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

Optimal. Leaf size=795 \[ -\frac{(4 e-3 f x) \left (c x^2+a\right )^{3/2}}{12 f^2}-\frac{\left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f \left (3 a f^2+4 c \left (e^2-d f\right )\right ) x\right ) \sqrt{c x^2+a}}{8 f^4}+\frac{\left (3 a^2 f^4+12 a c \left (e^2-d f\right ) f^2+8 c^2 \left (e^4-3 d f e^2+d^2 f^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{c x^2+a}}\right )}{8 \sqrt{c} f^5}-\frac{\left (a^2 \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right ) f^4+2 a c \left (e^4-\sqrt{e^2-4 d f} e^3-4 d f e^2+2 d f \sqrt{e^2-4 d f} e+2 d^2 f^2\right ) f^2+c^2 \left (e^6-\sqrt{e^2-4 d f} e^5-6 d f e^4+4 d f \sqrt{e^2-4 d f} e^3+9 d^2 f^2 e^2-3 d^2 f^2 \sqrt{e^2-4 d f} e-2 d^3 f^3\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c \left (e-\sqrt{e^2-4 d f}\right ) x}{\sqrt{2} \sqrt{2 a f^2+c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )} \sqrt{c x^2+a}}\right )}{\sqrt{2} f^5 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )}}+\frac{\left (a^2 \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right ) f^4+2 a c \left (e^4+\sqrt{e^2-4 d f} e^3-4 d f e^2-2 d f \sqrt{e^2-4 d f} e+2 d^2 f^2\right ) f^2+c^2 \left (e^6+\sqrt{e^2-4 d f} e^5-6 d f e^4-4 d f \sqrt{e^2-4 d f} e^3+9 d^2 f^2 e^2+3 d^2 f^2 \sqrt{e^2-4 d f} e-2 d^3 f^3\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c \left (e+\sqrt{e^2-4 d f}\right ) x}{\sqrt{2} \sqrt{2 a f^2+c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )} \sqrt{c x^2+a}}\right )}{\sqrt{2} f^5 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )}} \]

[Out]

-((8*e*(a*f^2 + c*(e^2 - 2*d*f)) - f*(3*a*f^2 + 4*c*(e^2 - d*f))*x)*Sqrt[a + c*x
^2])/(8*f^4) - ((4*e - 3*f*x)*(a + c*x^2)^(3/2))/(12*f^2) + ((3*a^2*f^4 + 12*a*c
*f^2*(e^2 - d*f) + 8*c^2*(e^4 - 3*d*e^2*f + d^2*f^2))*ArcTanh[(Sqrt[c]*x)/Sqrt[a
+ c*x^2]])/(8*Sqrt[c]*f^5) - ((a^2*f^4*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + 2*
a*c*f^2*(e^4 - 4*d*e^2*f + 2*d^2*f^2 - e^3*Sqrt[e^2 - 4*d*f] + 2*d*e*f*Sqrt[e^2
- 4*d*f]) + c^2*(e^6 - 6*d*e^4*f + 9*d^2*e^2*f^2 - 2*d^3*f^3 - e^5*Sqrt[e^2 - 4*
d*f] + 4*d*e^3*f*Sqrt[e^2 - 4*d*f] - 3*d^2*e*f^2*Sqrt[e^2 - 4*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*Sq
rt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*f^5*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2
+ c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) + ((a^2*f^4*(e^2 - 2*d*f + e*Sqrt[e^2
- 4*d*f]) + 2*a*c*f^2*(e^4 - 4*d*e^2*f + 2*d^2*f^2 + e^3*Sqrt[e^2 - 4*d*f] - 2*
d*e*f*Sqrt[e^2 - 4*d*f]) + c^2*(e^6 - 6*d*e^4*f + 9*d^2*e^2*f^2 - 2*d^3*f^3 + e^
5*Sqrt[e^2 - 4*d*f] - 4*d*e^3*f*Sqrt[e^2 - 4*d*f] + 3*d^2*e*f^2*Sqrt[e^2 - 4*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]*f^5*Sqrt[e^2 - 4*d*
f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])])

