\(\int \frac {x^2 \sqrt {a+c x^2}}{d+e x+f x^2} \, dx\) [19]

Optimal result

Integrand size = 27, antiderivative size = 462 \[ \int \frac {x^2 \sqrt {a+c x^2}}{d+e x+f x^2} \, dx=-\frac {e \sqrt {a+c x^2}}{f^2}+\frac {x \sqrt {a+c x^2}}{2 f}+\frac {\left (a f^2+2 c \left (e^2-d f\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c} f^3}-\frac {\left (e \left (e-\sqrt {e^2-4 d f}\right ) \left (a f^2+c \left (e^2-2 d f\right )\right )-2 d f \left (a f^2+c \left (e^2-d f\right )\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}+\frac {\left (e \left (e+\sqrt {e^2-4 d f}\right ) \left (a f^2+c \left (e^2-2 d f\right )\right )-2 d f \left (a f^2+c \left (e^2-d f\right )\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}} \] Output:

-e*(c*x^2+a)^(1/2)/f^2+1/2*x*(c*x^2+a)^(1/2)/f+1/2*(a*f^2+2*c*(-d*f+e^2))* 
arctanh(c^(1/2)*x/(c*x^2+a)^(1/2))/c^(1/2)/f^3-1/2*(e*(e-(-4*d*f+e^2)^(1/2 
))*(a*f^2+c*(-2*d*f+e^2))-2*d*f*(a*f^2+c*(-d*f+e^2)))*arctanh(1/2*(2*a*f-c 
*(e-(-4*d*f+e^2)^(1/2))*x)*2^(1/2)/(2*a*f^2+c*(e^2-2*d*f-e*(-4*d*f+e^2)^(1 
/2)))^(1/2)/(c*x^2+a)^(1/2))*2^(1/2)/f^3/(-4*d*f+e^2)^(1/2)/(2*a*f^2+c*(e^ 
2-2*d*f-e*(-4*d*f+e^2)^(1/2)))^(1/2)+1/2*(e*(e+(-4*d*f+e^2)^(1/2))*(a*f^2+ 
c*(-2*d*f+e^2))-2*d*f*(a*f^2+c*(-d*f+e^2)))*arctanh(1/2*(2*a*f-c*(e+(-4*d* 
f+e^2)^(1/2))*x)*2^(1/2)/(2*a*f^2+c*(e^2-2*d*f+e*(-4*d*f+e^2)^(1/2)))^(1/2 
)/(c*x^2+a)^(1/2))*2^(1/2)/f^3/(-4*d*f+e^2)^(1/2)/(2*a*f^2+c*(e^2-2*d*f+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 = 1.18 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.33 \[ \int \frac {x^2 \sqrt {a+c x^2}}{d+e x+f x^2} \, dx=\frac {f (-2 e+f x) \sqrt {a+c x^2}+\frac {2 \left (a f^2+2 c \left (e^2-d f\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+c x^2}}\right )}{\sqrt {c}}-2 \text {RootSum}\left [c^2 d+2 \sqrt {a} c e \text {$\#$1}-2 c d \text {$\#$1}^2+4 a f \text {$\#$1}^2-2 \sqrt {a} e \text {$\#$1}^3+d \text {$\#$1}^4\&,\frac {c^2 d e^2 \log (x)-c^2 d^2 f \log (x)+a c d f^2 \log (x)-c^2 d e^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right )+c^2 d^2 f \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right )-a c d f^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right )+2 \sqrt {a} c e^3 \log (x) \text {$\#$1}-4 \sqrt {a} c d e f \log (x) \text {$\#$1}+2 a^{3/2} e f^2 \log (x) \text {$\#$1}-2 \sqrt {a} c e^3 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}+4 \sqrt {a} c d e f \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}-2 a^{3/2} e f^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}-c d e^2 \log (x) \text {$\#$1}^2+c d^2 f \log (x) \text {$\#$1}^2-a d f^2 \log (x) \text {$\#$1}^2+c d e^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-c d^2 f \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+a d f^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-\sqrt {a} c e+2 c d \text {$\#$1}-4 a f \text {$\#$1}+3 \sqrt {a} e \text {$\#$1}^2-2 d \text {$\#$1}^3}\&\right ]}{2 f^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2139, 2145, 27, 224, 219, 1367, 488, 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.

