\(\int \frac {x^2}{(a+b x+c x^2)^{3/2} (d+e x+f x^2)} \, dx\) [150]

Optimal result

Integrand size = 30, antiderivative size = 609 \[ \int \frac {x^2}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=-\frac {2 \left (a (b c d-2 a c e+a b f)+c \left (b^2 d-a b e-2 a (c d-a f)\right ) x\right )}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {a+b x+c x^2}}-\frac {f \left (2 d (c d-a f)-(b d-a e) \left (e-\sqrt {e^2-4 d f}\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}}}+\frac {f \left (2 d (c d-a f)-(b d-a e) \left (e+\sqrt {e^2-4 d f}\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}}} \] Output:

(-2*a*(a*b*f-2*a*c*e+b*c*d)-2*c*(b^2*d-a*b*e-2*a*(-a*f+c*d))*x)/(-4*a*c+b^ 
2)/((-a*f+c*d)^2-(-a*e+b*d)*(-b*f+c*e))/(c*x^2+b*x+a)^(1/2)-1/2*f*(2*d*(-a 
*f+c*d)-(-a*e+b*d)*(e-(-4*d*f+e^2)^(1/2)))*arctanh(1/4*(4*a*f-b*(e-(-4*d*f 
+e^2)^(1/2))+2*(b*f-c*(e-(-4*d*f+e^2)^(1/2)))*x)*2^(1/2)/(c*e^2-2*c*d*f-b* 
e*f+2*a*f^2-(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)/(c*x^2+b*x+a)^(1/2))*2^(1 
/2)/(-4*d*f+e^2)^(1/2)/((-a*f+c*d)^2-(-a*e+b*d)*(-b*f+c*e))/(c*e^2-2*c*d*f 
-b*e*f+2*a*f^2-(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)+1/2*f*(2*d*(-a*f+c*d)- 
(-a*e+b*d)*(e+(-4*d*f+e^2)^(1/2)))*arctanh(1/4*(4*a*f-b*(e+(-4*d*f+e^2)^(1 
/2))+2*(b*f-c*(e+(-4*d*f+e^2)^(1/2)))*x)*2^(1/2)/(c*e^2-2*c*d*f-b*e*f+2*a* 
f^2+(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)/(c*x^2+b*x+a)^(1/2))*2^(1/2)/(-4* 
d*f+e^2)^(1/2)/((-a*f+c*d)^2-(-a*e+b*d)*(-b*f+c*e))/(c*e^2-2*c*d*f-b*e*f+2 
*a*f^2+(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 1.37 (sec) , antiderivative size = 534, normalized size of antiderivative = 0.88 \[ \int \frac {x^2}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\frac {-2 \left (b^2 c d x+a c (b d-2 c d x-b e x)+a^2 (-2 c e+b f+2 c f x)\right )-\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)} \text {RootSum}\left [b^2 d-a b e+a^2 f-4 b \sqrt {c} d \text {$\#$1}+2 a \sqrt {c} e \text {$\#$1}+4 c d \text {$\#$1}^2+b e \text {$\#$1}^2-2 a f \text {$\#$1}^2-2 \sqrt {c} e \text {$\#$1}^3+f \text {$\#$1}^4\&,\frac {b c d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 a b d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+a^2 e f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 c^{3/2} d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 a \sqrt {c} d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+b d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a e f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{2 b \sqrt {c} d-a \sqrt {c} e-4 c d \text {$\#$1}-b e \text {$\#$1}+2 a f \text {$\#$1}+3 \sqrt {c} e \text {$\#$1}^2-2 f \text {$\#$1}^3}\&\right ]}{\left (b^2-4 a c\right ) \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right ) \sqrt {a+x (b+c x)}} \] Input:

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

Output:

(-2*(b^2*c*d*x + a*c*(b*d - 2*c*d*x - b*e*x) + a^2*(-2*c*e + b*f + 2*c*f*x 
)) - (b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)]*RootSum[b^2*d - a*b*e + a^2*f - 4 
*b*Sqrt[c]*d*#1 + 2*a*Sqrt[c]*e*#1 + 4*c*d*#1^2 + b*e*#1^2 - 2*a*f*#1^2 - 
2*Sqrt[c]*e*#1^3 + f*#1^4 & , (b*c*d^2*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c 
*x^2] - #1] - 2*a*b*d*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] + a 
^2*e*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] - 2*c^(3/2)*d^2*Log[ 
-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 + 2*a*Sqrt[c]*d*f*Log[-(Sqrt 
[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 + b*d*f*Log[-(Sqrt[c]*x) + Sqrt[a 
+ b*x + c*x^2] - #1]*#1^2 - a*e*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] 
 - #1]*#1^2)/(2*b*Sqrt[c]*d - a*Sqrt[c]*e - 4*c*d*#1 - b*e*#1 + 2*a*f*#1 + 
 3*Sqrt[c]*e*#1^2 - 2*f*#1^3) & ])/((b^2 - 4*a*c)*(c^2*d^2 - b*c*d*e + f*( 
b^2*d - a*b*e + a^2*f) + a*c*(e^2 - 2*d*f))*Sqrt[a + x*(b + c*x)])
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 580, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2135, 27, 1365, 1154, 219}

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.

