Integrand size = 30, antiderivative size = 494 \[ \int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^{5/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {10 e \sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {5 \sqrt {2} e (2 c d-b e) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c \sqrt {b^2-4 a c} \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {20 \sqrt {2} e \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
-2/3*(e*x+d)^(5/2)/(c*x^2+b*x+a)^(3/2)-10/3*e*(b*d-2*a*e+(-b*e+2*c*d)*x)*( e*x+d)^(1/2)/(-4*a*c+b^2)/(c*x^2+b*x+a)^(1/2)+5/3*e*(-b*e+2*c*d)*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*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)*(e*x+ d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/c/(-4*a*c+b^2)^(1/2)/(c*x^2 +b*x+a)^(1/2)/(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)-20/3*e*(a *e^2-b*d*e+c*d^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*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)*(-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/(-4*a*c+b^2)^(1/2)/(e*x+d)^(1/2)/(c *x^2+b*x+a)^(1/2)
Result contains complex when optimal does not.
Time = 19.70 (sec) , antiderivative size = 1150, normalized size of antiderivative = 2.33 \[ \int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {\sqrt {d+e x} \left (a+b x+c x^2\right )^3 \left (-\frac {2 \left (c d^2-a e^2+2 c d e x-b e^2 x\right )}{3 c \left (a+b x+c x^2\right )^2}+\frac {2 \left (5 b c d e+b^2 e^2-14 a c e^2+10 c^2 d e x-5 b c e^2 x\right )}{3 c \left (-b^2+4 a c\right ) \left (a+b x+c x^2\right )}\right )}{(a+x (b+c x))^{5/2}}+\frac {5 (d+e x)^{3/2} \left (a+b x+c x^2\right )^{5/2} \left (-4 (-2 c d+b e) \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}}} \left (c \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {a e}{d+e x}\right )}{d+e x}\right )+\frac {i \sqrt {2} (-2 c d+b e) \left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}-\frac {2 a e^2}{d+e x}-2 c d \left (-1+\frac {d}{d+e x}\right )+b e \left (-1+\frac {2 d}{d+e x}\right )}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}+\frac {2 a e^2}{d+e x}+2 c d \left (-1+\frac {d}{d+e x}\right )+b \left (e-\frac {2 d e}{d+e x}\right )}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} E\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 {d+e x}}-\frac {i \sqrt {2} \left (-b^2 e^2+4 a c e^2-2 c d \sqrt {\left (b^2-4 a c\right ) e^2}+b e \sqrt {\left (b^2-4 a c\right ) e^2}\right ) \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}-\frac {2 a e^2}{d+e x}-2 c d \left (-1+\frac {d}{d+e x}\right )+b e \left (-1+\frac {2 d}{d+e x}\right )}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}+\frac {2 a e^2}{d+e x}+2 c d \left (-1+\frac {d}{d+e x}\right )+b \left (e-\frac {2 d e}{d+e x}\right )}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \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 {d+e x}}\right )}{6 c \left (b^2-4 a c\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}}} (a+x (b+c x))^{5/2} \sqrt {\frac {(d+e x)^2 \left (c \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {a e}{d+e x}\right )}{d+e x}\right )}{e^2}}} \]
(Sqrt[d + e*x]*(a + b*x + c*x^2)^3*((-2*(c*d^2 - a*e^2 + 2*c*d*e*x - b*e^2 *x))/(3*c*(a + b*x + c*x^2)^2) + (2*(5*b*c*d*e + b^2*e^2 - 14*a*c*e^2 + 10 *c^2*d*e*x - 5*b*c*e^2*x))/(3*c*(-b^2 + 4*a*c)*(a + b*x + c*x^2))))/(a + x *(b + c*x))^(5/2) + (5*(d + e*x)^(3/2)*(a + b*x + c*x^2)^(5/2)*(-4*(-2*c*d + b*e)*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c) *e^2])]*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x )))/(d + e*x)) + (I*Sqrt[2]*(-2*c*d + b*e)*(2*c*d - b*e + Sqrt[(b^2 - 4*a* c)*e^2])*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a *c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/(d + e*x) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*c*d + b*e + Sqrt[(b^2 - 4 *a*c)*e^2])]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2 *c*d + b*e + Sqrt[(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[d + e*x] - (I*Sqrt[2]*(-(b^2*e^2) + 4*a*c*e^2 - 2*c*d*Sqrt[(b^2 - 4*a*c)*e^ 2] + b*e*Sqrt[(b^2 - 4*a*c)*e^2])*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2 )/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c* d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e ^2)/(d + e*x) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2* c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt...
