Integrand size = 30, antiderivative size = 540 \[ \int \frac {(b+2 c x) (d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (d+e x)^{7/2}}{\sqrt {a+b x+c x^2}}+\frac {56 e^2 (2 c d-b e) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{15 c^2}+\frac {14 e^2 (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}+\frac {7 \sqrt {2} \sqrt {b^2-4 a c} e \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 c^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {56 \sqrt {2} \sqrt {b^2-4 a c} e (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 c^3 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
-2*(e*x+d)^(7/2)/(c*x^2+b*x+a)^(1/2)+14/5*e^2*(e*x+d)^(3/2)*(c*x^2+b*x+a)^ (1/2)/c+56/15*e^2*(-b*e+2*c*d)*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^2+7/15* e*(23*c^2*d^2+8*b^2*e^2-c*e*(9*a*e+23*b*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)*(-4*a*c+b^2)^(1/2)*(e*x+d) ^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/c^3/(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)-56/15*e*(-b*e+2*c*d)*(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)*(-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^3/(e*x+d)^(1/2)/(c*x^ 2+b*x+a)^(1/2)
Result contains complex when optimal does not.
Time = 18.19 (sec) , antiderivative size = 780, normalized size of antiderivative = 1.44 \[ \int \frac {(b+2 c x) (d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {d+e x} \left (c \left (-28 b e^3 (a+b x)+c^2 \left (-15 d^3-45 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )+7 c e^2 (b x (11 d-e x)+a (11 d+3 e x))\right )+7 (d+e x) \left (\frac {e^2 \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) (a+x (b+c x))}{(d+e x)^2}-\frac {i \sqrt {1-\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {1+\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \left (\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )+\left (-30 c^3 d^3+8 b^2 e^2 \left (b e-\sqrt {\left (b^2-4 a c\right ) e^2}\right )-c^2 d \left (-45 b d e-34 a e^2+23 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+c e \left (-31 b^2 d e-17 a b e^2+23 b d \sqrt {\left (b^2-4 a c\right ) e^2}+9 a e \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{2 \sqrt {2} \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {d+e x}}\right )\right )}{15 c^3 \sqrt {a+x (b+c x)}} \]
(2*Sqrt[d + e*x]*(c*(-28*b*e^3*(a + b*x) + c^2*(-15*d^3 - 45*d^2*e*x + 32* d*e^2*x^2 + 6*e^3*x^3) + 7*c*e^2*(b*x*(11*d - e*x) + a*(11*d + 3*e*x))) + 7*(d + e*x)*((e^2*(23*c^2*d^2 + 8*b^2*e^2 - c*e*(23*b*d + 9*a*e))*(a + x*( b + c*x)))/(d + e*x)^2 - ((I/2)*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))]*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))]*((2 *c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(23*c^2*d^2 + 8*b^2*e^2 - c*e*(23*b* d + 9*a*e))*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 + S qrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] + (-30*c ^3*d^3 + 8*b^2*e^2*(b*e - Sqrt[(b^2 - 4*a*c)*e^2]) - c^2*d*(-45*b*d*e - 34 *a*e^2 + 23*d*Sqrt[(b^2 - 4*a*c)*e^2]) + c*e*(-31*b^2*d*e - 17*a*b*e^2 + 2 3*b*d*Sqrt[(b^2 - 4*a*c)*e^2] + 9*a*e*Sqrt[(b^2 - 4*a*c)*e^2]))*EllipticF[ 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[2]*Sqrt[(c*d^2 + e*(-( b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[d + e*x]))))/( 15*c^3*Sqrt[a + x*(b + c*x)])
Time = 0.89 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1222, 1166, 27, 1236, 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.
