Integrand size = 24, antiderivative size = 515 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} (8 c d-7 b e-6 c e x) \sqrt {a+b x+c x^2}}{5 e^3}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{5 c e^4 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {16 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{5 c e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
-2*(c*x^2+b*x+a)^(3/2)/e/(e*x+d)^(1/2)-2/5*(-6*c*e*x-7*b*e+8*c*d)*(e*x+d)^ (1/2)*(c*x^2+b*x+a)^(1/2)/e^3+1/5*(16*c^2*d^2+b^2*e^2-4*c*e*(-3*a*e+4*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/e^4/(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)-16/5*(-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/e^4/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)
Result contains complex when optimal does not.
Time = 32.00 (sec) , antiderivative size = 1088, normalized size of antiderivative = 2.11 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {\sqrt {d+e x} \left (\frac {4 (a+x (b+c x)) \left (e (7 b d-5 a e+2 b e x)+c \left (-8 d^2-2 d e x+e^2 x^2\right )\right )}{e^3 (d+e x)}+\frac {(d+e x) \left (\frac {4 e^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}}} \left (16 c^2 d^2+b^2 e^2+4 c e (-4 b d+3 a e)\right ) (a+x (b+c x))}{(d+e x)^2}-\frac {i \sqrt {2} \left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (16 c^2 d^2+b^2 e^2+4 c e (-4 b d+3 a e)\right ) \sqrt {\frac {-2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}+2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (d-e x)}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}-2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (-d+e x)}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} 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^3 e^3+b^2 e^2 \left (2 c d+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+4 b \left (a c e^3-4 c d e \sqrt {\left (b^2-4 a c\right ) e^2}\right )+4 c \left (4 c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}+a e^2 \left (-2 c d+3 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )\right ) \sqrt {\frac {-2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}+2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (d-e x)}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}-2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (-d+e x)}{\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 {d+e x}}\right )}{c e^5 \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}}}}\right )}{10 \sqrt {a+x (b+c x)}} \]
(Sqrt[d + e*x]*((4*(a + x*(b + c*x))*(e*(7*b*d - 5*a*e + 2*b*e*x) + c*(-8* d^2 - 2*d*e*x + e^2*x^2)))/(e^3*(d + e*x)) + ((d + e*x)*((4*e^2*Sqrt[(c*d^ 2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(16*c^2*d^ 2 + b^2*e^2 + 4*c*e*(-4*b*d + 3*a*e))*(a + x*(b + c*x)))/(d + e*x)^2 - (I* Sqrt[2]*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(16*c^2*d^2 + b^2*e^2 + 4* c*e*(-4*b*d + 3*a*e))*Sqrt[(-2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] + 2*c*d*e *x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(d - e*x))/((2*c*d - b*e + Sqrt[(b^ 2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] - 2 *c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(-d + e*x))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*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^3*e^3) + b^2*e^2*(2*c*d + Sqrt[(b^2 - 4*a*c)*e^2]) + 4*b*(a*c*e^3 - 4*c*d*e*Sqrt[(b^2 - 4*a*c)*e^2 ]) + 4*c*(4*c*d^2*Sqrt[(b^2 - 4*a*c)*e^2] + a*e^2*(-2*c*d + 3*Sqrt[(b^2 - 4*a*c)*e^2])))*Sqrt[(-2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] + 2*c*d*e*x + e* Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(d - e*x))/((2*c*d - b*e + Sqrt[(b^2 - 4*a *c)*e^2])*(d + e*x))]*Sqrt[(2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] - 2*c*d*e* x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(-d + e*x))/((-2*c*d + b*e + Sqrt[(b ^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 ...
Time = 0.72 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1161, 1231, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1161 |
| \(\displaystyle \frac {3 \int \frac {(b+2 c x) \sqrt {c x^2+b x+a}}{\sqrt {d+e x}}dx}{e}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {3 \left (-\frac {2 \int \frac {c \left (7 d e b^2-8 \left (c d^2+a e^2\right ) b+4 a c d e-\left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{15 c e^2}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} (-7 b e+8 c d-6 c e x)}{15 e^2}\right )}{e}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \left (-\frac {\int \frac {7 d e b^2-8 \left (c d^2+a e^2\right ) b+4 a c d e-\left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{15 e^2}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} (-7 b e+8 c d-6 c e x)}{15 e^2}\right )}{e}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {8 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}}{15 e^2}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} (-7 b e+8 c d-6 c e x)}{15 e^2}\right )}{e}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {16 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}}{15 e^2}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} (-7 b e+8 c d-6 c e x)}{15 e^2}\right )}{e}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {16 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}}{15 e^2}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} (-7 b e+8 c d-6 c e x)}{15 e^2}\right )}{e}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {3 \left (-\frac {\frac {16 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}}{15 e^2}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} (-7 b e+8 c d-6 c e x)}{15 e^2}\right )}{e}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{e \sqrt {d+e x}}\) |
(-2*(a + b*x + c*x^2)^(3/2))/(e*Sqrt[d + e*x]) + (3*((-2*Sqrt[d + e*x]*(8* c*d - 7*b*e - 6*c*e*x)*Sqrt[a + b*x + c*x^2])/(15*e^2) - (-((Sqrt[2]*Sqrt[ b^2 - 4*a*c]*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*a*e))*Sqrt[d + e*x]* Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sq rt[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])) + (16*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*e*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2]))/(15*e^2)))/e
3.25.50.3.1 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)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si mp[p/(e*(m + 1)) Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] && !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[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[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[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]
Leaf count of result is larger than twice the leaf count of optimal. \(1176\) vs. \(2(453)=906\).
