2} \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}\right )}{3 g^2}-\frac {2}{3} f \sqrt {a+c x^2} \sqrt {f+g x}\right )}{5 g}+\frac {2 \sqrt {a+c x^2} (f+g x)^{3/2}}{5 g}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2 \left (-\frac {2 \left (\frac {\left (a g^2+c f^2\right )^{3/4} \left (-\sqrt {c} f \sqrt {a g^2+c f^2}-3 a g^2+c f^2\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}-\frac {\left (c f^2-3 a g^2\right ) \sqrt {a g^2+c f^2} \left (\frac {\sqrt [4]{a g^2+c f^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}-\frac {\sqrt {f+g x} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )}\right )}{\sqrt {c}}\right )}{3 g^2}-\frac {2}{3} f \sqrt {a+c x^2} \sqrt {f+g x}\right )}{5 g}+\frac {2 \sqrt {a+c x^2} (f+g x)^{3/2}}{5 g}\) |
(2*(f + g*x)^(3/2)*Sqrt[a + c*x^2])/(5*g) + (2*((-2*f*Sqrt[f + g*x]*Sqrt[a + c*x^2])/3 - (2*(-(((c*f^2 - 3*a*g^2)*Sqrt[c*f^2 + a*g^2]*(-((Sqrt[f + g *x]*Sqrt[a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2])/( (a + (c*f^2)/g^2)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2]))) + ((c*f^ 2 + a*g^2)^(1/4)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])*Sqrt[(a + ( c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2)/((a + (c*f^2)/g^ 2)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])^2)]*EllipticE[2*ArcTan[(c ^(1/4)*Sqrt[f + g*x])/(c*f^2 + a*g^2)^(1/4)], (1 + (Sqrt[c]*f)/Sqrt[c*f^2 + a*g^2])/2])/(c^(1/4)*Sqrt[a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*( f + g*x)^2)/g^2])))/Sqrt[c]) + ((c*f^2 + a*g^2)^(3/4)*(c*f^2 - 3*a*g^2 - S qrt[c]*f*Sqrt[c*f^2 + a*g^2])*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2] )*Sqrt[(a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2)/((a + (c*f^2)/g^2)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])^2)]*Elliptic F[2*ArcTan[(c^(1/4)*Sqrt[f + g*x])/(c*f^2 + a*g^2)^(1/4)], (1 + (Sqrt[c]*f )/Sqrt[c*f^2 + a*g^2])/2])/(2*c^(3/4)*Sqrt[a + (c*f^2)/g^2 - (2*c*f*(f + g *x))/g^2 + (c*(f + g*x)^2)/g^2])))/(3*g^2)))/(5*g)
3.7.25.3.1 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[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (c + d*x)^(n + 1)*((a + b*x^2)^p/(d*(n + 2*p + 1))), x] + Simp[2*(p/(d*(n + 2*p + 1))) Int[(c + d*x)^n*(a + b*x^2)^(p - 1)*(a*d - b*c*x), x], x] /; FreeQ[{a, b, c, d, n}, x] && GtQ[p, 0] && NeQ[n + 2*p + 1, 0] && ( !Rationa lQ[n] || LtQ[n, 1]) && !ILtQ[n + 2*p, 0] && IntQuadraticQ[a, 0, b, c, d, n , p, x]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[-2/d^2 Subst[Int[(B*c - A*d - B*x^2)/Sqrt[(b*c^2 + a *d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2)], x], x, Sqrt[c + d*x]], x] /; Fr eeQ[{a, b, c, d, A, B}, x] && PosQ[b/a]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) ), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp [c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x ] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Leaf count of result is larger than twice the leaf count of optimal. \(601\) vs. \(2(290)=580\).
Time = 0.93 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.66
| method | result | size |
| risch | \(\frac {2 \left (3 g x +f \right ) \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{15 g}+\frac {2 \left (\frac {8 a f g \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}+\frac {2 \left (3 a \,g^{2}-c \,f^{2}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}\right ) \sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}}{15 g \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\) | \(602\) |
| elliptic | \(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 x \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{5}+\frac {2 f \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{15 g}+\frac {16 f a \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{15 \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}+\frac {2 \left (\frac {2 a g}{5}-\frac {2 f^{2} c}{15 g}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\) | \(623\) |
| default | \(\text {Expression too large to display}\) | \(1162\) |
2/15*(3*g*x+f)*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/g+2/15/g*(8*a*f*g*(f/g-(-a*c) ^(1/2)/c)*((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-f/g-( -a*c)^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(-f/g+(-a*c)^(1/2)/c))^(1/2)/(c* g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)*EllipticF(((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1 /2),((-f/g+(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2))+2*(3*a*g^2-c*f^2) *(f/g-(-a*c)^(1/2)/c)*((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2 )/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(-f/g+(-a*c)^(1/2)/c ))^(1/2)/(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)*((-f/g-(-a*c)^(1/2)/c)*Elliptic E(((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2),((-f/g+(-a*c)^(1/2)/c)/(-f/g-(-a*c) ^(1/2)/c))^(1/2))+(-a*c)^(1/2)/c*EllipticF(((x+f/g)/(f/g-(-a*c)^(1/2)/c))^ (1/2),((-f/g+(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2))))*((g*x+f)*(c*x ^2+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.63 \[ \int \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\frac {2 \, {\left (2 \, {\left (c f^{3} + 9 \, a f g^{2}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) + 6 \, {\left (c f^{2} g - 3 \, a g^{3}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right ) + 3 \, {\left (3 \, c g^{3} x + c f g^{2}\right )} \sqrt {c x^{2} + a} \sqrt {g x + f}\right )}}{45 \, c g^{3}} \]
2/45*(2*(c*f^3 + 9*a*f*g^2)*sqrt(c*g)*weierstrassPInverse(4/3*(c*f^2 - 3*a *g^2)/(c*g^2), -8/27*(c*f^3 + 9*a*f*g^2)/(c*g^3), 1/3*(3*g*x + f)/g) + 6*( c*f^2*g - 3*a*g^3)*sqrt(c*g)*weierstrassZeta(4/3*(c*f^2 - 3*a*g^2)/(c*g^2) , -8/27*(c*f^3 + 9*a*f*g^2)/(c*g^3), weierstrassPInverse(4/3*(c*f^2 - 3*a* g^2)/(c*g^2), -8/27*(c*f^3 + 9*a*f*g^2)/(c*g^3), 1/3*(3*g*x + f)/g)) + 3*( 3*c*g^3*x + c*f*g^2)*sqrt(c*x^2 + a)*sqrt(g*x + f))/(c*g^3)
\[ \int \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\int \sqrt {a + c x^{2}} \sqrt {f + g x}\, dx \]
\[ \int \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\int { \sqrt {c x^{2} + a} \sqrt {g x + f} \,d x } \]
\[ \int \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\int { \sqrt {c x^{2} + a} \sqrt {g x + f} \,d x } \]
Timed out. \[ \int \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\int \sqrt {f+g\,x}\,\sqrt {c\,x^2+a} \,d x \]