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

Optimal. Leaf size=679 \[ \frac{\left (\left (e-\sqrt{e^2-4 d f}\right ) (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-2 f \left (-f^2 \left (b^2 d-a^2 f\right )+2 c d f (b e-a f)+c^2 (-d) \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (e-\sqrt{e^2-4 d f}\right )\right )-b \left (e-\sqrt{e^2-4 d f}\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{\left (\left (\sqrt{e^2-4 d f}+e\right ) (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-2 f \left (-f^2 \left (b^2 d-a^2 f\right )+2 c d f (b e-a f)+c^2 (-d) \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (\sqrt{e^2-4 d f}+e\right )\right )-b \left (\sqrt{e^2-4 d f}+e\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )}{8 \sqrt{c} f^3}-\frac{\sqrt{a+b x+c x^2} (-5 b f+4 c e-2 c f x)}{4 f^2} \]

[Out]

-((4*c*e - 5*b*f - 2*c*f*x)*Sqrt[a + b*x + c*x^2])/(4*f^2) + ((3*b^2*f^2 - 12*c*
f*(b*e - a*f) + 8*c^2*(e^2 - d*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x +
c*x^2])])/(8*Sqrt[c]*f^3) + (((c*e - b*f)*(e - Sqrt[e^2 - 4*d*f])*(f*(b*e - 2*a
*f) - c*(e^2 - 2*d*f)) - 2*f*(2*c*d*f*(b*e - a*f) - f^2*(b^2*d - a^2*f) - c^2*d*
(e^2 - 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]*f^3*Sqrt[e^2 - 4*d*f]*Sqr
t[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*e -
b*f)*(e + Sqrt[e^2 - 4*d*f])*(f*(b*e - 2*a*f) - c*(e^2 - 2*d*f)) - 2*f*(2*c*d*f
*(b*e - a*f) - f^2*(b^2*d - a^2*f) - c^2*d*(e^2 - 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]*f^3*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]])

_______________________________________________________________________________________

Rubi [A]  time = 18.25, antiderivative size = 678, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{\left (\left (e-\sqrt{e^2-4 d f}\right ) (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-2 f \left (-f^2 \left (b^2 d-a^2 f\right )+2 c d f (b e-a f)+c^2 (-d) \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (e-\sqrt{e^2-4 d f}\right )\right )-b \left (e-\sqrt{e^2-4 d f}\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac{\left (-2 f^3 \left (b^2 d-a^2 f\right )-\left (\sqrt{e^2-4 d f}+e\right ) (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+4 c d f^2 (b e-a f)-2 c^2 d f \left (e^2-d f\right )\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (\sqrt{e^2-4 d f}+e\right )\right )-b \left (\sqrt{e^2-4 d f}+e\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )}{8 \sqrt{c} f^3}-\frac{\sqrt{a+b x+c x^2} (-5 b f+4 c e-2 c f x)}{4 f^2} \]

Warning: Unable to verify antiderivative.

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

[Out]

-((4*c*e - 5*b*f - 2*c*f*x)*Sqrt[a + b*x + c*x^2])/(4*f^2) + ((3*b^2*f^2 - 12*c*
f*(b*e - a*f) + 8*c^2*(e^2 - d*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x +
c*x^2])])/(8*Sqrt[c]*f^3) + (((c*e - b*f)*(e - Sqrt[e^2 - 4*d*f])*(f*(b*e - 2*a
*f) - c*(e^2 - 2*d*f)) - 2*f*(2*c*d*f*(b*e - a*f) - f^2*(b^2*d - a^2*f) - c^2*d*
(e^2 - 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]*f^3*Sqrt[e^2 - 4*d*f]*Sqr
t[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]) + ((4*c*d*
f^2*(b*e - a*f) - 2*f^3*(b^2*d - a^2*f) - 2*c^2*d*f*(e^2 - d*f) - (c*e - b*f)*(e
+ Sqrt[e^2 - 4*d*f])*(f*(b*e - 2*a*f) - c*(e^2 - 2*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]*f^3*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]])

