Integrand size = 30, antiderivative size = 216 \[ \int \frac {(b+2 c x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x}}{\sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
-2*(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)+2*e*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*e*(-4*a*c+b^2)^(1/2)/(2*c *d-e*(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*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^( 1/2)/c/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)
Result contains complex when optimal does not.
Time = 12.67 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.47 \[ \int \frac {(b+2 c x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {-2 \sqrt {d+e x}+\frac {i (d+e x) \sqrt {2-\frac {4 \left (c d^2+e (-b d+a e)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {1+\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )}{\sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}}{\sqrt {a+x (b+c x)}} \]
(-2*Sqrt[d + e*x] + (I*(d + e*x)*Sqrt[2 - (4*(c*d^2 + e*(-(b*d) + a*e)))/( (2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[1 + (2*(c*d^2 + e *(-(b*d) + a*e)))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*El lipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sq rt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a *c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[(c*d^2 + e*(-(b* d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[a + x*(b + c*x) ]
Time = 0.33 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1222, 1172, 321}
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.
| \(\displaystyle \int \frac {(b+2 c x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1222 |
| \(\displaystyle e \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx-\frac {2 \sqrt {d+e x}}{\sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {2 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {d+e x}}{\sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {d+e x}}{\sqrt {a+b x+c x^2}}\) |
(-2*Sqrt[d + e*x])/Sqrt[a + b*x + c*x^2] + (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*e* Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*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]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(c*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])
3.17.46.3.1 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[((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*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] - Simp[e*g*(m/(2*c*(p + 1))) Int[(d + e*x)^(m - 1)* (a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ [2*c*f - b*g, 0] && LtQ[p, -1] && GtQ[m, 0]
Time = 1.12 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.68
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {2 \left (x c e +c d \right )}{c \sqrt {\left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right ) \left (x c e +c d \right )}}+\frac {2 e \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{\sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}\) | \(363\) |
| default | \(\frac {\left (-\sqrt {-4 a c +b^{2}}\, \sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{e \sqrt {-4 a c +b^{2}}+b e -2 c d}}\, \sqrt {\frac {\left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right ) e}{e \sqrt {-4 a c +b^{2}}-b e +2 c d}}\, \sqrt {\frac {\left (b +2 c x +\sqrt {-4 a c +b^{2}}\right ) e}{e \sqrt {-4 a c +b^{2}}+b e -2 c d}}\, F\left (\sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{e \sqrt {-4 a c +b^{2}}+b e -2 c d}}, \sqrt {-\frac {e \sqrt {-4 a c +b^{2}}+b e -2 c d}{e \sqrt {-4 a c +b^{2}}-b e +2 c d}}\right ) e -\sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{e \sqrt {-4 a c +b^{2}}+b e -2 c d}}\, \sqrt {\frac {\left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right ) e}{e \sqrt {-4 a c +b^{2}}-b e +2 c d}}\, \sqrt {\frac {\left (b +2 c x +\sqrt {-4 a c +b^{2}}\right ) e}{e \sqrt {-4 a c +b^{2}}+b e -2 c d}}\, F\left (\sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{e \sqrt {-4 a c +b^{2}}+b e -2 c d}}, \sqrt {-\frac {e \sqrt {-4 a c +b^{2}}+b e -2 c d}{e \sqrt {-4 a c +b^{2}}-b e +2 c d}}\right ) b e +2 \sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{e \sqrt {-4 a c +b^{2}}+b e -2 c d}}\, \sqrt {\frac {\left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right ) e}{e \sqrt {-4 a c +b^{2}}-b e +2 c d}}\, \sqrt {\frac {\left (b +2 c x +\sqrt {-4 a c +b^{2}}\right ) e}{e \sqrt {-4 a c +b^{2}}+b e -2 c d}}\, F\left (\sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{e \sqrt {-4 a c +b^{2}}+b e -2 c d}}, \sqrt {-\frac {e \sqrt {-4 a c +b^{2}}+b e -2 c d}{e \sqrt {-4 a c +b^{2}}-b e +2 c d}}\right ) c d -2 x c e -2 c d \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}{\left (c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d \right ) c}\) | \(705\) |
((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*(-2*(c*e*x +c*d)/c/((a/c+b/c*x+x^2)*(c*e*x+c*d))^(1/2)+2*e*(d/e-1/2*(b+(-4*a*c+b^2)^( 1/2))/c)*((x+d/e)/(d/e-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)))/(-d/e-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)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*e*x ^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)*EllipticF(((x+d/e)/(d/e-1/2*(b+( -4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-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 = 173, normalized size of antiderivative = 0.80 \[ \int \frac {(b+2 c x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \, {\left ({\left (c x^{2} + b x + a\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) - \sqrt {c x^{2} + b x + a} \sqrt {e x + d} c\right )}}{c^{2} x^{2} + b c x + a c} \]
2*((c*x^2 + b*x + a)*sqrt(c*e)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2* c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e)) - sqrt(c*x^2 + b*x + a)*sqrt(e*x + d)*c)/(c^2*x^2 + b*c*x + a*c)
\[ \int \frac {(b+2 c x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (b + 2 c x\right ) \sqrt {d + e x}}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(b+2 c x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (2 \, c x + b\right )} \sqrt {e x + d}}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(b+2 c x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (2 \, c x + b\right )} \sqrt {e x + d}}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(b+2 c x) \sqrt {d+e x}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (b+2\,c\,x\right )\,\sqrt {d+e\,x}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]