Integrand size = 25, antiderivative size = 373 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{\sqrt {x}} \, dx=-\frac {2 \left (2 b^2 B-5 A b c-6 a B c\right ) \sqrt {x} \sqrt {a+b x+c x^2}}{15 c^{3/2} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {2 \sqrt {x} (b B+5 A c+3 B c x) \sqrt {a+b x+c x^2}}{15 c}+\frac {2 \sqrt [4]{a} \left (2 b^2 B-5 A b c-6 a B c\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{15 c^{7/4} \sqrt {a+b x+c x^2}}-\frac {\sqrt [4]{a} \left (b+2 \sqrt {a} \sqrt {c}\right ) \left (2 b B-3 \sqrt {a} B \sqrt {c}-5 A c\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{15 c^{7/4} \sqrt {a+b x+c x^2}} \]
2/15*(3*B*c*x+5*A*c+B*b)*x^(1/2)*(c*x^2+b*x+a)^(1/2)/c-2/15*(-5*A*b*c-6*B* a*c+2*B*b^2)*x^(1/2)*(c*x^2+b*x+a)^(1/2)/c^(3/2)/(a^(1/2)+x*c^(1/2))+2/15* a^(1/4)*(-5*A*b*c-6*B*a*c+2*B*b^2)*(cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4))) ^2)^(1/2)/cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4)))*EllipticE(sin(2*arctan(c^ (1/4)*x^(1/2)/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*(a^(1/2)+x*c^(1/2 ))*((c*x^2+b*x+a)/(a^(1/2)+x*c^(1/2))^2)^(1/2)/c^(7/4)/(c*x^2+b*x+a)^(1/2) -1/15*a^(1/4)*(cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4)))^2)^(1/2)/cos(2*arcta n(c^(1/4)*x^(1/2)/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x^(1/2)/a^(1/4) )),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*(a^(1/2)+x*c^(1/2))*(b+2*a^(1/2)*c^(1/ 2))*(2*B*b-5*A*c-3*B*a^(1/2)*c^(1/2))*((c*x^2+b*x+a)/(a^(1/2)+x*c^(1/2))^2 )^(1/2)/c^(7/4)/(c*x^2+b*x+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.52 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.47 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{\sqrt {x}} \, dx=\frac {\frac {4 \sqrt {x} (b B+5 A c+3 B c x) (a+x (b+c x))}{c}+\frac {x \left (-\frac {4 \left (2 b^2 B-5 A b c-6 a B c\right ) (a+x (b+c x))}{x^{3/2}}+\frac {i \left (2 b^2 B-5 A b c-6 a B c\right ) \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}+\frac {i \left (2 b^3 B-b^2 \left (5 A c+2 B \sqrt {b^2-4 a c}\right )+2 a c \left (10 A c+3 B \sqrt {b^2-4 a c}\right )+b \left (-8 a B c+5 A c \sqrt {b^2-4 a c}\right )\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}\right )}{c^2}}{30 \sqrt {a+x (b+c x)}} \]
((4*Sqrt[x]*(b*B + 5*A*c + 3*B*c*x)*(a + x*(b + c*x)))/c + (x*((-4*(2*b^2* B - 5*A*b*c - 6*a*B*c)*(a + x*(b + c*x)))/x^(3/2) + (I*(2*b^2*B - 5*A*b*c - 6*a*B*c)*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[b^2 - 4*a*c] )*x)]*Sqrt[(2*a + b*x - Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]* EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[a/(b + Sqrt[b^2 - 4*a *c])] + (I*(2*b^3*B - b^2*(5*A*c + 2*B*Sqrt[b^2 - 4*a*c]) + 2*a*c*(10*A*c + 3*B*Sqrt[b^2 - 4*a*c]) + b*(-8*a*B*c + 5*A*c*Sqrt[b^2 - 4*a*c]))*Sqrt[2 + (4*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*Sqrt[(2*a + b*x - Sqrt[b^2 - 4*a*c]*x )/(b*x - Sqrt[b^2 - 4*a*c]*x)]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sq rt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c ])])/Sqrt[a/(b + Sqrt[b^2 - 4*a*c])]))/c^2)/(30*Sqrt[a + x*(b + c*x)])
Time = 0.52 (sec) , antiderivative size = 362, normalized size of antiderivative = 0.97, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1231, 27, 1240, 1511, 27, 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.
| \(\displaystyle \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{\sqrt {x}} \, dx\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a+b x+c x^2} (5 A c+b B+3 B c x)}{15 c}-\frac {2 \int \frac {a (b B-10 A c)+\left (2 B b^2-5 A c b-6 a B c\right ) x}{2 \sqrt {x} \sqrt {c x^2+b x+a}}dx}{15 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a+b x+c x^2} (5 A c+b B+3 B c x)}{15 c}-\frac {\int \frac {a (b B-10 A c)+\left (2 B b^2-5 A c b-6 a B c\right ) x}{\sqrt {x} \sqrt {c x^2+b x+a}}dx}{15 c}\) |
| \(\Big \downarrow \) 1240 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a+b x+c x^2} (5 A c+b B+3 B c x)}{15 c}-\frac {2 \int \frac {a (b B-10 A c)+\left (2 B b^2-5 A c b-6 a B c\right ) x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{15 c}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a+b x+c x^2} (5 A c+b B+3 B c x)}{15 c}-\frac {2 \left (\frac {\sqrt {a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (-3 \sqrt {a} B \sqrt {c}-5 A c+2 b B\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}-\frac {\sqrt {a} \left (-6 a B c-5 A b c+2 b^2 B\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{15 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a+b x+c x^2} (5 A c+b B+3 B c x)}{15 c}-\frac {2 \left (\frac {\sqrt {a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (-3 \sqrt {a} B \sqrt {c}-5 A c+2 b B\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}-\frac {\left (-6 a B c-5 A b c+2 b^2 B\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{15 c}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a+b x+c x^2} (5 A c+b B+3 B c x)}{15 c}-\frac {2 \left (\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (-3 \sqrt {a} B \sqrt {c}-5 A c+2 b B\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x+c x^2}}-\frac {\left (-6 a B c-5 A b c+2 b^2 B\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{15 c}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a+b x+c x^2} (5 A c+b B+3 B c x)}{15 c}-\frac {2 \left (\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (-3 \sqrt {a} B \sqrt {c}-5 A c+2 b B\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x+c x^2}}-\frac {\left (-6 a B c-5 A b c+2 b^2 B\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {\sqrt {x} \sqrt {a+b x+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {c}}\right )}{15 c}\) |
(2*Sqrt[x]*(b*B + 5*A*c + 3*B*c*x)*Sqrt[a + b*x + c*x^2])/(15*c) - (2*(-(( (2*b^2*B - 5*A*b*c - 6*a*B*c)*(-((Sqrt[x]*Sqrt[a + b*x + c*x^2])/(Sqrt[a] + Sqrt[c]*x)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqr t[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b /(Sqrt[a]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[a + b*x + c*x^2])))/Sqrt[c]) + (a^(1 /4)*(b + 2*Sqrt[a]*Sqrt[c])*(2*b*B - 3*Sqrt[a]*B*Sqrt[c] - 5*A*c)*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2* ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*c^(3/4 )*Sqrt[a + b*x + c*x^2])))/(15*c)
3.11.30.3.1 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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[2 Subst[Int[(f + g*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, b, c, f, g}, x]
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. \(794\) vs. \(2(361)=722\).
Time = 0.88 (sec) , antiderivative size = 795, normalized size of antiderivative = 2.13
| method | result | size |
| elliptic | \(\frac {\sqrt {x \left (c \,x^{2}+b x +a \right )}\, \left (\frac {2 B x \sqrt {c \,x^{3}+b \,x^{2}+a x}}{5}+\frac {2 \left (A c +\frac {B b}{5}\right ) \sqrt {c \,x^{3}+b \,x^{2}+a x}}{3 c}+\frac {\left (a A -\frac {a \left (A c +\frac {B b}{5}\right )}{3 c}\right ) \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, F\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{c \sqrt {c \,x^{3}+b \,x^{2}+a x}}+\frac {\left (A b +\frac {2 B a}{5}-\frac {2 b \left (A c +\frac {B b}{5}\right )}{3 c}\right ) \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \left (\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) E\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) F\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{2 c}\right )}{c \sqrt {c \,x^{3}+b \,x^{2}+a x}}\right )}{\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}\) | \(795\) |
| risch | \(\text {Expression too large to display}\) | \(1022\) |
| default | \(\text {Expression too large to display}\) | \(2012\) |
(x*(c*x^2+b*x+a))^(1/2)/x^(1/2)/(c*x^2+b*x+a)^(1/2)*(2/5*B*x*(c*x^3+b*x^2+ a*x)^(1/2)+2/3*(A*c+1/5*B*b)/c*(c*x^3+b*x^2+a*x)^(1/2)+(a*A-1/3*a/c*(A*c+1 /5*B*b))*(b+(-4*a*c+b^2)^(1/2))/c*2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c )/(b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-1/2 *(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*(-2*c*x/(b +(-4*a*c+b^2)^(1/2)))^(1/2)/(c*x^3+b*x^2+a*x)^(1/2)*EllipticF(2^(1/2)*((x+ 1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^(1/2))*c)^(1/2),1/2*(-2*(b+( -4*a*c+b^2)^(1/2))/c/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2) ^(1/2))))^(1/2))+(A*b+2/5*B*a-2/3*b/c*(A*c+1/5*B*b))*(b+(-4*a*c+b^2)^(1/2) )/c*2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^(1/2))*c)^(1 /2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/ c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*(-2*c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)/(c *x^3+b*x^2+a*x)^(1/2)*((-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^ 2)^(1/2)))*EllipticE(2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+ b^2)^(1/2))*c)^(1/2),1/2*(-2*(b+(-4*a*c+b^2)^(1/2))/c/(-1/2*(b+(-4*a*c+b^2 )^(1/2))/c-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(2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^ (1/2))*c)^(1/2),1/2*(-2*(b+(-4*a*c+b^2)^(1/2))/c/(-1/2*(b+(-4*a*c+b^2)^(1/ 2))/c-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 = 215, normalized size of antiderivative = 0.58 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{\sqrt {x}} \, dx=\frac {2 \, {\left ({\left (2 \, B b^{3} + 30 \, A a c^{2} - {\left (9 \, B a b + 5 \, A b^{2}\right )} c\right )} \sqrt {c} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right ) + 3 \, {\left (2 \, B b^{2} c - {\left (6 \, B a + 5 \, A b\right )} c^{2}\right )} \sqrt {c} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right )\right ) + 3 \, {\left (3 \, B c^{3} x + B b c^{2} + 5 \, A c^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x}\right )}}{45 \, c^{3}} \]
2/45*((2*B*b^3 + 30*A*a*c^2 - (9*B*a*b + 5*A*b^2)*c)*sqrt(c)*weierstrassPI nverse(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 9*a*b*c)/c^3, 1/3*(3*c*x + b) /c) + 3*(2*B*b^2*c - (6*B*a + 5*A*b)*c^2)*sqrt(c)*weierstrassZeta(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 9*a*b*c)/c^3, weierstrassPInverse(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 9*a*b*c)/c^3, 1/3*(3*c*x + b)/c)) + 3*(3*B*c^3 *x + B*b*c^2 + 5*A*c^3)*sqrt(c*x^2 + b*x + a)*sqrt(x))/c^3
\[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{\sqrt {x}} \, dx=\int \frac {\left (A + B x\right ) \sqrt {a + b x + c x^{2}}}{\sqrt {x}}\, dx \]
\[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{\sqrt {x}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} {\left (B x + A\right )}}{\sqrt {x}} \,d x } \]
\[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{\sqrt {x}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} {\left (B x + A\right )}}{\sqrt {x}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{\sqrt {x}} \, dx=\int \frac {\left (A+B\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{\sqrt {x}} \,d x \]