Integrand size = 34, antiderivative size = 712 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{5/2}} \, dx=-\frac {2 \left (e (b d-a e) (7 C d-3 B e)-c d \left (8 C d^2-e (4 B d-A e)\right )+e^2 \left (B c d+b C d-\frac {2 c C d^2}{e}-A c e-a C e\right ) x\right ) \sqrt {a+b x+c x^2}}{3 e^3 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{3 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 \left (4 c d-\frac {b e}{2}\right ) \left (B c d+b C d-\frac {2 c C d^2}{e}-A c e-a C e\right )+6 c (b d (C d-B e)+e (A c d-a C d+a B e))\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c e^3 \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (e (8 b C d-3 b B e-2 a C e)-2 c \left (8 C d^2-e (4 B d-A e)\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
-2/3*(C*d^2-e*(-A*e+B*d))*(c*x^2+b*x+a)^(3/2)/e/(a*e^2-b*d*e+c*d^2)/(e*x+d )^(3/2)-2/3*(e*(-a*e+b*d)*(-3*B*e+7*C*d)-c*d*(8*C*d^2-e*(-A*e+4*B*d))+e^2* (B*c*d+C*b*d-2*c*C*d^2/e-A*c*e-C*a*e)*x)*(c*x^2+b*x+a)^(1/2)/e^3/(a*e^2-b* d*e+c*d^2)/(e*x+d)^(1/2)+1/3*(2*(4*c*d-1/2*b*e)*(B*c*d+C*b*d-2*c*C*d^2/e-A *c*e-C*a*e)+6*c*(b*d*(-B*e+C*d)+e*(A*c*d+B*a*e-C*a*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/e^3/(a*e^2-b*d* e+c*d^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)-2/3*(e*(-3*B*b*e-2*C*a*e+8*C*b*d)-2*c*(8*C*d^2-e*(-A*e+4*B*d)))*Elli pticF(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^4/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)
Result contains complex when optimal does not.
Time = 35.08 (sec) , antiderivative size = 8456, normalized size of antiderivative = 11.88 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{5/2}} \, dx=\text {Result too large to show} \]
Time = 1.30 (sec) , antiderivative size = 722, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2181, 27, 1230, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle -\frac {2 \int -\frac {3 \left (b d (C d-B e)+e (A c d-a C d+a B e)-e \left (-\frac {2 c C d^2}{e}+B c d+b C d-A c e-a C e\right ) x\right ) \sqrt {c x^2+b x+a}}{2 e (d+e x)^{3/2}}dx}{3 \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{3 e (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (b d (C d-B e)+e (A c d-a C d+a B e)-e \left (-\frac {2 c C d^2}{e}+B c d+b C d-A c e-a C e\right ) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^{3/2}}dx}{e \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{3 e (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {-\frac {2 \int \frac {2 (2 b d-a e) \left (2 c C d^2-C e (b d-a e)-c e (B d-A e)\right )-3 b e (b d (C d-B e)+e (A c d-a C d+a B e))+\left ((8 c d-b e) \left (2 c C d^2-C e (b d-a e)-c e (B d-A e)\right )-6 c e (b d (C d-B e)+e (A c d-a C d+a B e))\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 e^2}-\frac {2 \sqrt {a+b x+c x^2} \left (e^2 x \left (-a C e-A c e+b C d+B c d-\frac {2 c C d^2}{e}\right )+e (b d-a e) (7 C d-3 B e)-c \left (8 C d^3-d e (4 B d-A e)\right )\right )}{3 e^2 \sqrt {d+e x}}}{e \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{3 e (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {2 (2 b d-a e) \left (2 c C d^2-C e (b d-a e)-c e (B d-A e)\right )-3 b e (b d (C d-B e)+e (A c d-a C d+a B e))+\left ((8 c d-b e) \left (2 c C d^2-C e (b d-a e)-c e (B d-A e)\right )-6 c e (b d (C d-B e)+e (A c d-a C d+a B e))\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 e^2}-\frac {2 \sqrt {a+b x+c x^2} \left (e^2 x \left (-a C e-A c e+b C d+B c d-\frac {2 c C d^2}{e}\right )+e (b d-a e) (7 C d-3 B e)-c \left (8 C d^3-d e (4 B d-A e)\right )\right )}{3 e^2 \sqrt {d+e x}}}{e \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{3 e (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {-\frac {\frac {\left (-c e \left (-2 a e (7 C d-3 B e)-b e (7 B d-A e)+16 b C d^2\right )+b C e^2 (b d-a e)+2 c^2 \left (8 C d^3-d e (4 B d-A e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}+\frac {\left (a