\(\int \frac {(f+g x)^2}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx\) [574]

Optimal result

Integrand size = 31, antiderivative size = 487 \[ \int \frac {(f+g x)^2}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=-\frac {2 (e f-d g)^2 \sqrt {a+b x+c x^2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (e (b d-a e) g^2-c \left (e^2 f^2-2 d e f g+2 d^2 g^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e^2 \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {4 \sqrt {2} \sqrt {b^2-4 a c} g (e f-d g) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

-2*(-d*g+e*f)^2*(c*x^2+b*x+a)^(1/2)/e/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(1/2)-2^ 
(1/2)*(-4*a*c+b^2)^(1/2)*(e*(-a*e+b*d)*g^2-c*(2*d^2*g^2-2*d*e*f*g+e^2*f^2) 
)*(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/e^2/(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*x^2+b*x+a)^(1/2)+4*2^(1/2)*(-4*a*c+ 
b^2)^(1/2)*g*(-d*g+e*f)*(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/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 = 29.24 (sec) , antiderivative size = 770, normalized size of antiderivative = 1.58 \[ \int \frac {(f+g x)^2}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\frac {2 \left (-e^2 (e f-d g)^2 (a+x (b+c x))+\frac {e^2 \left (e (-b d+a e) g^2+c \left (e^2 f^2-2 d e f g+2 d^2 g^2\right )\right ) (a+x (b+c x))}{c}-\frac {i (d+e x)^{3/2} \sqrt {1-\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {1+\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \left (\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (e (-b d+a e) g^2+c \left (e^2 f^2-2 d e f g+2 d^2 g^2\right )\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )+\left (-2 c^2 d e^2 f^2+e (-b d+a e) \left (b e-\sqrt {\left (b^2-4 a c\right ) e^2}\right ) g^2-c \left (4 a e^3 f g-2 d e \sqrt {\left (b^2-4 a c\right ) e^2} f g+2 d^2 \sqrt {\left (b^2-4 a c\right ) e^2} g^2-b e^2 f (e f+2 d g)+e^2 \left (\sqrt {\left (b^2-4 a c\right ) e^2} f^2-2 a d g^2\right )\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{2 \sqrt {2} c \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}\right )}{e^3 \left (c d^2+e (-b d+a e)\right ) \sqrt {d+e x} \sqrt {a+x (b+c x)}} \] Input:

Integrate[(f + g*x)^2/((d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2]),x]
 

Output:

(2*(-(e^2*(e*f - d*g)^2*(a + x*(b + c*x))) + (e^2*(e*(-(b*d) + a*e)*g^2 + 
c*(e^2*f^2 - 2*d*e*f*g + 2*d^2*g^2))*(a + x*(b + c*x)))/c - ((I/2)*(d + e* 
x)^(3/2)*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])*(e*(-(b*d) + a*e)*g^2 + c*(e^2*f^2 - 2*d*e*f*g + 2*d^2*g^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]))] + (-2*c^2*d*e^2*f^ 
2 + e*(-(b*d) + a*e)*(b*e - Sqrt[(b^2 - 4*a*c)*e^2])*g^2 - c*(4*a*e^3*f*g 
- 2*d*e*Sqrt[(b^2 - 4*a*c)*e^2]*f*g + 2*d^2*Sqrt[(b^2 - 4*a*c)*e^2]*g^2 - 
b*e^2*f*(e*f + 2*d*g) + e^2*(Sqrt[(b^2 - 4*a*c)*e^2]*f^2 - 2*a*d*g^2)))*El 
lipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sq 
rt[(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]*c*Sqrt[(c*d 
^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])))/(e^3*( 
c*d^2 + e*(-(b*d) + a*e))*Sqrt[d + e*x]*Sqrt[a + x*(b + c*x)])
 

Rubi [A] (verified)

Time = 1.41 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {1290, 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.

\(\displaystyle \int \frac {(f+g x)^2}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx\)

\(\Big \downarrow \) 1290

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {2 g (e f-d g) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e^2}-\frac {\left (e g^2 (b d-a e)-c \left (2 d^2 g^2-2 d e f g+e^2 f^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e^2}}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} (e f-d g)^2}{e \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1172

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

Input:

Int[(f + g*x)^2/((d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2]),x]
 

Output:

(-2*(e*f - d*g)^2*Sqrt[a + b*x + c*x^2])/(e*(c*d^2 - b*d*e + a*e^2)*Sqrt[d 
 + e*x]) + (-((Sqrt[2]*Sqrt[b^2 - 4*a*c]*(e*(b*d - a*e)*g^2 - c*(e^2*f^2 - 
 2*d*e*f*g + 2*d^2*g^2))*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^2*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqr 
t[a + b*x + c*x^2])) + (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2 
)*g*(e*f - d*g)*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sq 
rt[-((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^2*Sqrt[d + e*x]*Sqrt[a + b*x 
 + c*x^2]))/(c*d^2 - b*d*e + a*e^2)
 

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 1290

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(f + g*x)^ 
n, d + e*x, x], R = PolynomialRemainder[(f + g*x)^n, 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)*Q + 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, f, g, p}, x] && IGtQ[n, 1] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(988\) vs. \(2(439)=878\).