Time = 2.29 (sec) , antiderivative size = 1177, normalized size of antiderivative = 2.29
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1177\) |
| risch | \(\text {Expression too large to display}\) | \(1750\) |
| default | \(\text {Expression too large to display}\) | \(4365\) |
((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 ^2+b*e*x+a*e)*(a*e^2-b*d*e+c*d^2)/e^4/((x+d/e)*(c*e*x^2+b*e*x+a*e))^(1/2)+ 2/5*c/e^2*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*(c/e^2*(2* b*e-c*d)-2/5*c/e^2*(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*((2*a*b*e^3-2*a*c*d*e^2-b^2*d*e^2+2*b*c*d^2*e-c^2*d^3)/e^4-( a*e^2-b*d*e+c*d^2)/e^4*(b*e-c*d)+b/e^3*(a*e^2-b*d*e+c*d^2)-2/5*c/e^2*a*d-2 /3*(c/e^2*(2*b*e-c*d)-2/5*c/e^2*(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/e^3*(2*a*c *e^2+b^2*e^2-2*b*c*d*e+c^2*d^2)+(a*e^2-b*d*e+c*d^2)/e^3*c-2/5*c/e^2*(3/2*a *e+3/2*b*d)-2/3*(c/e^2*(2*b*e-c*d)-2/5*c/e^2*(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)^(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...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 630, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left ({\left (16 \, c^{3} d^{4} - 24 \, b c^{2} d^{3} e + 6 \, {\left (b^{2} c + 4 \, a c^{2}\right )} d^{2} e^{2} + {\left (b^{3} - 12 \, a b c\right )} d e^{3} + {\left (16 \, c^{3} d^{3} e - 24 \, b c^{2} d^{2} e^{2} + 6 \, {\left (b^{2} c + 4 \, a c^{2}\right )} d e^{3} + {\left (b^{3} - 12 \, a b c\right )} e^{4}\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 ) + 3 \, {\left (16 \, c^{3} d^{3} e - 16 \, b c^{2} d^{2} e^{2} + {\left (b^{2} c + 12 \, a c^{2}\right )} d e^{3} + {\left (16 \, c^{3} d^{2} e^{2} - 16 \, b c^{2} d e^{3} + {\left (b^{2} c + 12 \, a c^{2}\right )} e^{4}\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 (c^{3} e^{4} x^{2} - 8 \, c^{3} d^{2} e^{2} + 7 \, b c^{2} d e^{3} - 5 \, a c^{2} e^{4} - 2 \, {\left (c^{3} d e^{3} - b c^{2} e^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}\right )}}{15 \, {\left (c^{2} e^{6} x + c^{2} d e^{5}\right )}} \]
-2/15*((16*c^3*d^4 - 24*b*c^2*d^3*e + 6*(b^2*c + 4*a*c^2)*d^2*e^2 + (b^3 - 12*a*b*c)*d*e^3 + (16*c^3*d^3*e - 24*b*c^2*d^2*e^2 + 6*(b^2*c + 4*a*c^2)* d*e^3 + (b^3 - 12*a*b*c)*e^4)*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^3*d^3*e - 16*b*c^2*d^2*e^2 + (b^2*c + 12*a*c^2)*d*e^3 + (16*c^3*d^2*e^2 - 16*b*c^2*d*e^3 + (b^2*c + 12*a*c^2)*e^ 4)*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*(c^3*e^4*x^2 - 8*c^3*d^2*e^2 + 7*b*c^2*d*e^3 - 5*a *c^2*e^4 - 2*(c^3*d*e^3 - b*c^2*e^4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d ))/(c^2*e^6*x + c^2*d*e^5)
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d+e\,x\right )}^{3/2}} \,d x \]