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

\(\Big \downarrow \) 2139

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

\(\Big \downarrow \) 2145

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1367

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-1/2*((2*e - f*x)*Sqrt[a + c*x^2])/f^2 - (-((Sqrt[c]*(a*f^2 + 2*c*(e^2 - d 
*f))*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/f) + (2*c*(((e*(e - Sqrt[e^2 - 
4*d*f])*(a*f^2 + c*(e^2 - 2*d*f)) - 2*d*f*(a*f^2 + c*(e^2 - d*f)))*ArcTanh 
[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2* 
d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]* 
Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) - ((e*(e + Sqrt[e^2 
 - 4*d*f])*(a*f^2 + c*(e^2 - 2*d*f)) - 2*d*f*(a*f^2 + c*(e^2 - d*f)))*ArcT 
anh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 
 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d* 
f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])])))/f)/(2*c*f^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 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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 488

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

rule 1367

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

rule 2139

Int[(Px_)*((a_) + (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[P 
x, x, 2]}, Simp[(B*c*f*(2*p + 2*q + 3) + C*((-c)*e*(2*p + q + 2)) + 2*c*C*f 
*(p + q + 1)*x)*(a + c*x^2)^p*((d + e*x + f*x^2)^(q + 1)/(2*c*f^2*(p + q + 
1)*(2*p + 2*q + 3))), x] - Simp[1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3))   I 
nt[(a + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Simp[p*((-a)*e)*(C*(c*e)*(q + 1) 
 - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(a*c*(C*(2*d*f - e^2*(2*p + 
 q + 2)) + f*(B*e - 2*A*f)*(2*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*(c*e)*(q 
 + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*e*f*p*(-4*a*c)))*x 
+ (p*(c*e)*(C*(c*e)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)* 
(C*f^2*p*(-4*a*c) - c^2*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2 
*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x]] /; FreeQ[{a, c, d, e, f, q}, x] & 
& PolyQ[Px, x, 2] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2*p + 2*q + 3, 0 
] &&  !IGtQ[p, 0] &&  !IGtQ[q, 0]
 

rule 2145

Int[(Px_)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (f_.)*(x_)^2]), 
x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, 
x, 2]}, Simp[C/c   Int[1/Sqrt[d + f*x^2], x], x] + Simp[1/c   Int[(A*c - a* 
C + (B*c - b*C)*x)/((a + b*x + c*x^2)*Sqrt[d + f*x^2]), x], x]] /; FreeQ[{a 
, b, c, d, f}, x] && PolyQ[Px, x, 2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(833\) vs. \(2(411)=822\).

Time = 2.42 (sec) , antiderivative size = 834, normalized size of antiderivative = 1.81

