\(\int \frac {(a+c x^2)^{3/2}}{d+e x+f x^2} \, dx\) [27]

Optimal result

Integrand size = 24, antiderivative size = 482 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=-\frac {c (2 e-f x) \sqrt {a+c x^2}}{2 f^2}+\frac {\sqrt {c} \left (2 c e^2-2 c d f+3 a f^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 f^3}+\frac {\left (4 a c d f^3-2 a^2 f^4+2 c^2 d f \left (e^2-d f\right )-c e \left (e-\sqrt {e^2-4 d f}\right ) \left (2 a f^2+c \left (e^2-2 d f\right )\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}-\frac {\left (4 a c d f^3-2 a^2 f^4+2 c^2 d f \left (e^2-d f\right )-c e \left (e+\sqrt {e^2-4 d f}\right ) \left (2 a f^2+c \left (e^2-2 d f\right )\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}} \] Output:

-1/2*c*(-f*x+2*e)*(c*x^2+a)^(1/2)/f^2+1/2*c^(1/2)*(3*a*f^2-2*c*d*f+2*c*e^2 
)*arctanh(c^(1/2)*x/(c*x^2+a)^(1/2))/f^3+1/2*(4*a*c*d*f^3-2*a^2*f^4+2*c^2* 
d*f*(-d*f+e^2)-c*e*(e-(-4*d*f+e^2)^(1/2))*(2*a*f^2+c*(-2*d*f+e^2)))*arctan 
h(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*(4*a*c*d*f^3-2*a^2* 
f^4+2*c^2*d*f*(-d*f+e^2)-c*e*(e+(-4*d*f+e^2)^(1/2))*(2*a*f^2+c*(-2*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 = 0.77 (sec) , antiderivative size = 542, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=\frac {c f (-2 e+f x) \sqrt {a+c x^2}+\sqrt {c} \left (-2 c e^2+2 c d f-3 a f^2\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )+2 \text {RootSum}\left [a^2 f+2 a \sqrt {c} e \text {$\#$1}+4 c d \text {$\#$1}^2-2 a f \text {$\#$1}^2-2 \sqrt {c} e \text {$\#$1}^3+f \text {$\#$1}^4\&,\frac {a c^2 e^3 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )-2 a c^2 d e f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )+2 a^2 c e f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )+2 c^{5/2} d e^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 c^{5/2} d^2 f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+4 a c^{3/2} d f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 a^2 \sqrt {c} f^3 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-c^2 e^3 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+2 c^2 d e f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-2 a c e f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{a \sqrt {c} e+4 c d \text {$\#$1}-2 a f \text {$\#$1}-3 \sqrt {c} e \text {$\#$1}^2+2 f \text {$\#$1}^3}\&\right ]}{2 f^3} \] Input:

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

Output:

(c*f*(-2*e + f*x)*Sqrt[a + c*x^2] + Sqrt[c]*(-2*c*e^2 + 2*c*d*f - 3*a*f^2) 
*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]] + 2*RootSum[a^2*f + 2*a*Sqrt[c]*e*#1 
+ 4*c*d*#1^2 - 2*a*f*#1^2 - 2*Sqrt[c]*e*#1^3 + f*#1^4 & , (a*c^2*e^3*Log[- 
(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1] - 2*a*c^2*d*e*f*Log[-(Sqrt[c]*x) + Sqr 
t[a + c*x^2] - #1] + 2*a^2*c*e*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1 
] + 2*c^(5/2)*d*e^2*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1 - 2*c^(5/2 
)*d^2*f*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1 + 4*a*c^(3/2)*d*f^2*Lo 
g[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1 - 2*a^2*Sqrt[c]*f^3*Log[-(Sqrt[c 
]*x) + Sqrt[a + c*x^2] - #1]*#1 - c^2*e^3*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^ 
2] - #1]*#1^2 + 2*c^2*d*e*f*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1^2 
- 2*a*c*e*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1^2)/(a*Sqrt[c]*e 
+ 4*c*d*#1 - 2*a*f*#1 - 3*Sqrt[c]*e*#1^2 + 2*f*#1^3) & ])/(2*f^3)
 

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1310, 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.

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

\(\Big \downarrow \) 1310

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

\(\Big \downarrow \) 2145

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {2 \int \frac {-a^2 f^3+2 a c d f^2+c^2 d \left (e^2-d f\right )+c e \left (2 a f^2+c \left (e^2-2 d f\right )\right ) x}{\sqrt {c x^2+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a f^2+2 c \left (e^2-d f\right )\right )}{f}}{2 f^2}-\frac {c \sqrt {a+c x^2} (2 e-f x)}{2 f^2}\)

\(\Big \downarrow \) 1367

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

\(\Big \downarrow \) 488

\(\displaystyle -\frac {\frac {2 \left (\frac {\left (-2 a^2 f^4-c e \left (\sqrt {e^2-4 d f}+e\right ) \left (2 a f^2+c \left (e^2-2 d f\right )\right )+4 a c d f^3+2 c^2 d f \left (e^2-d f\right )\right ) \int \frac {1}{4 a f^2+c \left (e+\sqrt {e^2-4 d f}\right )^2-\frac {\left (2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x\right )^2}{c x^2+a}}d\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {c x^2+a}}}{\sqrt {e^2-4 d f}}-\frac {\left (-2 a^2 f^4-c e \left (e-\sqrt {e^2-4 d f}\right ) \left (2 a f^2+c \left (e^2-2 d f\right )\right )+4 a c d f^3+2 c^2 d f \left (e^2-d f\right )\right ) \int \frac {1}{4 a f^2+c \left (e-\sqrt {e^2-4 d f}\right )^2-\frac {\left (2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x\right )^2}{c x^2+a}}d\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {c x^2+a}}}{\sqrt {e^2-4 d f}}\right )}{f}-\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a f^2+2 c \left (e^2-d f\right )\right )}{f}}{2 f^2}-\frac {c \sqrt {a+c x^2} (2 e-f x)}{2 f^2}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {2 \left (\frac {\left (-2 a^2 f^4-c e \left (\sqrt {e^2-4 d f}+e\right ) \left (2 a f^2+c \left (e^2-2 d f\right )\right )+4 a c d f^3+2 c^2 d f \left (e^2-d f\right )\right ) \text {arctanh}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {\left (-2 a^2 f^4-c e \left (e-\sqrt {e^2-4 d f}\right ) \left (2 a f^2+c \left (e^2-2 d f\right )\right )+4 a c d f^3+2 c^2 d f \left (e^2-d f\right )\right ) \text {arctanh}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{f}-\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a f^2+2 c \left (e^2-d f\right )\right )}{f}}{2 f^2}-\frac {c \sqrt {a+c x^2} (2 e-f x)}{2 f^2}\)

Input:

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

Output:

-1/2*(c*(2*e - f*x)*Sqrt[a + c*x^2])/f^2 - (-((Sqrt[c]*(3*a*f^2 + 2*c*(e^2 
 - d*f))*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/f) + (2*(-(((4*a*c*d*f^3 - 
2*a^2*f^4 + 2*c^2*d*f*(e^2 - d*f) - c*e*(e - Sqrt[e^2 - 4*d*f])*(2*a*f^2 + 
 c*(e^2 - 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])])) + ((4*a*c*d*f^3 - 2*a^2*f^4 + 2*c^2*d*f*(e^2 - d*f) - c*e*(e + Sqr 
t[e^2 - 4*d*f])*(2*a*f^2 + c*(e^2 - 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])])))/f)/(2*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 1310

Int[((a_.) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x 
_Symbol] :> Simp[(-c)*(e*(2*p + q) - 2*f*(p + q)*x)*(a + c*x^2)^(p - 1)*((d 
 + e*x + f*x^2)^(q + 1)/(2*f^2*(p + q)*(2*p + 2*q + 1))), x] - Simp[1/(2*f^ 
2*(p + q)*(2*p + 2*q + 1))   Int[(a + c*x^2)^(p - 2)*(d + e*x + f*x^2)^q*Si 
mp[(-a)*c*e^2*(1 - p)*(2*p + q) + a*(p + q)*(-2*a*f^2*(2*p + 2*q + 1) + c*( 
2*d*f - e^2*(2*p + q))) + (2*(c*d - a*f)*(c*e)*(1 - p)*(2*p + q) + 4*a*c*e* 
f*(1 - p)*(p + q))*x + (p*c^2*e^2*(1 - p) - c*(p + q)*(2*a*f^2*(4*p + 2*q - 
 1) + c*(2*d*f*(1 - 2*p) + e^2*(3*p + q - 1))))*x^2, x], x], x] /; FreeQ[{a 
, c, d, e, f, q}, x] && NeQ[e^2 - 4*d*f, 0] && GtQ[p, 1] && NeQ[p + q, 0] & 
& NeQ[2*p + 2*q + 1, 0] &&  !IGtQ[p, 0] &&  !IGtQ[q, 0]
 

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 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. \(880\) vs. \(2(433)=866\).

Time = 2.23 (sec) , antiderivative size = 881, normalized size of antiderivative = 1.83

method	result	size
risch	\(-\frac {c \left (-f x +2 e \right ) \sqrt {c \,x^{2}+a}}{2 f^{2}}+\frac {\frac {\sqrt {c}\, \left (3 a \,f^{2}-2 d f c +2 c \,e^{2}\right ) \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{f}+\frac {\left (2 a c e \,f^{2} \sqrt {-4 d f +e^{2}}-2 c^{2} d e f \sqrt {-4 d f +e^{2}}+c^{2} e^{3} \sqrt {-4 d f +e^{2}}+2 a^{2} f^{4}-4 a c d \,f^{3}+2 a c \,e^{2} f^{2}+2 f^{2} c^{2} d^{2}-4 c^{2} d \,e^{2} f +c^{2} 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 (2 a c e \,f^{2} \sqrt {-4 d f +e^{2}}-2 c^{2} d e f \sqrt {-4 d f +e^{2}}+c^{2} e^{3} \sqrt {-4 d f +e^{2}}-2 a^{2} f^{4}+4 a c d \,f^{3}-2 a c \,e^{2} f^{2}-2 f^{2} c^{2} d^{2}+4 c^{2} d \,e^{2} f -c^{2} 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}}\)	\(881\)
default	\(\text {Expression too large to display}\)	\(2261\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((c*x^2+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(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 8*sqrt(a + c*x**2)*a*d*e*f**3 + 8*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 - 3*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 + 3*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**2*d**2*f**4 - 8*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 + 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)*a*c*d*e*f**4 - 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)*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((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*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...