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [B]  time = 6.26447, size = 1934, normalized size = 2.85 \[ -\frac{\left (-c^2 e^4+2 b c f e^3+c^2 \sqrt{e^2-4 d f} e^3-b^2 f^2 e^2-2 a c f^2 e^2+4 c^2 d f e^2-2 b c f \sqrt{e^2-4 d f} e^2+2 a b f^3 e-6 b c d f^2 e+b^2 f^2 \sqrt{e^2-4 d f} e+2 a c f^2 \sqrt{e^2-4 d f} e-2 c^2 d f \sqrt{e^2-4 d f} e-2 a^2 f^4+2 b^2 d f^3+4 a c d f^3-2 c^2 d^2 f^2-2 a b f^3 \sqrt{e^2-4 d f}+2 b c d f^2 \sqrt{e^2-4 d f}\right ) \log \left (-e-2 f x+\sqrt{e^2-4 d f}\right ) (a+x (b+c x))^{3/2}}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{c e^2-b f e-c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt{e^2-4 d f}} \left (c x^2+b x+a\right )^{3/2}}-\frac{\left (c^2 e^4-2 b c f e^3+c^2 \sqrt{e^2-4 d f} e^3+b^2 f^2 e^2+2 a c f^2 e^2-4 c^2 d f e^2-2 b c f \sqrt{e^2-4 d f} e^2-2 a b f^3 e+6 b c d f^2 e+b^2 f^2 \sqrt{e^2-4 d f} e+2 a c f^2 \sqrt{e^2-4 d f} e-2 c^2 d f \sqrt{e^2-4 d f} e+2 a^2 f^4-2 b^2 d f^3-4 a c d f^3+2 c^2 d^2 f^2-2 a b f^3 \sqrt{e^2-4 d f}+2 b c d f^2 \sqrt{e^2-4 d f}\right ) \log \left (e+2 f x+\sqrt{e^2-4 d f}\right ) (a+x (b+c x))^{3/2}}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{c e^2-b f e+c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt{e^2-4 d f}} \left (c x^2+b x+a\right )^{3/2}}+\frac{\left (8 e^2 c^2-8 d f c^2+12 a f^2 c-12 b e f c+3 b^2 f^2\right ) \log \left (b+2 c x+2 \sqrt{c} \sqrt{c x^2+b x+a}\right ) (a+x (b+c x))^{3/2}}{8 \sqrt{c} f^3 \left (c x^2+b x+a\right )^{3/2}}+\frac{\left (c^2 e^4-2 b c f e^3+c^2 \sqrt{e^2-4 d f} e^3+b^2 f^2 e^2+2 a c f^2 e^2-4 c^2 d f e^2-2 b c f \sqrt{e^2-4 d f} e^2-2 a b f^3 e+6 b c d f^2 e+b^2 f^2 \sqrt{e^2-4 d f} e+2 a c f^2 \sqrt{e^2-4 d f} e-2 c^2 d f \sqrt{e^2-4 d f} e+2 a^2 f^4-2 b^2 d f^3-4 a c d f^3+2 c^2 d^2 f^2-2 a b f^3 \sqrt{e^2-4 d f}+2 b c d f^2 \sqrt{e^2-4 d f}\right ) \log \left (-b e^2-2 c x e^2-2 c \sqrt{e^2-4 d f} x e-b \sqrt{e^2-4 d f} e+4 b d f+8 c d f x+2 b f \sqrt{e^2-4 d f} x+4 a f \sqrt{e^2-4 d f}+2 \sqrt{2} \sqrt{e^2-4 d f} \sqrt{c e^2-b f e+c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}\right ) (a+x (b+c x))^{3/2}}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{c e^2-b f e+c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt{e^2-4 d f}} \left (c x^2+b x+a\right )^{3/2}}+\frac{\left (-c^2 e^4+2 b c f e^3+c^2 \sqrt{e^2-4 d f} e^3-b^2 f^2 e^2-2 a c f^2 e^2+4 c^2 d f e^2-2 b c f \sqrt{e^2-4 d f} e^2+2 a b f^3 e-6 b c d f^2 e+b^2 f^2 \sqrt{e^2-4 d f} e+2 a c f^2 \sqrt{e^2-4 d f} e-2 c^2 d f \sqrt{e^2-4 d f} e-2 a^2 f^4+2 b^2 d f^3+4 a c d f^3-2 c^2 d^2 f^2-2 a b f^3 \sqrt{e^2-4 d f}+2 b c d f^2 \sqrt{e^2-4 d f}\right ) \log \left (b e^2+2 c x e^2-2 c \sqrt{e^2-4 d f} x e-b \sqrt{e^2-4 d f} e-4 b d f-8 c d f x+2 b f \sqrt{e^2-4 d f} x+4 a f \sqrt{e^2-4 d f}+2 \sqrt{2} \sqrt{e^2-4 d f} \sqrt{c e^2-b f e-c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}\right ) (a+x (b+c x))^{3/2}}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{c e^2-b f e-c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt{e^2-4 d f}} \left (c x^2+b x+a\right )^{3/2}}+\frac{\left (\frac{5 b f-4 c e}{4 f^2}+\frac{c x}{2 f}\right ) (a+x (b+c x))^{3/2}}{c x^2+b x+a} \]