Time = 0.67 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {1222, 1164, 27, 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.
| \(\displaystyle \int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1222 |
| \(\displaystyle \frac {5}{3} e \int \frac {(d+e x)^{3/2}}{\left (c x^2+b x+a\right )^{3/2}}dx-\frac {2 (d+e x)^{5/2}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1164 |
| \(\displaystyle \frac {5}{3} e \left (-\frac {2 \int -\frac {e (b d-2 a e+(2 c d-b e) x)}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{b^2-4 a c}-\frac {2 \sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )-\frac {2 (d+e x)^{5/2}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{3} e \left (\frac {e \int \frac {b d-2 a e+(2 c d-b e) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{b^2-4 a c}-\frac {2 \sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )-\frac {2 (d+e x)^{5/2}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {5}{3} e \left (\frac {e \left (\frac {(2 c d-b e) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {2 \left (a e^2-b d e+c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}\right )}{b^2-4 a c}-\frac {2 \sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )-\frac {2 (d+e x)^{5/2}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {5}{3} e \left (\frac {e \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}\right )}{b^2-4 a c}-\frac {2 \sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )-\frac {2 (d+e x)^{5/2}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {5}{3} e \left (\frac {e \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}\right )}{b^2-4 a c}-\frac {2 \sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )-\frac {2 (d+e x)^{5/2}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {5}{3} e \left (\frac {e \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}\right )}{b^2-4 a c}-\frac {2 \sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )-\frac {2 (d+e x)^{5/2}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
(-2*(d + e*x)^(5/2))/(3*(a + b*x + c*x^2)^(3/2)) + (5*e*((-2*Sqrt[d + e*x] *(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) + (e*((Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*Sqrt[d + e*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]*e)/(2*c*d - ( b + Sqrt[b^2 - 4*a*c])*e)])/(c*e*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^ 2 - b*d*e + a*e^2)*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 + S qrt[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*e*Sqrt[d + e*x]*Sqrt[a + b* x + c*x^2])))/(b^2 - 4*a*c)))/3
3.17.50.3.1 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[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_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* c)) Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int QuadraticQ[a, b, c, d, e, m, p, x]
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1066\) vs. \(2(430)=860\).
Time = 3.81 (sec) , antiderivative size = 1067, normalized size of antiderivative = 2.16
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1067\) |
| default | \(\text {Expression too large to display}\) | \(5517\) |
((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*((2/3*(b*e -2*c*d)/c^3*e*x+2/3*(a*e^2-c*d^2)/c^3)*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d* x+a*d)^(1/2)/(a/c+b/c*x+x^2)^2-2*(c*e*x+c*d)*(5/3*(b*e-2*c*d)/c*e/(4*a*c-b ^2)*x+1/3*(14*a*c*e-b^2*e-5*b*c*d)*e/c^2/(4*a*c-b^2))/((a/c+b/c*x+x^2)*(c* e*x+c*d))^(1/2)+2*(2*e^3/c-1/3*(28*a*c*e^2-7*b^2*e^2+10*b*c*d*e-20*c^2*d^2 )*e/c/(4*a*c-b^2)+1/3/c*e^2*(14*a*c*e-b^2*e-5*b*c*d)/(4*a*c-b^2)+10/3*d*(b *e-2*c*d)*e/(4*a*c-b^2))*(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))+10/3*e^2*(b*e-2*c*d)/(4*a*c-b^2)*(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)*((-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/ 