method	result	size
risch	\(-\frac {\left (-f x +2 e \right ) \sqrt {c \,x^{2}+a}}{2 f^{2}}+\frac {\frac {\left (a \,f^{2}-2 d f c +2 c \,e^{2}\right ) \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{f \sqrt {c}}+\frac {\left (a e \,f^{2} \sqrt {-4 d f +e^{2}}-2 c d e f \sqrt {-4 d f +e^{2}}+e^{3} c \sqrt {-4 d f +e^{2}}+2 a d \,f^{3}-a \,e^{2} f^{2}-2 c \,d^{2} f^{2}+4 c d \,e^{2} f -c \,e^{4}\right ) \sqrt {2}\, \ln \left (\frac {\frac {-\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 d f c +c \,e^{2}}{f^{2}}-\frac {c \left (e -\sqrt {-4 d f +e^{2}}\right ) \left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {\sqrt {2}\, \sqrt {\frac {-\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 d f c +c \,e^{2}}{f^{2}}}\, \sqrt {4 c {\left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}^{2}-\frac {4 c \left (e -\sqrt {-4 d f +e^{2}}\right ) \left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {-2 \sqrt {-4 d f +e^{2}}\, c e +4 a \,f^{2}-4 d f c +2 c \,e^{2}}{f^{2}}}}{2}}{x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}}\right )}{f^{2} \sqrt {-4 d f +e^{2}}\, \sqrt {\frac {-\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 d f c +c \,e^{2}}{f^{2}}}}+\frac {\left (a e \,f^{2} \sqrt {-4 d f +e^{2}}-2 c d e f \sqrt {-4 d f +e^{2}}+e^{3} c \sqrt {-4 d f +e^{2}}-2 a d \,f^{3}+a \,e^{2} f^{2}+2 c \,d^{2} f^{2}-4 c d \,e^{2} f +c \,e^{4}\right ) \sqrt {2}\, \ln \left (\frac {\frac {\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 d f c +c \,e^{2}}{f^{2}}-\frac {c \left (e +\sqrt {-4 d f +e^{2}}\right ) \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {\sqrt {2}\, \sqrt {\frac {\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 d f c +c \,e^{2}}{f^{2}}}\, \sqrt {4 c {\left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}^{2}-\frac {4 c \left (e +\sqrt {-4 d f +e^{2}}\right ) \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {2 \sqrt {-4 d f +e^{2}}\, c e +4 a \,f^{2}-4 d f c +2 c \,e^{2}}{f^{2}}}}{2}}{x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}}\right )}{f^{2} \sqrt {-4 d f +e^{2}}\, \sqrt {\frac {\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 d f c +c \,e^{2}}{f^{2}}}}}{2 f^{2}}\)	\(834\)
default	\(\text {Expression too large to display}\)	\(1300\)

Input:

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

Output:

-1/2*(-f*x+2*e)*(c*x^2+a)^(1/2)/f^2+1/2/f^2*(1/f*(a*f^2-2*c*d*f+2*c*e^2)*l 
n(c^(1/2)*x+(c*x^2+a)^(1/2))/c^(1/2)+1/f^2*(a*e*f^2*(-4*d*f+e^2)^(1/2)-2*c 
*d*e*f*(-4*d*f+e^2)^(1/2)+e^3*c*(-4*d*f+e^2)^(1/2)+2*a*d*f^3-a*e^2*f^2-2*c 
*d^2*f^2+4*c*d*e^2*f-c*e^4)/(-4*d*f+e^2)^(1/2)*2^(1/2)/((-(-4*d*f+e^2)^(1/ 
2)*c*e+2*a*f^2-2*d*f*c+c*e^2)/f^2)^(1/2)*ln(((-(-4*d*f+e^2)^(1/2)*c*e+2*a* 
f^2-2*d*f*c+c*e^2)/f^2-c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2/f*(-e+(-4*d*f+e^2 
)^(1/2)))+1/2*2^(1/2)*((-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*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*(e-(-4*d*f+e^2)^(1/2) 
)/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+2*(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2 
*d*f*c+c*e^2)/f^2)^(1/2))/(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2))))+1/f^2*(a*e*f^ 
2*(-4*d*f+e^2)^(1/2)-2*c*d*e*f*(-4*d*f+e^2)^(1/2)+e^3*c*(-4*d*f+e^2)^(1/2) 
-2*a*d*f^3+a*e^2*f^2+2*c*d^2*f^2-4*c*d*e^2*f+c*e^4)/(-4*d*f+e^2)^(1/2)*2^( 
1/2)/(((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*d*f*c+c*e^2)/f^2)^(1/2)*ln((((-4*d 
*f+e^2)^(1/2)*c*e+2*a*f^2-2*d*f*c+c*e^2)/f^2-c*(e+(-4*d*f+e^2)^(1/2))/f*(x 
+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*2^(1/2)*(((-4*d*f+e^2)^(1/2)*c*e+2*a*f^ 
2-2*d*f*c+c*e^2)/f^2)^(1/2)*(4*c*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2-4*c*(e 
+(-4*d*f+e^2)^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+2*((-4*d*f+e^2)^(1 
/2)*c*e+2*a*f^2-2*d*f*c+c*e^2)/f^2)^(1/2))/(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f 
)))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2 \sqrt {a+c x^2}}{d+e x+f x^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^2*(c*x^2+a)^(1/2)/(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(-2)]