Warning: Unable to verify antiderivative.

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

[Out]

(((-4*c*e + 5*b*f)/(4*f^2) + (c*x)/(2*f))*(a + x*(b + c*x))^(3/2))/(a + b*x + c*
x^2) - ((-(c^2*e^4) + 4*c^2*d*e^2*f + 2*b*c*e^3*f - 2*c^2*d^2*f^2 - 6*b*c*d*e*f^
2 - b^2*e^2*f^2 - 2*a*c*e^2*f^2 + 2*b^2*d*f^3 + 4*a*c*d*f^3 + 2*a*b*e*f^3 - 2*a^
2*f^4 + c^2*e^3*Sqrt[e^2 - 4*d*f] - 2*c^2*d*e*f*Sqrt[e^2 - 4*d*f] - 2*b*c*e^2*f*
Sqrt[e^2 - 4*d*f] + 2*b*c*d*f^2*Sqrt[e^2 - 4*d*f] + b^2*e*f^2*Sqrt[e^2 - 4*d*f]
+ 2*a*c*e*f^2*Sqrt[e^2 - 4*d*f] - 2*a*b*f^3*Sqrt[e^2 - 4*d*f])*(a + x*(b + c*x))
^(3/2)*Log[-e + Sqrt[e^2 - 4*d*f] - 2*f*x])/(Sqrt[2]*f^3*Sqrt[e^2 - 4*d*f]*Sqrt[
c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - c*e*Sqrt[e^2 - 4*d*f] + b*f*Sqrt[e^2 - 4*d*f
]]*(a + b*x + c*x^2)^(3/2)) - ((c^2*e^4 - 4*c^2*d*e^2*f - 2*b*c*e^3*f + 2*c^2*d^
2*f^2 + 6*b*c*d*e*f^2 + b^2*e^2*f^2 + 2*a*c*e^2*f^2 - 2*b^2*d*f^3 - 4*a*c*d*f^3
- 2*a*b*e*f^3 + 2*a^2*f^4 + c^2*e^3*Sqrt[e^2 - 4*d*f] - 2*c^2*d*e*f*Sqrt[e^2 - 4
*d*f] - 2*b*c*e^2*f*Sqrt[e^2 - 4*d*f] + 2*b*c*d*f^2*Sqrt[e^2 - 4*d*f] + b^2*e*f^
2*Sqrt[e^2 - 4*d*f] + 2*a*c*e*f^2*Sqrt[e^2 - 4*d*f] - 2*a*b*f^3*Sqrt[e^2 - 4*d*f
])*(a + x*(b + c*x))^(3/2)*Log[e + Sqrt[e^2 - 4*d*f] + 2*f*x])/(Sqrt[2]*f^3*Sqrt
[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + c*e*Sqrt[e^2 - 4*d*f] - b
*f*Sqrt[e^2 - 4*d*f]]*(a + b*x + c*x^2)^(3/2)) + ((8*c^2*e^2 - 8*c^2*d*f - 12*b*
c*e*f + 3*b^2*f^2 + 12*a*c*f^2)*(a + x*(b + c*x))^(3/2)*Log[b + 2*c*x + 2*Sqrt[c
]*Sqrt[a + b*x + c*x^2]])/(8*Sqrt[c]*f^3*(a + b*x + c*x^2)^(3/2)) + ((c^2*e^4 -
4*c^2*d*e^2*f - 2*b*c*e^3*f + 2*c^2*d^2*f^2 + 6*b*c*d*e*f^2 + b^2*e^2*f^2 + 2*a*
c*e^2*f^2 - 2*b^2*d*f^3 - 4*a*c*d*f^3 - 2*a*b*e*f^3 + 2*a^2*f^4 + c^2*e^3*Sqrt[e