2)))*EllipticE(((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))+ 1/2/c*(-b+(-4*a*c+b^2)^(1/2))*EllipticF(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 959, normalized size of antiderivative = 1.94 \[ \int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (5 \, {\left (2 \, a^{2} c^{2} d^{2} - 2 \, a^{2} b c d e + {\left (2 \, c^{4} d^{2} - 2 \, b c^{3} d e - {\left (b^{2} c^{2} - 6 \, a c^{3}\right )} e^{2}\right )} x^{4} + 2 \, {\left (2 \, b c^{3} d^{2} - 2 \, b^{2} c^{2} d e - {\left (b^{3} c - 6 \, a b c^{2}\right )} e^{2}\right )} x^{3} - {\left (a^{2} b^{2} - 6 \, a^{3} c\right )} e^{2} + {\left (2 \, {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} - 2 \, {\left (b^{3} c + 2 \, a b c^{2}\right )} d e - {\left (b^{4} - 4 \, a b^{2} c - 12 \, a^{2} c^{2}\right )} e^{2}\right )} x^{2} + 2 \, {\left (2 \, a b c^{2} d^{2} - 2 \, a b^{2} c d e - {\left (a b^{3} - 6 \, a^{2} b c\right )} e^{2}\right )} x\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 ) + 15 \, {\left (2 \, a^{2} c^{2} d e - a^{2} b c e^{2} + {\left (2 \, c^{4} d e - b c^{3} e^{2}\right )} x^{4} + 2 \, {\left (2 \, b c^{3} d e - b^{2} c^{2} e^{2}\right )} x^{3} + {\left (2 \, {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d e - {\left (b^{3} c + 2 \, a b c^{2}\right )} e^{2}\right )} x^{2} + 2 \, {\left (2 \, a b c^{2} d e - a b^{2} c e^{2}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\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}}, {\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 )\right ) + 3 \, {\left (5 \, a b c^{2} d e - 10 \, a^{2} c^{2} e^{2} + 5 \, {\left (2 \, c^{4} d e - b c^{3} e^{2}\right )} x^{3} + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} + {\left (15 \, b c^{3} d e - 2 \, {\left (2 \, b^{2} c^{2} + 7 \, a c^{3}\right )} e^{2}\right )} x^{2} - {\left (15 \, a b c^{2} e^{2} - {\left (7 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}\right )}}{9 \, {\left (a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3} + {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 2 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + {\left (b^{4} c^{2} - 2 \, a b^{2} c^{3} - 8 \, a^{2} c^{4}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} - 4 \, a^{2} b c^{3}\right )} x\right )}} \]
-2/9*(5*(2*a^2*c^2*d^2 - 2*a^2*b*c*d*e + (2*c^4*d^2 - 2*b*c^3*d*e - (b^2*c ^2 - 6*a*c^3)*e^2)*x^4 + 2*(2*b*c^3*d^2 - 2*b^2*c^2*d*e - (b^3*c - 6*a*b*c ^2)*e^2)*x^3 - (a^2*b^2 - 6*a^3*c)*e^2 + (2*(b^2*c^2 + 2*a*c^3)*d^2 - 2*(b ^3*c + 2*a*b*c^2)*d*e - (b^4 - 4*a*b^2*c - 12*a^2*c^2)*e^2)*x^2 + 2*(2*a*b *c^2*d^2 - 2*a*b^2*c*d*e - (a*b^3 - 6*a^2*b*c)*e^2)*x)*sqrt(c*e)*weierstra ssPInverse(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)) + 15*(2*a^2*c^2*d*e - a^2* b*c*e^2 + (2*c^4*d*e - b*c^3*e^2)*x^4 + 2*(2*b*c^3*d*e - b^2*c^2*e^2)*x^3 + (2*(b^2*c^2 + 2*a*c^3)*d*e - (b^3*c + 2*a*b*c^2)*e^2)*x^2 + 2*(2*a*b*c^2 *d*e - a*b^2*c*e^2)*x)*sqrt(c*e)*weierstrassZeta(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), 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))) + 3*(5*a*b*c^2*d*e - 10*a^2*c^2*e^2 + 5*(2*c^4*d*e - b*c^3*e^2)*x^3 + (b^2*c^2 - 4*a*c^3)*d^2 + (15*b*c^3*d*e - 2*(2*b^2*c^2 + 7*a*c^3)*e^2)*x^2 - (15*a*b*c^2*e^2 - (7*b^2*c^2 + 2*a*c^3 )*d*e)*x)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d))/(a^2*b^2*c^2 - 4*a^3*c^3 + (b^2*c^4 - 4*a*c^5)*x^4 + 2*(b^3*c^3 - 4*a*b*c^4)*x^3 + (b^4*c^2 - 2*a*...
Timed out. \[ \int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (2 \, c x + b\right )} {\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (2 \, c x + b\right )} {\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {\left (b+2\,c\,x\right )\,{\left (d+e\,x\right )}^{5/2}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]