Integrand size = 31, antiderivative size = 479 \[ \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\frac {2 e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{3 c g}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} e (c e f-3 c d g+b e g) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3 c^2 g^2 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (e^2 g (b f-a g)+c \left (2 e^2 f^2-6 d e f g+3 d^2 g^2\right )\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3 c^2 g^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \]
2/3*e^2*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/c/g-2/3*e*(b*e*g-3*c*d*g+c*e*f)* EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1 /2),(-2*g*(-4*a*c+b^2)^(1/2)/(2*c*f-g*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1 /2)*(-4*a*c+b^2)^(1/2)*(g*x+f)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2) /c^2/g^2/(c*x^2+b*x+a)^(1/2)/(c*(g*x+f)/(2*c*f-g*(b+(-4*a*c+b^2)^(1/2))))^ (1/2)+2/3*(e^2*g*(-a*g+b*f)+c*(3*d^2*g^2-6*d*e*f*g+2*e^2*f^2))*EllipticF(1 /2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*g*( -4*a*c+b^2)^(1/2)/(2*c*f-g*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-4*a*c +b^2)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*(c*(g*x+f)/(2*c*f-g*(b+( -4*a*c+b^2)^(1/2))))^(1/2)/c^2/g^2/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2)
Result contains complex when optimal does not.
Time = 30.40 (sec) , antiderivative size = 981, normalized size of antiderivative = 2.05 \[ \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {f+g x} \left (2 c e^2 g^2 (a+x (b+c x))+\frac {(f+g x) \left (-\frac {4 e g^2 (c e f-3 c d g+b e g) \sqrt {\frac {c f^2+g (-b f+a g)}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}} (a+x (b+c x))}{(f+g x)^2}+\frac {i \sqrt {2} e (c e f-3 c d g+b e g) \left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) \sqrt {\frac {-2 a g^2+f \sqrt {\left (b^2-4 a c\right ) g^2}+2 c f g x+g \sqrt {\left (b^2-4 a c\right ) g^2} x+b g (f-g x)}{\left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \sqrt {\frac {2 a g^2+f \sqrt {\left (b^2-4 a c\right ) g^2}-2 c f g x+g \sqrt {\left (b^2-4 a c\right ) g^2} x+b g (-f+g x)}{\left (-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b f g+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right )|-\frac {-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )}{\sqrt {f+g x}}+\frac {i \sqrt {2} \left (3 c^2 d^2 g^2+b e^2 g \left (b g-\sqrt {\left (b^2-4 a c\right ) g^2}\right )-c e \left (3 b d g^2+a e g^2+\sqrt {\left (b^2-4 a c\right ) g^2} (e f-3 d g)\right )\right ) \sqrt {\frac {-2 a g^2+f \sqrt {\left (b^2-4 a c\right ) g^2}+2 c f g x+g \sqrt {\left (b^2-4 a c\right ) g^2} x+b g (f-g x)}{\left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \sqrt {\frac {2 a g^2+f \sqrt {\left (b^2-4 a c\right ) g^2}-2 c f g x+g \sqrt {\left (b^2-4 a c\right ) g^2} x+b g (-f+g x)}{\left (-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b f g+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right ),-\frac {-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )}{\sqrt {f+g x}}\right )}{\sqrt {\frac {c f^2+g (-b f+a g)}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}\right )}{3 c^2 g^3 \sqrt {a+x (b+c x)}} \]
(Sqrt[f + g*x]*(2*c*e^2*g^2*(a + x*(b + c*x)) + ((f + g*x)*((-4*e*g^2*(c*e *f - 3*c*d*g + b*e*g)*Sqrt[(c*f^2 + g*(-(b*f) + a*g))/(-2*c*f + b*g + Sqrt [(b^2 - 4*a*c)*g^2])]*(a + x*(b + c*x)))/(f + g*x)^2 + (I*Sqrt[2]*e*(c*e*f - 3*c*d*g + b*e*g)*(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*Sqrt[(-2*a*g^2 + f*Sqrt[(b^2 - 4*a*c)*g^2] + 2*c*f*g*x + g*Sqrt[(b^2 - 4*a*c)*g^2]*x + b *g*(f - g*x))/((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*Sqrt[(2 *a*g^2 + f*Sqrt[(b^2 - 4*a*c)*g^2] - 2*c*f*g*x + g*Sqrt[(b^2 - 4*a*c)*g^2] *x + b*g*(-f + g*x))/((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))] *EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))])/Sqrt[f + g*x] + (I *Sqrt[2]*(3*c^2*d^2*g^2 + b*e^2*g*(b*g - Sqrt[(b^2 - 4*a*c)*g^2]) - c*e*(3 *b*d*g^2 + a*e*g^2 + Sqrt[(b^2 - 4*a*c)*g^2]*(e*f - 3*d*g)))*Sqrt[(-2*a*g^ 2 + f*Sqrt[(b^2 - 4*a*c)*g^2] + 2*c*f*g*x + g*Sqrt[(b^2 - 4*a*c)*g^2]*x + b*g*(f - g*x))/((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*Sqrt[( 2*a*g^2 + f*Sqrt[(b^2 - 4*a*c)*g^2] - 2*c*f*g*x + g*Sqrt[(b^2 - 4*a*c)*g^2 ]*x + b*g*(-f + g*x))/((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x)) ]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))])/Sqrt[f + g*x])...