^2 - 4*d*f] - 2*c^2*d*e*f*Sqrt[e^2 - 4*d*f] - 2*b*c*e^2*f*Sqrt[e^2 - 4*d*f] + 2*
b*c*d*f^2*Sqrt[e^2 - 4*d*f] + b^2*e*f^2*Sqrt[e^2 - 4*d*f] + 2*a*c*e*f^2*Sqrt[e^2
- 4*d*f] - 2*a*b*f^3*Sqrt[e^2 - 4*d*f])*(a + x*(b + c*x))^(3/2)*Log[-(b*e^2) +
4*b*d*f - b*e*Sqrt[e^2 - 4*d*f] + 4*a*f*Sqrt[e^2 - 4*d*f] - 2*c*e^2*x + 8*c*d*f*
x - 2*c*e*Sqrt[e^2 - 4*d*f]*x + 2*b*f*Sqrt[e^2 - 4*d*f]*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*Sqrt[e^2 - 4*d*f] - b*f*Sqr
t[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2]])/(Sqrt[2]*f^3*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^
2 - 2*c*d*f - b*e*f + 2*a*f^2 + c*e*Sqrt[e^2 - 4*d*f] - b*f*Sqrt[e^2 - 4*d*f]]*(
a + b*x + c*x^2)^(3/2)) + ((-(c^2*e^4) + 4*c^2*d*e^2*f + 2*b*c*e^3*f - 2*c^2*d^2
*f^2 - 6*b*c*d*e*f^2 - b^2*e^2*f^2 - 2*a*c*e^2*f^2 + 2*b^2*d*f^3 + 4*a*c*d*f^3 +
2*a*b*e*f^3 - 2*a^2*f^4 + c^2*e^3*Sqrt[e^2 - 4*d*f] - 2*c^2*d*e*f*Sqrt[e^2 - 4*
d*f] - 2*b*c*e^2*f*Sqrt[e^2 - 4*d*f] + 2*b*c*d*f^2*Sqrt[e^2 - 4*d*f] + b^2*e*f^2
*Sqrt[e^2 - 4*d*f] + 2*a*c*e*f^2*Sqrt[e^2 - 4*d*f] - 2*a*b*f^3*Sqrt[e^2 - 4*d*f]
)*(a + x*(b + c*x))^(3/2)*Log[b*e^2 - 4*b*d*f - b*e*Sqrt[e^2 - 4*d*f] + 4*a*f*Sq
rt[e^2 - 4*d*f] + 2*c*e^2*x - 8*c*d*f*x - 2*c*e*Sqrt[e^2 - 4*d*f]*x + 2*b*f*Sqrt
[e^2 - 4*d*f]*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*Sqrt[e^2 - 4*d*f] + b*f*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2]])/(S
qrt[2]*f^3*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - c*e*Sqrt[e
^2 - 4*d*f] + b*f*Sqrt[e^2 - 4*d*f]]*(a + b*x + c*x^2)^(3/2))

_______________________________________________________________________________________

Maple [B]  time = 0.045, size = 22523, normalized size = 33.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError