\(\int \frac {x^2 \sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx\) [89]

Optimal result

Integrand size = 27, antiderivative size = 496 \[ \int \frac {x^2 \sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=-\frac {4 (c d+2 b e) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{15 c^2 e}+\frac {2 (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c e}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 c^2 d^2-8 b^2 e^2+3 c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 c^3 e^2 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {4 \sqrt {2} \sqrt {b^2-4 a c} (c d+2 b e) \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 c^3 e^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

-4/15*(2*b*e+c*d)*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^2/e+2/5*(e*x+d)^(3/2 
)*(c*x^2+b*x+a)^(1/2)/c/e-1/15*2^(1/2)*(-4*a*c+b^2)^(1/2)*(2*c^2*d^2-8*b^2 
*e^2+3*c*e*(3*a*e+b*d))*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2 
)*EllipticE(1/2*(1+(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4*a*c 
+b^2)^(1/2)*e/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2))/c^3/e^2/(c*(e*x+d)/ 
(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2)/(c*x^2+b*x+a)^(1/2)+4/15*2^(1/2)*( 
-4*a*c+b^2)^(1/2)*(2*b*e+c*d)*(a*e^2-b*d*e+c*d^2)*(c*(e*x+d)/(2*c*d-(b+(-4 
*a*c+b^2)^(1/2))*e))^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*EllipticF 
(1/2*(1+(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1/2) 
*e/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2))/c^3/e^2/(e*x+d)^(1/2)/(c*x^2+b 
*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 32.48 (sec) , antiderivative size = 707, normalized size of antiderivative = 1.43 \[ \int \frac {x^2 \sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=\frac {-\frac {4 e^2 \left (2 c^2 d^2-8 b^2 e^2+3 c e (b d+3 a e)\right ) (a+x (b+c x))}{\sqrt {d+e x}}+4 c e^2 \sqrt {d+e x} (a+x (b+c x)) (-4 b e+c (d+3 e x))+\frac {i (d+e x) \sqrt {1-\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {2+\frac {4 \left (c d^2+e (-b d+a e)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \left (-\left (\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (-2 c^2 d^2+8 b^2 e^2-3 c e (b d+3 a e)\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )+\left (-8 b^3 e^3+b^2 e^2 \left (11 c d+8 \sqrt {\left (b^2-4 a c\right ) e^2}\right )+b c e \left (17 a e^2-3 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )-c \left (2 c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}+a e^2 \left (14 c d+9 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{\sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}}{30 c^3 e^3 \sqrt {a+x (b+c x)}} \] Input:

Integrate[(x^2*Sqrt[d + e*x])/Sqrt[a + b*x + c*x^2],x]
 

Output:

((-4*e^2*(2*c^2*d^2 - 8*b^2*e^2 + 3*c*e*(b*d + 3*a*e))*(a + x*(b + c*x)))/ 
Sqrt[d + e*x] + 4*c*e^2*Sqrt[d + e*x]*(a + x*(b + c*x))*(-4*b*e + c*(d + 3 
*e*x)) + (I*(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))]*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))]*(-((2*c*d - b 
*e + Sqrt[(b^2 - 4*a*c)*e^2])*(-2*c^2*d^2 + 8*b^2*e^2 - 3*c*e*(b*d + 3*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 + Sqrt[(b^2 
- 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]) + (-8*b^3*e^3 + 
b^2*e^2*(11*c*d + 8*Sqrt[(b^2 - 4*a*c)*e^2]) + b*c*e*(17*a*e^2 - 3*d*Sqrt[ 
(b^2 - 4*a*c)*e^2]) - c*(2*c*d^2*Sqrt[(b^2 - 4*a*c)*e^2] + a*e^2*(14*c*d + 
 9*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[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4* 
a*c)*e^2])])/(30*c^3*e^3*Sqrt[a + x*(b + c*x)])
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1283, 2184, 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.