e^2-b d e+c d^2\right ) \left (e (-2 a C e-3 b B e+8 b C d)-2 c \left (8 C d^2-e (4 B d-A e)\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}}{3 e^2}-\frac {2 \sqrt {a+b x+c x^2} \left (e^2 x \left (-a C e-A c e+b C d+B c d-\frac {2 c C d^2}{e}\right )+e (b d-a e) (7 C d-3 B e)-c \left (8 C d^3-d e (4 B d-A e)\right )\right )}{3 e^2 \sqrt {d+e x}}}{e \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{3 e (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {-\frac {2 \sqrt {c x^2+b x+a} \left (\left (-\frac {2 c C d^2}{e}+B c d+b C d-A c e-a C e\right ) x e^2+(b d-a e) (7 C d-3 B e) e-c \left (8 C d^3-d e (4 B d-A e)\right )\right )}{3 e^2 \sqrt {d+e x}}-\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (e (8 b C d-3 b B e-2 a C e)-2 c \left (8 C d^2-e (4 B d-A e)\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 \left (8 C d^3-d e (4 B d-A e)\right ) c^2-e \left (16 b C d^2-b e (7 B d-A e)-2 a e (7 C d-3 B e)\right ) c+b C e^2 (b d-a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}}{3 e^2}}{e \left (c d^2-b e d+a e^2\right )}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (c x^2+b x+a\right )^{3/2}}{3 e \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {-\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e \left (-2 a e (7 C d-3 B e)-b e (7 B d-A e)+16 b C d^2\right )+b C e^2 (b d-a e)+2 c^2 \left (8 C d^3-d e (4 B d-A e)\right )\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \left (e (-2 a C e-3 b B e+8 b C d)-2 c \left (8 C d^2-e (4 B d-A e)\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{3 e^2}-\frac {2 \sqrt {a+b x+c x^2} \left (e^2 x \left (-a C e-A c e+b C d+B c d-\frac {2 c C d^2}{e}\right )+e (b d-a e) (7 C d-3 B e)-c \left (8 C d^3-d e (4 B d-A e)\right )\right )}{3 e^2 \sqrt {d+e x}}}{e \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{3 e (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {-\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e \left (-2 a e (7 C d-3 B e)-b e (7 B d-A e)+16 b C d^2\right )+b C e^2 (b d-a e)+2 c^2 \left (8 C d^3-d e (4 B d-A e)\right )\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \left (e (-2 a C e-3 b B e+8 b C d)-2 c \left (8 C d^2-e (4 B d-A e)\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{3 e^2}-\frac {2 \sqrt {a+b x+c x^2} \left (e^2 x \left (-a C e-A c e+b C d+B c d-\frac {2 c C d^2}{e}\right )+e (b d-a e) (7 C d-3 B e)-c \left (8 C d^3-d e (4 B d-A e)\right )\right )}{3 e^2 \sqrt {d+e x}}}{e \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{3 e (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}\) |
(-2*(C*d^2 - e*(B*d - A*e))*(a + b*x + c*x^2)^(3/2))/(3*e*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(3/2)) + ((-2*(e*(b*d - a*e)*(7*C*d - 3*B*e) - c*(8*C*d^ 3 - d*e*(4*B*d - A*e)) + e^2*(B*c*d + b*C*d - (2*c*C*d^2)/e - A*c*e - a*C* e)*x)*Sqrt[a + b*x + c*x^2])/(3*e^2*Sqrt[d + e*x]) - ((Sqrt[2]*Sqrt[b^2 - 4*a*c]*(b*C*e^2*(b*d - a*e) + 2*c^2*(8*C*d^3 - d*e*(4*B*d - A*e)) - c*e*(1 6*b*C*d^2 - b*e*(7*B*d - A*e) - 2*a*e*(7*C*d - 3*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]) + (2*Sqrt[2]*Sqrt[b^2 - 4 *a*c]*(c*d^2 - b*d*e + a*e^2)*(e*(8*b*C*d - 3*b*B*e - 2*a*C*e) - 2*c*(8*C* d^2 - e*(4*B*d - A*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*e*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2]))/(3*e^2))/(e*(c*d^2 - b*d*e + a*e^2))
3.3.62.3.1 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_)/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[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ [m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ ), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi alRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*x^2) ^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Qx + c*d*R*(m + 1) - b*e*R*(m + p + 2) - c*e*R *(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1339\) vs. \(2(646)=1292\).
