Integrand size = 25, antiderivative size = 353 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{5/2}} \, dx=-\frac {2 (a A+(A b+3 a B) x) \sqrt {a+b x+c x^2}}{3 a x^{3/2}}+\frac {2 (A b+6 a B) \sqrt {c} \sqrt {x} \sqrt {a+b x+c x^2}}{3 a \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {2 (A b+6 a B) \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 a^{3/4} \sqrt {a+b x+c x^2}}+\frac {\left ((A b+6 a B) \sqrt {c}+\sqrt {a} (3 b B+2 A c)\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 a^{3/4} \sqrt [4]{c} \sqrt {a+b x+c x^2}} \]
-2/3*(a*A+(A*b+3*B*a)*x)*(c*x^2+b*x+a)^(1/2)/a/x^(3/2)+2/3*(A*b+6*B*a)*c^( 1/2)*x^(1/2)*(c*x^2+b*x+a)^(1/2)/a/(a^(1/2)+x*c^(1/2))-2/3*(A*b+6*B*a)*c^( 1/4)*(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)/a^(3/4)/(c*x^2+b*x+a)^(1/2)+1/3*(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)))*EllipticF(s in(2*arctan(c^(1/4)*x^(1/2)/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*((2 *A*c+3*B*b)*a^(1/2)+(A*b+6*B*a)*c^(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)/a^(3/4)/c^(1/4)/(c*x^2+b*x+a)^(1/2)
Result contains complex when optimal does not.
Time = 23.76 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.41 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{5/2}} \, dx=\frac {-4 (A b x+a (A+3 B x)) (a+x (b+c x))+\frac {x \left (4 (A b+6 a B) \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}} (a+x (b+c x))+i (A b+6 a B) \left (b-\sqrt {b^2-4 a c}\right ) \sqrt {1+\frac {2 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{3/2} \sqrt {\frac {4 a+2 b x-2 \sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+i \left (6 a B \sqrt {b^2-4 a c}+A \left (-b^2+4 a c+b \sqrt {b^2-4 a c}\right )\right ) \sqrt {1+\frac {2 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{3/2} \sqrt {\frac {4 a+2 b x-2 \sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}}{6 a x^{3/2} \sqrt {a+x (b+c x)}} \]
(-4*(A*b*x + a*(A + 3*B*x))*(a + x*(b + c*x)) + (x*(4*(A*b + 6*a*B)*Sqrt[a /(b + Sqrt[b^2 - 4*a*c])]*(a + x*(b + c*x)) + I*(A*b + 6*a*B)*(b - Sqrt[b^ 2 - 4*a*c])*Sqrt[1 + (2*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*x^(3/2)*Sqrt[(4*a + 2*b*x - 2*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])] + I*(6*a*B*Sqrt[b^2 - 4*a*c] + A*(-b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c]))*Sqrt[1 + (2*a)/((b + Sqrt[b^2 - 4*a*c])*x) ]*x^(3/2)*Sqrt[(4*a + 2*b*x - 2*Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a *c]*x)]*EllipticF[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])])/(6*a*x^(3/2)*Sqrt[a + x*(b + c*x)])
Time = 0.50 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1229, 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.
| \(\displaystyle \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle -\frac {2 \int -\frac {a (3 b B+2 A c)+(A b+6 a B) c x}{2 \sqrt {x} \sqrt {c x^2+b x+a}}dx}{3 a}-\frac {2 \sqrt {a+b x+c x^2} (x (3 a B+A b)+a A)}{3 a x^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a (3 b B+2 A c)+(A b+6 a B) c x}{\sqrt {x} \sqrt {c x^2+b x+a}}dx}{3 a}-\frac {2 \sqrt {a+b x+c x^2} (x (3 a B+A b)+a A)}{3 a x^{3/2}}\) |
| \(\Big \downarrow \) 1240 |
| \(\displaystyle \frac {2 \int \frac {a (3 b B+2 A c)+(A b+6 a B) c x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{3 a}-\frac {2 \sqrt {a+b x+c x^2} (x (3 a B+A b)+a A)}{3 a x^{3/2}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {2 \left (\sqrt {a} \left (\sqrt {c} (6 a B+A b)+\sqrt {a} (2 A c+3 b B)\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}-\sqrt {a} \sqrt {c} (6 a B+A b) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+b x+a}}d\sqrt {x}\right )}{3 a}-\frac {2 \sqrt {a+b x+c x^2} (x (3 a B+A b)+a A)}{3 a x^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\sqrt {a} \left (\sqrt {c} (6 a B+A b)+\sqrt {a} (2 A c+3 b B)\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}-\sqrt {c} (6 a B+A b) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}\right )}{3 a}-\frac {2 \sqrt {a+b x+c x^2} (x (3 a B+A b)+a A)}{3 a x^{3/2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {2 \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\sqrt {c} (6 a B+A b)+\sqrt {a} (2 A c+3 b B)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x+c x^2}}-\sqrt {c} (6 a B+A b) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}\right )}{3 a}-\frac {2 \sqrt {a+b x+c x^2} (x (3 a B+A b)+a A)}{3 a x^{3/2}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2 \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\sqrt {c} (6 a B+A b)+\sqrt {a} (2 A c+3 b B)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x+c x^2}}-\sqrt {c} (6 a B+A b) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {\sqrt {x} \sqrt {a+b x+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )\right )}{3 a}-\frac {2 \sqrt {a+b x+c x^2} (x (3 a B+A b)+a A)}{3 a x^{3/2}}\) |
(-2*(a*A + (A*b + 3*a*B)*x)*Sqrt[a + b*x + c*x^2])/(3*a*x^(3/2)) + (2*(-(( A*b + 6*a*B)*Sqrt[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)/(Sqrt[a] + Sq rt[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]))) + (a^(1/4)*((A*b + 6*a*B) *Sqrt[c] + Sqrt[a]*(3*b*B + 2*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^(1/4)*Sqrt[a + b*x + c*x^2])))/(3* a)
3.11.32.3.1 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))*((a + b*x + c*x^2 )^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 )*(m + 2)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 )^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c *(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g }, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3, 0]
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. \(790\) vs. \(2(345)=690\).
Time = 1.13 (sec) , antiderivative size = 791, normalized size of antiderivative = 2.24
| method | result | size |
| elliptic | \(\frac {\sqrt {x \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {2 A \sqrt {c \,x^{3}+b \,x^{2}+a x}}{3 x^{2}}-\frac {2 \left (c \,x^{2}+b x +a \right ) \left (A b +3 B a \right )}{3 a \sqrt {x \left (c \,x^{2}+b x +a \right )}}+\frac {\left (\frac {2 A c}{3}+B b \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 (B c +\frac {c \left (A b +3 B a \right )}{3 a}\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}}\) | \(791\) |
| risch | \(\text {Expression too large to display}\) | \(1015\) |
| default | \(\text {Expression too large to display}\) | \(1687\) |
(x*(c*x^2+b*x+a))^(1/2)/x^(1/2)/(c*x^2+b*x+a)^(1/2)*(-2/3*A*(c*x^3+b*x^2+a *x)^(1/2)/x^2-2/3*(c*x^2+b*x+a)*(A*b+3*B*a)/a/(x*(c*x^2+b*x+a))^(1/2)+(2/3 *A*c+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))+(B*c+1/3*c*(A*b+3*B*a)/a)*(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))*Ell ipticF(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 = 0.10 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.56 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{5/2}} \, dx=-\frac {2 \, {\left (3 \, {\left (6 \, B a + A b\right )} c^{\frac {3}{2}} x^{2} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right )\right ) - {\left (3 \, B a b - A b^{2} + 6 \, A a c\right )} \sqrt {c} x^{2} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right ) + 3 \, {\left (A a c + {\left (3 \, B a + A b\right )} c x\right )} \sqrt {c x^{2} + b x + a} \sqrt {x}\right )}}{9 \, a c x^{2}} \]
-2/9*(3*(6*B*a + A*b)*c^(3/2)*x^2*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*B*a*b - A*b^2 + 6*A*a*c )*sqrt(c)*x^2*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*(A*a*c + (3*B*a + A*b)*c*x)*sqrt(c*x^2 + b*x + a)*sqrt(x))/(a*c*x^2)
\[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{5/2}} \, dx=\int \frac {\left (A + B x\right ) \sqrt {a + b x + c x^{2}}}{x^{\frac {5}{2}}}\, dx \]
\[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{5/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} {\left (B x + A\right )}}{x^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{5/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} {\left (B x + A\right )}}{x^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{x^{5/2}} \,d x \]