Time = 0.75 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {1278, 2004, 1269, 1172, 321, 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.
| \(\displaystyle \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx\) |
| \(\Big \downarrow \) 1278 |
| \(\displaystyle \frac {2 e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{3 c g}-\frac {\int \frac {2 e^2 (c e f-3 c d g+b e g) x^2+e (c d (2 e f-9 d g)+e (b e f+2 b d g+a e g)) x+d \left (-3 c g d^2+b e^2 f+a e^2 g\right )}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{3 c g}\) |
| \(\Big \downarrow \) 2004 |
| \(\displaystyle \frac {2 e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{3 c g}-\frac {\int \frac {-3 c g d^2+b e^2 f+a e^2 g+2 e (c e f-3 c d g+b e g) x}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{3 c g}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {2 e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{3 c g}-\frac {\frac {2 e (b e g-3 c d g+c e f) \int \frac {\sqrt {f+g x}}{\sqrt {c x^2+b x+a}}dx}{g}-\frac {\left (e^2 g (b f-a g)+c \left (3 d^2 g^2-6 d e f g+2 e^2 f^2\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{g}}{3 c g}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {2 e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{3 c g}-\frac {\frac {2 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (b e g-3 c d g+c e f) \int \frac {\sqrt {\frac {g \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c g \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \left (e^2 g (b f-a g)+c \left (3 d^2 g^2-6 d e f g+2 e^2 f^2\right )\right ) \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {g \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c g \sqrt {f+g x} \sqrt {a+b x+c x^2}}}{3 c g}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2 e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{3 c g}-\frac {\frac {2 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (b e g-3 c d g+c e f) \int \frac {\sqrt {\frac {g \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c g \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \left (e^2 g (b f-a g)+c \left (3 d^2 g^2-6 d e f g+2 e^2 f^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c g \sqrt {f+g x} \sqrt {a+b x+c x^2}}}{3 c g}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2 e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{3 c g}-\frac {\frac {2 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (b e g-3 c d g+c e f) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c g \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \left (e^2 g (b f-a g)+c \left (3 d^2 g^2-6 d e f g+2 e^2 f^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c g \sqrt {f+g x} \sqrt {a+b x+c x^2}}}{3 c g}\) |
(2*e^2*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(3*c*g) - ((2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*e*(c*e*f - 3*c*d*g + b*e*g)*Sqrt[f + g*x]*Sqrt[-((c*(a + b*x + c *x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c* x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - (b + Sqr t[b^2 - 4*a*c])*g)])/(c*g*Sqrt[(c*(f + g*x))/(2*c*f - (b + Sqrt[b^2 - 4*a* c])*g)]*Sqrt[a + b*x + c*x^2]) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(e^2*g*(b*f - a*g) + c*(2*e^2*f^2 - 6*d*e*f*g + 3*d^2*g^2))*Sqrt[(c*(f + g*x))/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))] *EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/ Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/( c*g*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))/(3*c*g)
3.10.10.3.1 Defintions of rubi rules used
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[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 )/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e *Rt[b^2 - 4*a*c, 2])))^m)) Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e }, x] && EqQ[m^2, 1/4]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)* (x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[2*e^2*(d + e*x)^(m - 2)*Sqrt[f + g *x]*(Sqrt[a + b*x + c*x^2]/(c*g*(2*m - 1))), x] - Simp[1/(c*g*(2*m - 1)) Int[((d + e*x)^(m - 3)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))*Simp[b*d*e^2* f + a*e^2*(d*g + 2*e*f*(m - 2)) - c*d^3*g*(2*m - 1) + e*(e*(2*b*d*g + e*(b* f + a*g)*(2*m - 3)) + c*d*(2*e*f - 3*d*g*(2*m - 1)))*x + 2*e^2*(c*e*f - 3*c *d*g + b*e*g)*(m - 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[2*m] && GeQ[m, 2]