| \(\displaystyle \int \frac {(b+2 c x) (d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1222 |
| \(\displaystyle 7 e \int \frac {(d+e x)^{5/2}}{\sqrt {c x^2+b x+a}}dx-\frac {2 (d+e x)^{7/2}}{\sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1166 |
| \(\displaystyle 7 e \left (\frac {2 \int \frac {\sqrt {d+e x} \left (5 c d^2-e (b d+3 a e)+4 e (2 c d-b e) x\right )}{2 \sqrt {c x^2+b x+a}}dx}{5 c}+\frac {2 e (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}\right )-\frac {2 (d+e x)^{7/2}}{\sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 7 e \left (\frac {\int \frac {\sqrt {d+e x} \left (5 c d^2-e (b d+3 a e)+4 e (2 c d-b e) x\right )}{\sqrt {c x^2+b x+a}}dx}{5 c}+\frac {2 e (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}\right )-\frac {2 (d+e x)^{7/2}}{\sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle 7 e \left (\frac {\frac {2 \int \frac {15 c^2 d^3-c e (11 b d+17 a e) d+4 b e^2 (b d+a e)+e \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 c}+\frac {8 e \sqrt {d+e x} \sqrt {a+b x+c x^2} (2 c d-b e)}{3 c}}{5 c}+\frac {2 e (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}\right )-\frac {2 (d+e x)^{7/2}}{\sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 7 e \left (\frac {\frac {\int \frac {15 c^2 d^3-c e (11 b d+17 a e) d+4 b e^2 (b d+a e)+e \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 c}+\frac {8 e \sqrt {d+e x} \sqrt {a+b x+c x^2} (2 c d-b e)}{3 c}}{5 c}+\frac {2 e (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}\right )-\frac {2 (d+e x)^{7/2}}{\sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle 7 e \left (\frac {\frac {\left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx-4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 c}+\frac {8 e \sqrt {d+e x} \sqrt {a+b x+c x^2} (2 c d-b e)}{3 c}}{5 c}+\frac {2 e (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}\right )-\frac {2 (d+e x)^{7/2}}{\sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle 7 e \left (\frac {\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {8 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{3 c}+\frac {8 e \sqrt {d+e x} \sqrt {a+b x+c x^2} (2 c d-b e)}{3 c}}{5 c}+\frac {2 e (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}\right )-\frac {2 (d+e x)^{7/2}}{\sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle 7 e \left (\frac {\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {8 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{3 c}+\frac {8 e \sqrt {d+e x} \sqrt {a+b x+c x^2} (2 c d-b e)}{3 c}}{5 c}+\frac {2 e (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}\right )-\frac {2 (d+e x)^{7/2}}{\sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle 7 e \left (\frac {\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {8 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{3 c}+\frac {8 e \sqrt {d+e x} \sqrt {a+b x+c x^2} (2 c d-b e)}{3 c}}{5 c}+\frac {2 e (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}\right )-\frac {2 (d+e x)^{7/2}}{\sqrt {a+b x+c x^2}}\) |
(-2*(d + e*x)^(7/2))/Sqrt[a + b*x + c*x^2] + 7*e*((2*e*(d + e*x)^(3/2)*Sqr t[a + b*x + c*x^2])/(5*c) + ((8*e*(2*c*d - b*e)*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])/(3*c) + ((Sqrt[2]*Sqrt[b^2 - 4*a*c]*(23*c^2*d^2 + 8*b^2*e^2 - c *e*(23*b*d + 9*a*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*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b* x + c*x^2]) - (8*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(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 + 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*c))/(5*c))
3.17.43.3.1 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[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[1/(c*(m + 2*p + 1)) Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* (a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat icQ[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*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1 )*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 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. \(1345\) vs. \(2(472)=944\).
Time = 8.10 (sec) , antiderivative size = 1346, normalized size of antiderivative = 2.49
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1346\) |
