\(\int \frac {A+B x+C x^2}{(d+e x)^{3/2} (f+g x)^2 \sqrt {a+b x+c x^2}} \, dx\) [11]

Optimal result

Integrand size = 41, antiderivative size = 1097 \[ \int \frac {A+B x+C x^2}{(d+e x)^{3/2} (f+g x)^2 \sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Output:

-2*e*(C*d^2-e*(-A*e+B*d))*(c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c*d^2)/(-d*g+e* 
f)^2/(e*x+d)^(1/2)-g*(C*f^2-g*(-A*g+B*f))*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2 
)/(-d*g+e*f)^2/(a*g^2-b*f*g+c*f^2)/(g*x+f)-1/2*(-4*a*c+b^2)^(1/2)*(b*C*d*f 
*(2*d*g+e*f)-b*e*g*(-A*d*g-2*A*e*f+3*B*d*f)+a*e*g*(-3*A*e*g+2*B*d*g+B*e*f) 
-a*C*(2*d^2*g^2+e^2*f^2)-c*(3*C*d^2*f^2-B*d*f*(d*g+2*e*f)+A*(d^2*g^2+2*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))*2^(1/2)/(a*e^2-b*d*e+c*d^2)/(-d*g+e 
*f)^2/(c*f^2-g*(-a*g+b*f))/(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)+2^(1/2)*(-4*a*c+b^2)^(1/2)*(C*f^2-g*(-A*g+B*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))/g/(-d*g+ 
e*f)/(a*g^2-b*f*g+c*f^2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)-2*2^(1/2)*(-4*a 
*c+b^2)^(1/2)*(c*(C*f^3*(2*d*g+e*f)-f*g*(3*B*e*f^2-A*g*(-2*d*g+5*e*f)))-g^ 
2*(b*(3*C*d*f^2+A*g*(-d*g+4*e*f)-B*f*(d*g+2*e*f))+a*(C*f*(-4*d*g+e*f)+g*(- 
3*A*e*g+2*B*d*g+B*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)*EllipticPi(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)*g/(2*c*f-(b+(-4*a*c+b^2 
)^(1/2))*g),(-2*(-4*a*c+b^2)^(1/2)*e/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^...
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 44.27 (sec) , antiderivative size = 48666, normalized size of antiderivative = 44.36 \[ \int \frac {A+B x+C x^2}{(d+e x)^{3/2} (f+g x)^2 \sqrt {a+b x+c x^2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [F]

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.

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

\(\Big \downarrow \) 2154

\(\displaystyle \left (A+\frac {f (C f-B g)}{g^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x)^2 \sqrt {c x^2+b x+a}}dx+\int \frac {\frac {B}{g}+\frac {C x}{g}-\frac {C f}{g^2}}{(d+e x)^{3/2} (f+g x) \sqrt {c x^2+b x+a}}dx\)

\(\Big \downarrow \) 1292

\(\displaystyle \left (A+\frac {f (C f-B g)}{g^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x)^2 \sqrt {c x^2+b x+a}}dx+\int \frac {\frac {B}{g}+\frac {C x}{g}-\frac {C f}{g^2}}{(d+e x)^{3/2} (f+g x) \sqrt {c x^2+b x+a}}dx\)

\(\Big \downarrow \) 2154

\(\displaystyle \left (A+\frac {f (C f-B g)}{g^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x)^2 \sqrt {c x^2+b x+a}}dx-\frac {(2 C f-B g) \int \frac {1}{(d+e x)^{3/2} (f+g x) \sqrt {c x^2+b x+a}}dx}{g^2}+\int \frac {C}{g^2 (d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \left (A+\frac {f (C f-B g)}{g^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x)^2 \sqrt {c x^2+b x+a}}dx-\frac {(2 C f-B g) \int \frac {1}{(d+e x)^{3/2} (f+g x) \sqrt {c x^2+b x+a}}dx}{g^2}+\frac {C \int \frac {1}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{g^2}\)

\(\Big \downarrow \) 1167

\(\displaystyle \left (A+\frac {f (C f-B g)}{g^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x)^2 \sqrt {c x^2+b x+a}}dx-\frac {(2 C f-B g) \int \frac {1}{(d+e x)^{3/2} (f+g x) \sqrt {c x^2+b x+a}}dx}{g^2}+\frac {C \left (-\frac {2 \int -\frac {c \sqrt {d+e x}}{2 \sqrt {c x^2+b x+a}}dx}{a e^2-b d e+c d^2}-\frac {2 e \sqrt {a+b x+c x^2}}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\right )}{g^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \left (A+\frac {f (C f-B g)}{g^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x)^2 \sqrt {c x^2+b x+a}}dx-\frac {(2 C f-B g) \int \frac {1}{(d+e x)^{3/2} (f+g x) \sqrt {c x^2+b x+a}}dx}{g^2}+\frac {C \left (\frac {c \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{a e^2-b d e+c d^2}-\frac {2 e \sqrt {a+b x+c x^2}}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\right )}{g^2}\)

\(\Big \downarrow \) 1172

\(\displaystyle \left (A+\frac {f (C f-B g)}{g^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x)^2 \sqrt {c x^2+b x+a}}dx+\frac {C \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{\sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 e \sqrt {a+b x+c x^2}}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\right )}{g^2}-\frac {(2 C f-B g) \int \frac {1}{(d+e x)^{3/2} (f+g x) \sqrt {c x^2+b x+a}}dx}{g^2}\)

\(\Big \downarrow \) 327

\(\displaystyle \left (A+\frac {f (C f-B g)}{g^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x)^2 \sqrt {c x^2+b x+a}}dx-\frac {(2 C f-B g) \int \frac {1}{(d+e x)^{3/2} (f+g x) \sqrt {c x^2+b x+a}}dx}{g^2}+\frac {C \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 e \sqrt {a+b x+c x^2}}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\right )}{g^2}\)