\(\displaystyle \int \frac {x^2 \sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx\)

\(\Big \downarrow \) 1283

\(\displaystyle \frac {2 x \sqrt {d+e x} \sqrt {a+b x+c x^2}}{5 c}-\frac {\int \frac {-\left ((c d-4 b e) x^2\right )+3 (b d+a e) x+2 a d}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{5 c}\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {2 x \sqrt {d+e x} \sqrt {a+b x+c x^2}}{5 c}-\frac {\frac {2 \int \frac {e \left (-4 d e b^2+c d^2 b-4 a e^2 b+7 a c d e+\left (2 c^2 d^2-8 b^2 e^2+3 c e (b d+3 a e)\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 c e^2}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} (c d-4 b e)}{3 c e}}{5 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 x \sqrt {d+e x} \sqrt {a+b x+c x^2}}{5 c}-\frac {\frac {\int \frac {-4 d e b^2+c d^2 b-4 a e^2 b+7 a c d e+\left (2 c^2 d^2-8 b^2 e^2+3 c e (b d+3 a e)\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 c e}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} (c d-4 b e)}{3 c e}}{5 c}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {2 x \sqrt {d+e x} \sqrt {a+b x+c x^2}}{5 c}-\frac {\frac {\frac {\left (3 c e (3 a e+b d)-8 b^2 e^2+2 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {2 (2 b e+c d) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}}{3 c e}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} (c d-4 b e)}{3 c e}}{5 c}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {2 x \sqrt {d+e x} \sqrt {a+b x+c x^2}}{5 c}-\frac {\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (3 c e (3 a e+b d)-8 b^2 e^2+2 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 b e+c d) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{3 c e}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} (c d-4 b e)}{3 c e}}{5 c}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2 x \sqrt {d+e x} \sqrt {a+b x+c x^2}}{5 c}-\frac {\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (3 c e (3 a e+b d)-8 b^2 e^2+2 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 b e+c d) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{3 c e}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} (c d-4 b e)}{3 c e}}{5 c}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2 x \sqrt {d+e x} \sqrt {a+b x+c x^2}}{5 c}-\frac {\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (3 c e (3 a e+b d)-8 b^2 e^2+2 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 b e+c d) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{3 c e}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} (c d-4 b e)}{3 c e}}{5 c}\)

Input:

Int[(x^2*Sqrt[d + e*x])/Sqrt[a + b*x + c*x^2],x]
 

Output:

(2*x*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])/(5*c) - ((-2*(c*d - 4*b*e)*Sqrt[ 
d + e*x]*Sqrt[a + b*x + c*x^2])/(3*c*e) + ((Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c 
^2*d^2 - 8*b^2*e^2 + 3*c*e*(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 + 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*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]))/(3*c*e))/(5*c)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

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])
 

rule 327

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]
 

rule 1172

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]
 

rule 1269

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]
 

rule 1283

Int[(((d_.) + (e_.)*(x_))^(m_)*Sqrt[(f_.) + (g_.)*(x_)])/Sqrt[(a_.) + (b_.) 
*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2*e*(d + e*x)^(m - 1)*Sqrt[f + g*x 
]*(Sqrt[a + b*x + c*x^2]/(c*(2*m + 1))), x] - Simp[1/(c*(2*m + 1))   Int[(( 
d + e*x)^(m - 2)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))*Simp[e*(b*d*f + a*( 
d*g + 2*e*f*(m - 1))) - c*d^2*f*(2*m + 1) + (a*e^2*g*(2*m - 1) - c*d*(4*e*f 
*m + d*g*(2*m + 1)) + b*e*(2*d*g + e*f*(2*m - 1)))*x + e*(2*b*e*g*m - c*(e* 
f + d*g*(4*m - 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && I 
ntegerQ[2*m] && GtQ[m, 1]
 

rule 2184

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S 
imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q 
+ 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + 
b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 
1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c 
*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ 
q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol 
yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !(IGt 
Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(924\) vs. \(2(436)=872\).