| risch | \(\text {Expression too large to display}\) | \(1833\) |
| default | \(\text {Expression too large to display}\) | \(4802\) |
((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)*(-(a*c*e^2-b^2*e^2+3*b*c*d*e-3*c^2*d^2)/c^3*e*x+(a*b*e^3-3*a*c*d*e^2 +c^2*d^3)/c^3)/((a/c+b/c*x+x^2)*(c*e*x+c*d))^(1/2)+4/5/c*e^3*x*(c*e*x^3+b* e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2/3*(-1/c*e^3*(b*e-8*c*d)-4/5/c*e^3*( 2*b*e+2*c*d))/c/e*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2*(e*(3* a*b*c*e^3-8*a*c^2*d*e^2-b^3*e^3+4*b^2*c*d*e^2-6*b*c^2*d^2*e+8*c^3*d^3)/c^3 -(3*a*b*c*e^3-8*a*c^2*d*e^2-b^3*e^3+5*b^2*c*d*e^2-9*b*c^2*d^2*e+8*c^3*d^3) *e/c^3+1/c^2*e*(a*b*e^3-3*a*c*d*e^2+c^2*d^3)-2/c^2*d*(a*c*e^2-b^2*e^2+3*b* c*d*e-3*c^2*d^2)*e-4/5/c*e^3*a*d-2/3*(-1/c*e^3*(b*e-8*c*d)-4/5/c*e^3*(2*b* e+2*c*d))/c/e*(1/2*a*e+1/2*b*d))*(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))+2*(-1/c^2*e^2*(2*a*c*e^2-b^2*e^2+4*b*c*d*e-12*c^2*d^ 2)-(a*c*e^2-b^2*e^2+3*b*c*d*e-3*c^2*d^2)*e^2/c^2-4/5/c*e^3*(3/2*a*e+3/2*b* d)-2/3*(-1/c*e^3*(b*e-8*c*d)-4/5/c*e^3*(2*b*e+2*c*d))/c/e*(b*e+c*d))*(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)^...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.29 (sec) , antiderivative size = 805, normalized size of antiderivative = 1.49 \[ \int \frac {(b+2 c x) (d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (7 \, {\left (22 \, a c^{3} d^{3} - 33 \, a b c^{2} d^{2} e + 3 \, {\left (9 \, a b^{2} c - 14 \, a^{2} c^{2}\right )} d e^{2} - {\left (8 \, a b^{3} - 21 \, a^{2} b c\right )} e^{3} + {\left (22 \, c^{4} d^{3} - 33 \, b c^{3} d^{2} e + 3 \, {\left (9 \, b^{2} c^{2} - 14 \, a c^{3}\right )} d e^{2} - {\left (8 \, b^{3} c - 21 \, a b c^{2}\right )} e^{3}\right )} x^{2} + {\left (22 \, b c^{3} d^{3} - 33 \, b^{2} c^{2} d^{2} e + 3 \, {\left (9 \, b^{3} c - 14 \, a b c^{2}\right )} d e^{2} - {\left (8 \, b^{4} - 21 \, a b^{2} c\right )} e^{3}\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 ) - 21 \, {\left (23 \, a c^{3} d^{2} e - 23 \, a b c^{2} d e^{2} + {\left (8 \, a b^{2} c - 9 \, a^{2} c^{2}\right )} e^{3} + {\left (23 \, c^{4} d^{2} e - 23 \, b c^{3} d e^{2} + {\left (8 \, b^{2} c^{2} - 9 \, a c^{3}\right )} e^{3}\right )} x^{2} + {\left (23 \, b c^{3} d^{2} e - 23 \, b^{2} c^{2} d e^{2} + {\left (8 \, b^{3} c - 9 \, a b c^{2}\right )} e^{3}\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 (6 \, c^{4} e^{3} x^{3} - 15 \, c^{4} d^{3} + 77 \, a c^{3} d e^{2} - 28 \, a b c^{2} e^{3} + {\left (32 \, c^{4} d e^{2} - 7 \, b c^{3} e^{3}\right )} x^{2} - {\left (45 \, c^{4} d^{2} e - 77 \, b c^{3} d e^{2} + 7 \, {\left (4 \, b^{2} c^{2} - 3 \, a c^{3}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}\right )}}{45 \, {\left (c^{5} x^{2} + b c^{4} x + a c^{4}\right )}} \]
2/45*(7*(22*a*c^3*d^3 - 33*a*b*c^2*d^2*e + 3*(9*a*b^2*c - 14*a^2*c^2)*d*e^ 2 - (8*a*b^3 - 21*a^2*b*c)*e^3 + (22*c^4*d^3 - 33*b*c^3*d^2*e + 3*(9*b^2*c ^2 - 14*a*c^3)*d*e^2 - (8*b^3*c - 21*a*b*c^2)*e^3)*x^2 + (22*b*c^3*d^3 - 3 3*b^2*c^2*d^2*e + 3*(9*b^3*c - 14*a*b*c^2)*d*e^2 - (8*b^4 - 21*a*b^2*c)*e^ 3)*x)*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)) - 21*(23*a*c^3*d^2*e - 23*a*b*c^2*d*e^2 + (8*a*b^2*c - 9*a^2*c^2)*e^3 + (23 *c^4*d^2*e - 23*b*c^3*d*e^2 + (8*b^2*c^2 - 9*a*c^3)*e^3)*x^2 + (23*b*c^3*d ^2*e - 23*b^2*c^2*d*e^2 + (8*b^3*c - 9*a*b*c^2)*e^3)*x)*sqrt(c*e)*weierstr assZeta(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* (6*c^4*e^3*x^3 - 15*c^4*d^3 + 77*a*c^3*d*e^2 - 28*a*b*c^2*e^3 + (32*c^4*d* e^2 - 7*b*c^3*e^3)*x^2 - (45*c^4*d^2*e - 77*b*c^3*d*e^2 + 7*(4*b^2*c^2 - 3 *a*c^3)*e^3)*x)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d))/(c^5*x^2 + b*c^4*x + a*c^4)
Timed out. \[ \int \frac {(b+2 c x) (d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(b+2 c x) (d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (2 \, c x + b\right )} {\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(b+2 c x) (d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (2 \, c x + b\right )} {\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(b+2 c x) (d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (b+2\,c\,x\right )\,{\left (d+e\,x\right )}^{7/2}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]