Int[(u_)*((d_) + (e_.)*(x_))^(q_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.) , x_Symbol] :> Int[u*(d + e*x)^(p + q)*(a/d + (c/e)*x)^p, x] /; FreeQ[{a, b , c, d, e, q}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]
Time = 3.25 (sec) , antiderivative size = 834, normalized size of antiderivative = 1.74
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {2 e^{2} \sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a}}{3 c g}+\frac {2 \left (d^{2}-\frac {2 e^{2} \left (\frac {a g}{2}+\frac {b f}{2}\right )}{3 c g}\right ) \left (\frac {f}{g}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{\sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a}}+\frac {2 \left (2 d e -\frac {2 e^{2} \left (b g +c f \right )}{3 c g}\right ) \left (\frac {f}{g}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \left (\left (-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) E\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{2 c}\right )}{\sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}}\) | \(834\) |
| risch | \(\text {Expression too large to display}\) | \(1384\) |
| default | \(\text {Expression too large to display}\) | \(4295\) |
((g*x+f)*(c*x^2+b*x+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2)*(2/3*e^2/c /g*(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f*x+a*f)^(1/2)+2*(d^2-2/3*e^2/c/g*(1/2 *a*g+1/2*b*f))*(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+f/g)/(f/g-1/2*(b+(-4 *a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-f/g-1/2/c* (-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-f/g+1/ 2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f*x+a* f)^(1/2)*EllipticF(((x+f/g)/(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-f /g+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/ 2))+2*(2*d*e-2/3*e^2/c/g*(b*g+c*f))*(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x +f/g)/(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2) ^(1/2)))/(-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^ 2)^(1/2))/c)/(-f/g+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*g*x^3+b*g*x^2+c *f*x^2+a*g*x+b*f*x+a*f)^(1/2)*((-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*Ellipt icE(((x+f/g)/(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-f/g+1/2*(b+(-4*a *c+b^2)^(1/2))/c)/(-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+1/2/c*(-b+( -4*a*c+b^2)^(1/2))*EllipticF(((x+f/g)/(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^ (1/2),((-f/g+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1 /2))))^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 471, normalized size of antiderivative = 0.98 \[ \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\frac {2 \, {\left (3 \, \sqrt {c x^{2} + b x + a} \sqrt {g x + f} c^{2} e^{2} g^{2} + {\left (2 \, c^{2} e^{2} f^{2} - {\left (6 \, c^{2} d e - b c e^{2}\right )} f g + {\left (9 \, c^{2} d^{2} - 6 \, b c d e + {\left (2 \, b^{2} - 3 \, a c\right )} e^{2}\right )} g^{2}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, \frac {3 \, c g x + c f + b g}{3 \, c g}\right ) + 6 \, {\left (c^{2} e^{2} f g - {\left (3 \, c^{2} d e - b c e^{2}\right )} g^{2}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, \frac {3 \, c g x + c f + b g}{3 \, c g}\right )\right )\right )}}{9 \, c^{3} g^{3}} \]
2/9*(3*sqrt(c*x^2 + b*x + a)*sqrt(g*x + f)*c^2*e^2*g^2 + (2*c^2*e^2*f^2 - (6*c^2*d*e - b*c*e^2)*f*g + (9*c^2*d^2 - 6*b*c*d*e + (2*b^2 - 3*a*c)*e^2)* g^2)*sqrt(c*g)*weierstrassPInverse(4/3*(c^2*f^2 - b*c*f*g + (b^2 - 3*a*c)* g^2)/(c^2*g^2), -4/27*(2*c^3*f^3 - 3*b*c^2*f^2*g - 3*(b^2*c - 6*a*c^2)*f*g ^2 + (2*b^3 - 9*a*b*c)*g^3)/(c^3*g^3), 1/3*(3*c*g*x + c*f + b*g)/(c*g)) + 6*(c^2*e^2*f*g - (3*c^2*d*e - b*c*e^2)*g^2)*sqrt(c*g)*weierstrassZeta(4/3* (c^2*f^2 - b*c*f*g + (b^2 - 3*a*c)*g^2)/(c^2*g^2), -4/27*(2*c^3*f^3 - 3*b* c^2*f^2*g - 3*(b^2*c - 6*a*c^2)*f*g^2 + (2*b^3 - 9*a*b*c)*g^3)/(c^3*g^3), weierstrassPInverse(4/3*(c^2*f^2 - b*c*f*g + (b^2 - 3*a*c)*g^2)/(c^2*g^2), -4/27*(2*c^3*f^3 - 3*b*c^2*f^2*g - 3*(b^2*c - 6*a*c^2)*f*g^2 + (2*b^3 - 9 *a*b*c)*g^3)/(c^3*g^3), 1/3*(3*c*g*x + c*f + b*g)/(c*g))))/(c^3*g^3)
\[ \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (d + e x\right )^{2}}{\sqrt {f + g x} \sqrt {a + b x + c x^{2}}}\, dx \]
\[ \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{\sqrt {c x^{2} + b x + a} \sqrt {g x + f}} \,d x } \]
\[ \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{\sqrt {c x^{2} + b x + a} \sqrt {g x + f}} \,d x } \]
Timed out. \[ \int \frac {(d+e x)^2}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{\sqrt {f+g\,x}\,\sqrt {c\,x^2+b\,x+a}} \,d x \]