Time = 4.24 (sec) , antiderivative size = 925, normalized size of antiderivative = 1.86

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {2 x \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d}}{5 c}+\frac {2 \left (d -\frac {2 \left (2 b e +2 c d \right )}{5 c}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d}}{3 c e}+\frac {2 \left (-\frac {2 a d}{5 c}-\frac {2 \left (d -\frac {2 \left (2 b e +2 c d \right )}{5 c}\right ) \left (\frac {a e}{2}+\frac {b d}{2}\right )}{3 c e}\right ) \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}}}\, \operatorname {EllipticF}\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}}+\frac {2 \left (-\frac {2 \left (\frac {3 a e}{2}+\frac {3 b d}{2}\right )}{5 c}-\frac {2 \left (d -\frac {2 \left (2 b e +2 c d \right )}{5 c}\right ) \left (b e +c d \right )}{3 c e}\right ) \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}}}\, \left (\left (-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \operatorname {EllipticE}\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 )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \operatorname {EllipticF}\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 )}{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}}\)	\(925\)
risch	\(\text {Expression too large to display}\)	\(1691\)
default	\(\text {Expression too large to display}\)	\(4340\)

Input:

int(x^2*(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*(2/5*x/c*( 
c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2/3*(d-2/5/c*(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*(-2/5*a/c*d-2/3*(d- 
2/5/c*(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*(-2/5/c*(3/2*a*e+3/2*b*d)-2/3*(d-2/5/c* 
(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+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^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))))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 486, normalized size of antiderivative = 0.98 \[ \int \frac {x^2 \sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 \, {\left ({\left (2 \, c^{3} d^{3} + 2 \, b c^{2} d^{2} e + {\left (7 \, b^{2} c - 12 \, a c^{2}\right )} d e^{2} - {\left (8 \, b^{3} - 21 \, a b c\right )} e^{3}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left (2 \, c^{3} d^{2} e + 3 \, b c^{2} d e^{2} - {\left (8 \, b^{2} c - 9 \, a c^{2}\right )} e^{3}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (3 \, c^{3} e^{3} x + c^{3} d e^{2} - 4 \, b c^{2} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}\right )}}{45 \, c^{4} e^{3}} \] Input:

integrate(x^2*(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
 

Output:

2/45*((2*c^3*d^3 + 2*b*c^2*d^2*e + (7*b^2*c - 12*a*c^2)*d*e^2 - (8*b^3 - 2 
1*a*b*c)*e^3)*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*(2*c^3*d^2*e + 3*b*c^2*d*e^2 - (8*b^2*c - 9*a*c^2)*e^3)*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*(3*c^3*e^3*x + c^3*d*e^2 - 4*b*c^2*e^3)*sqrt(c*x^2 + b*x + a)*s 
qrt(e*x + d))/(c^4*e^3)
 

Sympy [F]

\[ \int \frac {x^2 \sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {x^{2} \sqrt {d + e x}}{\sqrt {a + b x + c x^{2}}}\, dx \] Input:

integrate(x**2*(e*x+d)**(1/2)/(c*x**2+b*x+a)**(1/2),x)
 

Output:

Integral(x**2*sqrt(d + e*x)/sqrt(a + b*x + c*x**2), x)
 

Maxima [F]

\[ \int \frac {x^2 \sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=\int { \frac {\sqrt {e x + d} x^{2}}{\sqrt {c x^{2} + b x + a}} \,d x } \] Input:

integrate(x^2*(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(e*x + d)*x^2/sqrt(c*x^2 + b*x + a), x)
 

Giac [F]

\[ \int \frac {x^2 \sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=\int { \frac {\sqrt {e x + d} x^{2}}{\sqrt {c x^{2} + b x + a}} \,d x } \] Input:

integrate(x^2*(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(e*x + d)*x^2/sqrt(c*x^2 + b*x + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {x^2\,\sqrt {d+e\,x}}{\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:

int((x^2*(d + e*x)^(1/2))/(a + b*x + c*x^2)^(1/2),x)
 

Output:

int((x^2*(d + e*x)^(1/2))/(a + b*x + c*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^2 \sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {x^{2} \sqrt {e x +d}}{\sqrt {c \,x^{2}+b x +a}}d x \] Input:

int(x^2*(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2),x)
 

Output:

int(x^2*(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2),x)