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

\(\Big \downarrow \) 2135

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1365

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(-2*(a*(b*c*d - 2*a*c*e + a*b*f) + c*(b^2*d - a*b*e - 2*a*(c*d - a*f))*x)) 
/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*Sqrt[a + b*x + c 
*x^2]) + (-((f*(2*d*(c*d - a*f) - (b*d - a*e)*(e - Sqrt[e^2 - 4*d*f]))*Arc 
Tanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f 
]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqr 
t[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*Sqrt[c 
*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]])) + (f*( 
2*d*(c*d - a*f) - (b*d - a*e)*(e + Sqrt[e^2 - 4*d*f]))*ArcTanh[(4*a*f - b* 
(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2 
]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]* 
Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - 
 b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]))/((c*d - a*f)^2 - (b*d 
- a*e)*(c*e - b*f))
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1365

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + ( 
e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Sim 
p[(2*c*g - h*(b - q))/q   Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x] 
, x] - Simp[(2*c*g - h*(b + q))/q   Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f 
*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0 
] && NeQ[e^2 - 4*d*f, 0] && PosQ[b^2 - 4*a*c]
 

rule 2135

Int[(Px_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_. 
)*(x_)^2)^(q_), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1] 
, C = Coeff[Px, x, 2]}, Simp[(a + b*x + c*x^2)^(p + 1)*((d + e*x + f*x^2)^( 
q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)))*( 
(A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b 
*e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b*c*d - 2*a*c* 
e + a*b*f) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x), x] + Simp[1/((b^2 - 4 
*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1))   Int[(a + b*x + c 
*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 
 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + 
 a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(a*f*(p 
+ 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B 
)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a 
*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p 
 + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*( 
c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4))) 
*x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d 
 - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x]] /; F 
reeQ[{a, b, c, d, e, f, q}, x] && PolyQ[Px, x, 2] && LtQ[p, -1] && NeQ[(c*d 
 - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) 
 &&  !IGtQ[q, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1989\) vs. \(2(561)=1122\).

Time = 2.72 (sec) , antiderivative size = 1990, normalized size of antiderivative = 3.27

method	result	size
default	\(\text {Expression too large to display}\)	\(1990\)

Input:

int(x^2/(c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d),x,method=_RETURNVERBOSE)
 

Output:

2/f*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-1/2/f^2*(e*(-4*d*f+e^2)^(1/2 
)+2*d*f-e^2)/(-4*d*f+e^2)^(1/2)*(2/(f*b*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1 
/2)*c*e+2*a*f^2-b*e*f-2*d*f*c+c*e^2)*f^2/(c*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2 
)))^2+(c*(-4*d*f+e^2)^(1/2)+f*b-c*e)/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+1 
/2*(f*b*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*d*f*c+c* 
e^2)/f^2)^(1/2)-2*(c*(-4*d*f+e^2)^(1/2)+f*b-c*e)*f/(f*b*(-4*d*f+e^2)^(1/2) 
-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*d*f*c+c*e^2)*(2*c*(x-1/2/f*(-e+(-4 
*d*f+e^2)^(1/2)))+(c*(-4*d*f+e^2)^(1/2)+f*b-c*e)/f)/(2*c*(f*b*(-4*d*f+e^2) 
^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*d*f*c+c*e^2)/f^2-(c*(-4*d*f+ 
e^2)^(1/2)+f*b-c*e)^2/f^2)/(c*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))^2+(c*(-4*d 
*f+e^2)^(1/2)+f*b-c*e)/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+1/2*(f*b*(-4*d* 
f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*d*f*c+c*e^2)/f^2)^(1/2 
)-2/(f*b*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*d*f*c+c 
*e^2)*f^2*2^(1/2)/((f*b*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2- 
b*e*f-2*d*f*c+c*e^2)/f^2)^(1/2)*ln(((f*b*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^( 
1/2)*c*e+2*a*f^2-b*e*f-2*d*f*c+c*e^2)/f^2+(c*(-4*d*f+e^2)^(1/2)+f*b-c*e)/f 
*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+1/2*2^(1/2)*((f*b*(-4*d*f+e^2)^(1/2)-(- 
4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*d*f*c+c*e^2)/f^2)^(1/2)*(4*c*(x-1/2/f 
*(-e+(-4*d*f+e^2)^(1/2)))^2+4*(c*(-4*d*f+e^2)^(1/2)+f*b-c*e)/f*(x-1/2/f*(- 
e+(-4*d*f+e^2)^(1/2)))+2*(f*b*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*d*f-e^2>0)', see `assume?` for 
 more deta
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\int \frac {\sqrt {c \,x^{2}+b x +a}\, x^{2}}{c^{2} f \,x^{6}+2 b c f \,x^{5}+c^{2} e \,x^{5}+2 a c f \,x^{4}+b^{2} f \,x^{4}+2 b c e \,x^{4}+c^{2} d \,x^{4}+2 a b f \,x^{3}+2 a c e \,x^{3}+b^{2} e \,x^{3}+2 b c d \,x^{3}+a^{2} f \,x^{2}+2 a b e \,x^{2}+2 a c d \,x^{2}+b^{2} d \,x^{2}+a^{2} e x +2 a b d x +a^{2} d}d x \] Input:

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

Output:

int((sqrt(a + b*x + c*x**2)*x**2)/(a**2*d + a**2*e*x + a**2*f*x**2 + 2*a*b 
*d*x + 2*a*b*e*x**2 + 2*a*b*f*x**3 + 2*a*c*d*x**2 + 2*a*c*e*x**3 + 2*a*c*f 
*x**4 + b**2*d*x**2 + b**2*e*x**3 + b**2*f*x**4 + 2*b*c*d*x**3 + 2*b*c*e*x 
**4 + 2*b*c*f*x**5 + c**2*d*x**4 + c**2*e*x**5 + c**2*f*x**6),x)