Time = 6.33 (sec) , antiderivative size = 989, normalized size of antiderivative = 2.03

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {2 \left (c e \,x^{2}+b e x +a e \right ) \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right )}{e^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x +a e \right )}}+\frac {2 \left (-\frac {g \left (d g -2 e f \right )}{e^{2}}-\frac {\left (b e -c d \right ) \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right )}{e^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )}+\frac {b \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right )}{e \left (a \,e^{2}-b d e +c \,d^{2}\right )}\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 {g^{2}}{e}+\frac {c \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right )}{e \left (a \,e^{2}-b d e +c \,d^{2}\right )}\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}}\)	\(989\)
default	\(\text {Expression too large to display}\)	\(6107\)

Input:

int((g*x+f)^2/(e*x+d)^(3/2)/(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*(c*e*x 
^2+b*e*x+a*e)/e^2/(a*e^2-b*d*e+c*d^2)*(d^2*g^2-2*d*e*f*g+e^2*f^2)/((x+d/e) 
*(c*e*x^2+b*e*x+a*e))^(1/2)+2*(-g*(d*g-2*e*f)/e^2-1/e^2*(b*e-c*d)*(d^2*g^2 
-2*d*e*f*g+e^2*f^2)/(a*e^2-b*d*e+c*d^2)+b/e/(a*e^2-b*d*e+c*d^2)*(d^2*g^2-2 
*d*e*f*g+e^2*f^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))+2*(g^2/e+c/e*(d^2*g^2-2*d*e*f*g+e^2*f^2)/(a*e^2-b*d*e+c*d^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^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 = 772, normalized size of antiderivative = 1.59 \[ \int \frac {(f+g x)^2}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Input:

integrate((g*x+f)^2/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas 
")
 

Output:

2/3*(((2*c^2*d^2*e^2 - b*c*d*e^3)*f^2 + 2*(c^2*d^3*e - 2*b*c*d^2*e^2 + 3*a 
*c*d*e^3)*f*g - (2*c^2*d^4 - 2*b*c*d^3*e + a*b*d*e^3 - (b^2 - 4*a*c)*d^2*e 
^2)*g^2 + ((2*c^2*d*e^3 - b*c*e^4)*f^2 + 2*(c^2*d^2*e^2 - 2*b*c*d*e^3 + 3* 
a*c*e^4)*f*g - (2*c^2*d^3*e - 2*b*c*d^2*e^2 + a*b*e^4 - (b^2 - 4*a*c)*d*e^ 
3)*g^2)*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*(c^2*d*e^3*f^2 - 2*c^2*d^2*e^2*f*g + (2*c^2*d^3*e - b*c*d^2*e^2 + 
a*c*d*e^3)*g^2 + (c^2*e^4*f^2 - 2*c^2*d*e^3*f*g + (2*c^2*d^2*e^2 - b*c*d*e 
^3 + a*c*e^4)*g^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*(c^2*e^4*f^2 - 2*c^2*d*e^3*f*g + c 
^2*d^2*e^2*g^2)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d))/(c^3*d^3*e^3 - b*c^2* 
d^2*e^4 + a*c^2*d*e^5 + (c^3*d^2*e^4 - b*c^2*d*e^5 + a*c^2*e^6)*x)
 

Sympy [F]

\[ \int \frac {(f+g x)^2}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (f + g x\right )^{2}}{\left (d + e x\right )^{\frac {3}{2}} \sqrt {a + b x + c x^{2}}}\, dx \] Input:

integrate((g*x+f)**2/(e*x+d)**(3/2)/(c*x**2+b*x+a)**(1/2),x)
 

Output:

Integral((f + g*x)**2/((d + e*x)**(3/2)*sqrt(a + b*x + c*x**2)), x)
 

Maxima [F]

\[ \int \frac {(f+g x)^2}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{2}}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((g*x+f)^2/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima 
")
 

Output:

integrate((g*x + f)^2/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(3/2)), x)
 

Giac [F]

\[ \int \frac {(f+g x)^2}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{2}}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((g*x+f)^2/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
 

Output:

integrate((g*x + f)^2/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(3/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x)^2}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {{\left (f+g\,x\right )}^2}{{\left (d+e\,x\right )}^{3/2}\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:

int((f + g*x)^2/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(1/2)),x)
 

Output:

int((f + g*x)^2/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(1/2)), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {(f+g x)^2}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (g x +f \right )^{2}}{\left (e x +d \right )^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}}d x \] Input:

int((g*x+f)^2/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x)
 

Output:

int((g*x+f)^2/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x)