Time = 4.86 (sec) , antiderivative size = 1340, normalized size of antiderivative = 1.88
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1340\) |
| risch | \(\text {Expression too large to display}\) | \(2764\) |
| default | \(\text {Expression too large to display}\) | \(21038\) |
((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*(A*e ^2-B*d*e+C*d^2)/e^5*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)/(x+d/e )^2-2/3*(c*e*x^2+b*e*x+a*e)/e^4/(a*e^2-b*d*e+c*d^2)*(A*b*e^3-2*A*c*d*e^2+3 *B*a*e^3-4*B*b*d*e^2+5*B*c*d^2*e-6*C*a*d*e^2+7*C*b*d^2*e-8*C*c*d^3)/((x+d/ e)*(c*e*x^2+b*e*x+a*e))^(1/2)+2/3*C/e^3*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d *x+a*d)^(1/2)+2*((A*c*e^2+B*b*e^2-2*B*c*d*e+C*a*e^2-2*C*b*d*e+3*C*c*d^2)/e ^4-1/3*(A*e^2-B*d*e+C*d^2)/e^4*c-1/3/e^4*(b*e-c*d)*(A*b*e^3-2*A*c*d*e^2+3* B*a*e^3-4*B*b*d*e^2+5*B*c*d^2*e-6*C*a*d*e^2+7*C*b*d^2*e-8*C*c*d^3)/(a*e^2- b*d*e+c*d^2)+1/3*b/e^3/(a*e^2-b*d*e+c*d^2)*(A*b*e^3-2*A*c*d*e^2+3*B*a*e^3- 4*B*b*d*e^2+5*B*c*d^2*e-6*C*a*d*e^2+7*C*b*d^2*e-8*C*c*d^3)-2/3*C/e^3*(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/e^3*(B*c*e+C*b*e-2*C*c*d)+1/3*c/e^3*(A*b*e^3-2*A*c*d*e^2+3*B*a*e^3 -4*B*b*d*e^2+5*B*c*d^2*e-6*C*a*d*e^2+7*C*b*d^2*e-8*C*c*d^3)/(a*e^2-b*d*e+c *d^2)-2/3*C/e^3*(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)...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 1385, normalized size of antiderivative = 1.95 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{5/2}} \, dx=\text {Too large to display} \]
2/9*((16*C*c^3*d^6 - 8*(3*C*b*c^2 + B*c^3)*d^5*e + (6*C*b^2*c + 2*A*c^3 + (26*C*a + 11*B*b)*c^2)*d^4*e^2 + (C*b^3 - 2*(6*B*a + A*b)*c^2 - 2*(7*C*a*b + B*b^2)*c)*d^3*e^3 - (C*a*b^2 - 6*A*a*c^2 - (6*C*a^2 + 3*B*a*b - A*b^2)* c)*d^2*e^4 + (16*C*c^3*d^4*e^2 - 8*(3*C*b*c^2 + B*c^3)*d^3*e^3 + (6*C*b^2* c + 2*A*c^3 + (26*C*a + 11*B*b)*c^2)*d^2*e^4 + (C*b^3 - 2*(6*B*a + A*b)*c^ 2 - 2*(7*C*a*b + B*b^2)*c)*d*e^5 - (C*a*b^2 - 6*A*a*c^2 - (6*C*a^2 + 3*B*a *b - A*b^2)*c)*e^6)*x^2 + 2*(16*C*c^3*d^5*e - 8*(3*C*b*c^2 + B*c^3)*d^4*e^ 2 + (6*C*b^2*c + 2*A*c^3 + (26*C*a + 11*B*b)*c^2)*d^3*e^3 + (C*b^3 - 2*(6* B*a + A*b)*c^2 - 2*(7*C*a*b + B*b^2)*c)*d^2*e^4 - (C*a*b^2 - 6*A*a*c^2 - ( 6*C*a^2 + 3*B*a*b - A*b^2)*c)*d*e^5)*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)) + 3*(16*C*c^3*d^5*e - 8*(2*C*b*c^2 + B*c^ 3)*d^4*e^2 + (C*b^2*c + 2*A*c^3 + 7*(2*C*a + B*b)*c^2)*d^3*e^3 - (C*a*b*c + (6*B*a + A*b)*c^2)*d^2*e^4 + (16*C*c^3*d^3*e^3 - 8*(2*C*b*c^2 + B*c^3)*d ^2*e^4 + (C*b^2*c + 2*A*c^3 + 7*(2*C*a + B*b)*c^2)*d*e^5 - (C*a*b*c + (6*B *a + A*b)*c^2)*e^6)*x^2 + 2*(16*C*c^3*d^4*e^2 - 8*(2*C*b*c^2 + B*c^3)*d^3* e^3 + (C*b^2*c + 2*A*c^3 + 7*(2*C*a + B*b)*c^2)*d^2*e^4 - (C*a*b*c + (6*B* a + A*b)*c^2)*d*e^5)*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^...
\[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{5/2}} \, dx=\int \frac {\left (A + B x + C x^{2}\right ) \sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{5/2}} \, dx=\int \frac {\left (C\,x^2+B\,x+A\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^{5/2}} \,d x \]