Exception generated. \[ \int \frac {x^2 \sqrt {a+c x^2}}{d+e x+f x^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2 \sqrt {a+c x^2}}{d+e x+f x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 4*sqrt(a + c*x**2)*a*d*e*f**3 + 4*sqrt(a + c*x**2)*a*e**3*f**2 + 4*sqr 
t(a + c*x**2)*c*d**2*e*f**2 + 2*sqrt(a + c*x**2)*c*d**2*f**3*x - 12*sqrt(a 
 + c*x**2)*c*d*e**3*f + 4*sqrt(a + c*x**2)*c*e**5 - sqrt(c)*log(sqrt(a + c 
*x**2) - sqrt(c)*x)*a*d**2*f**3 + 2*sqrt(c)*log(sqrt(a + c*x**2) - sqrt(c) 
*x)*c*d**3*f**2 - 2*sqrt(c)*log(sqrt(a + c*x**2) - sqrt(c)*x)*c*d**2*e**2* 
f + sqrt(c)*log(sqrt(a + c*x**2) + sqrt(c)*x)*a*d**2*f**3 - 2*sqrt(c)*log( 
sqrt(a + c*x**2) + sqrt(c)*x)*c*d**3*f**2 + 2*sqrt(c)*log(sqrt(a + c*x**2) 
 + sqrt(c)*x)*c*d**2*e**2*f - 4*int(sqrt(a + c*x**2)/(a*d + a*e*x + a*f*x* 
*2 + c*d*x**2 + c*e*x**3 + c*f*x**4),x)*a*c*d**3*f**3 + 4*int(sqrt(a + c*x 
**2)/(a*d + a*e*x + a*f*x**2 + c*d*x**2 + c*e*x**3 + c*f*x**4),x)*c**2*d** 
4*f**2 - 4*int(sqrt(a + c*x**2)/(a*d + a*e*x + a*f*x**2 + c*d*x**2 + c*e*x 
**3 + c*f*x**4),x)*c**2*d**3*e**2*f + 4*int((sqrt(a + c*x**2)*x**3)/(a*d + 
 a*e*x + a*f*x**2 + c*d*x**2 + c*e*x**3 + c*f*x**4),x)*a*c*d*e*f**4 - 4*in 
t((sqrt(a + c*x**2)*x**3)/(a*d + a*e*x + a*f*x**2 + c*d*x**2 + c*e*x**3 + 
c*f*x**4),x)*a*c*e**3*f**3 - 8*int((sqrt(a + c*x**2)*x**3)/(a*d + a*e*x + 
a*f*x**2 + c*d*x**2 + c*e*x**3 + c*f*x**4),x)*c**2*d**2*e*f**3 + 12*int((s 
qrt(a + c*x**2)*x**3)/(a*d + a*e*x + a*f*x**2 + c*d*x**2 + c*e*x**3 + c*f* 
x**4),x)*c**2*d*e**3*f**2 - 4*int((sqrt(a + c*x**2)*x**3)/(a*d + a*e*x + a 
*f*x**2 + c*d*x**2 + c*e*x**3 + c*f*x**4),x)*c**2*e**5*f + 4*int((sqrt(a + 
 c*x**2)*x**2)/(a*d + a*e*x + a*f*x**2 + c*d*x**2 + c*e*x**3 + c*f*x**4...