\(\Big \downarrow \) 1288

\(\displaystyle \left (A+\frac {f (C f-B g)}{g^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x)^2 \sqrt {c x^2+b x+a}}dx-\frac {(2 C f-B g) \int \left (\frac {e}{(e f-d g) (d+e x)^{3/2} \sqrt {c x^2+b x+a}}-\frac {g}{(e f-d g) \sqrt {d+e x} (f+g x) \sqrt {c x^2+b x+a}}\right )dx}{g^2}+\frac {C \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 e \sqrt {a+b x+c x^2}}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\right )}{g^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {C \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\left (c d^2-b e d+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {2 e \sqrt {c x^2+b x+a}}{\left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}\right )}{g^2}-\frac {(2 C f-B g) \left (-\frac {2 \sqrt {c x^2+b x+a} e^2}{\left (c d^2-b e d+a e^2\right ) (e f-d g) \sqrt {d+e x}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right ) e}{\left (c d^2-b e d+a e^2\right ) (e f-d g) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {\sqrt {2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} g \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \operatorname {EllipticPi}\left (-\frac {\left (2 c d-b e+\sqrt {b^2-4 a c} e\right ) g}{2 c (e f-d g)},\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right ),\frac {b-\sqrt {b^2-4 a c}-\frac {2 c d}{e}}{b+\sqrt {b^2-4 a c}-\frac {2 c d}{e}}\right )}{\sqrt {c} (e f-d g)^2 \sqrt {c x^2+b x+a}}\right )}{g^2}+\left (A+\frac {f (C f-B g)}{g^2}\right ) \int \frac {1}{(d+e x)^{3/2} (f+g x)^2 \sqrt {c x^2+b x+a}}dx\)

Input:

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

Output:

$Aborted
 

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 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 1167

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)/((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)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, 
 x]*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[m 
, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimp 
lerQ[m, 1] && IntegerQ[p]) || ILtQ[Simplify[m + 2*p + 3], 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 1288

Int[((f_.) + (g_.)*(x_))^(n_)/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2]), x_Symbol] :> Int[ExpandIntegrand[1/(Sqrt[f + g*x]*Sqrt[a 
 + b*x + c*x^2]), (f + g*x)^(n + 1/2)/(d + e*x), x], x] /; FreeQ[{a, b, c, 
d, e, f, g}, x] && IntegerQ[n + 1/2]
 

rule 1292

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Unintegrable[(d + e*x)^m*(f + g*x)^n* 
(a + b*x + c*x^2)^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2154

Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b 
_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, d + 
 e*x, x]*(d + e*x)^(m + 1)*(f + g*x)^n*(a + b*x + c*x^2)^p, x] + Simp[Polyn 
omialRemainder[Px, d + e*x, x]   Int[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x 
^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && PolynomialQ[Px, x 
] && LtQ[m, 0] &&  !IntegerQ[n] && IntegersQ[2*m, 2*n, 2*p]
 

Maple [A] (verified)

Time = 96.13 (sec) , antiderivative size = 1780, normalized size of antiderivative = 1.62

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1780\)
default	\(\text {Expression too large to display}\)	\(121389\)

Input:

int((C*x^2+B*x+A)/(e*x+d)^(3/2)/(g*x+f)^2/(c*x^2+b*x+a)^(1/2),x,method=_RE 
TURNVERBOSE)
 

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)/(a*e^2-b*d*e+c*d^2)*(A*e^2-B*d*e+C*d^2)/(d*g-e*f)^2/((x+d/e) 
*(c*e*x^2+b*e*x+a*e))^(1/2)-g/(a*d*g^3-a*e*f*g^2-b*d*f*g^2+b*e*f^2*g+c*d*f 
^2*g-c*e*f^3)*(A*g^2-B*f*g+C*f^2)/(d*g-e*f)*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x 
+b*d*x+a*d)^(1/2)/(g*x+f)+2*(-(b*e-c*d)*(A*e^2-B*d*e+C*d^2)/(a*e^2-b*d*e+c 
*d^2)/(d*g-e*f)^2+b*e/(a*e^2-b*d*e+c*d^2)*(A*e^2-B*d*e+C*d^2)/(d*g-e*f)^2+ 
1/2*c*e*f*(A*g^2-B*f*g+C*f^2)/(a*d*g^3-a*e*f*g^2-b*d*f*g^2+b*e*f^2*g+c*d*f 
^2*g-c*e*f^3)/(d*g-e*f)/g)*(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*(c*e*(A*e^2-B*d*e+C*d^2)/(a*e^2-b*d*e+c*d^2)/(d*g-e*f)^2 
+1/2*e*c*(A*g^2-B*f*g+C*f^2)/(a*d*g^3-a*e*f*g^2-b*d*f*g^2+b*e*f^2*g+c*d*f^ 
2*g-c*e*f^3)/(d*g-e*f))*(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...
 

Fricas [F(-1)]

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

integrate((C*x^2+B*x+A)/(e*x+d)^(3/2)/(g*x+f)^2/(c*x^2+b*x+a)^(1/2),x, alg 
orithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((C*x^2+B*x+A)/(e*x+d)^(3/2)/(g*x+f)^2/(c*x^2+b*x+a)^(1/2),x, alg 
orithm="maxima")
 

Output:

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

Giac [F]

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

integrate((C*x^2+B*x+A)/(e*x+d)^(3/2)/(g*x+f)^2/(c*x^2+b*x+a)^(1/2),x, alg 
orithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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