_______________________________________________________________________________________

Rubi [A]  time = 9.888, antiderivative size = 795, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ -\frac{(4 e-3 f x) \left (c x^2+a\right )^{3/2}}{12 f^2}-\frac{\left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f \left (3 a f^2+4 c \left (e^2-d f\right )\right ) x\right ) \sqrt{c x^2+a}}{8 f^4}+\frac{\left (3 a^2 f^4+12 a c \left (e^2-d f\right ) f^2+8 c^2 \left (e^4-3 d f e^2+d^2 f^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{c x^2+a}}\right )}{8 \sqrt{c} f^5}-\frac{\left (a^2 \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right ) f^4+2 a c \left (e^4-\sqrt{e^2-4 d f} e^3-4 d f e^2+2 d f \sqrt{e^2-4 d f} e+2 d^2 f^2\right ) f^2+c^2 \left (e^6-\sqrt{e^2-4 d f} e^5-6 d f e^4+4 d f \sqrt{e^2-4 d f} e^3+9 d^2 f^2 e^2-3 d^2 f^2 \sqrt{e^2-4 d f} e-2 d^3 f^3\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c \left (e-\sqrt{e^2-4 d f}\right ) x}{\sqrt{2} \sqrt{2 a f^2+c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )} \sqrt{c x^2+a}}\right )}{\sqrt{2} f^5 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )}}+\frac{\left (a^2 \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right ) f^4+2 a c \left (e^4+\sqrt{e^2-4 d f} e^3-4 d f e^2-2 d f \sqrt{e^2-4 d f} e+2 d^2 f^2\right ) f^2+c^2 \left (e^6+\sqrt{e^2-4 d f} e^5-6 d f e^4-4 d f \sqrt{e^2-4 d f} e^3+9 d^2 f^2 e^2+3 d^2 f^2 \sqrt{e^2-4 d f} e-2 d^3 f^3\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c \left (e+\sqrt{e^2-4 d f}\right ) x}{\sqrt{2} \sqrt{2 a f^2+c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )} \sqrt{c x^2+a}}\right )}{\sqrt{2} f^5 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )}} \]

Antiderivative was successfully verified.

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

[Out]

-((8*e*(a*f^2 + c*(e^2 - 2*d*f)) - f*(3*a*f^2 + 4*c*(e^2 - d*f))*x)*Sqrt[a + c*x
^2])/(8*f^4) - ((4*e - 3*f*x)*(a + c*x^2)^(3/2))/(12*f^2) + ((3*a^2*f^4 + 12*a*c
*f^2*(e^2 - d*f) + 8*c^2*(e^4 - 3*d*e^2*f + d^2*f^2))*ArcTanh[(Sqrt[c]*x)/Sqrt[a
+ c*x^2]])/(8*Sqrt[c]*f^5) - ((a^2*f^4*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + 2*
a*c*f^2*(e^4 - 4*d*e^2*f + 2*d^2*f^2 - e^3*Sqrt[e^2 - 4*d*f] + 2*d*e*f*Sqrt[e^2
- 4*d*f]) + c^2*(e^6 - 6*d*e^4*f + 9*d^2*e^2*f^2 - 2*d^3*f^3 - e^5*Sqrt[e^2 - 4*
d*f] + 4*d*e^3*f*Sqrt[e^2 - 4*d*f] - 3*d^2*e*f^2*Sqrt[e^2 - 4*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*Sq
rt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*f^5*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2
+ c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) + ((a^2*f^4*(e^2 - 2*d*f + e*Sqrt[e^2
- 4*d*f]) + 2*a*c*f^2*(e^4 - 4*d*e^2*f + 2*d^2*f^2 + e^3*Sqrt[e^2 - 4*d*f] - 2*
d*e*f*Sqrt[e^2 - 4*d*f]) + c^2*(e^6 - 6*d*e^4*f + 9*d^2*e^2*f^2 - 2*d^3*f^3 + e^
5*Sqrt[e^2 - 4*d*f] - 4*d*e^3*f*Sqrt[e^2 - 4*d*f] + 3*d^2*e*f^2*Sqrt[e^2 - 4*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]*f^5*Sqrt[e^2 - 4*d*
f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*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(x**2*(c*x**2+a)**(3/2)/(f*x**2+e*x+d),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 6.22491, size = 1565, normalized size = 1.97 \[ \sqrt{c x^2+a} \left (\frac{c x^3}{4 f}-\frac{c e x^2}{3 f^2}+\frac{\left (4 c e^2+5 a f^2-4 c d f\right ) x}{8 f^3}-\frac{e \left (3 c e^2+4 a f^2-6 c d f\right )}{3 f^4}\right )-\frac{\left (-c^2 e^6+c^2 \sqrt{e^2-4 d f} e^5-2 a c f^2 e^4+6 c^2 d f e^4+2 a c f^2 \sqrt{e^2-4 d f} e^3-4 c^2 d f \sqrt{e^2-4 d f} e^3-a^2 f^4 e^2+8 a c d f^3 e^2-9 c^2 d^2 f^2 e^2+a^2 f^4 \sqrt{e^2-4 d f} e-4 a c d f^3 \sqrt{e^2-4 d f} e+3 c^2 d^2 f^2 \sqrt{e^2-4 d f} e+2 a^2 d f^5-4 a c d^2 f^4+2 c^2 d^3 f^3\right ) \log \left (-e-2 f x+\sqrt{e^2-4 d f}\right )}{\sqrt{2} f^5 \sqrt{e^2-4 d f} \sqrt{c e^2-c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f}}-\frac{\left (c^2 e^6+c^2 \sqrt{e^2-4 d f} e^5+2 a c f^2 e^4-6 c^2 d f e^4+2 a c f^2 \sqrt{e^2-4 d f} e^3-4 c^2 d f \sqrt{e^2-4 d f} e^3+a^2 f^4 e^2-8 a c d f^3 e^2+9 c^2 d^2 f^2 e^2+a^2 f^4 \sqrt{e^2-4 d f} e-4 a c d f^3 \sqrt{e^2-4 d f} e+3 c^2 d^2 f^2 \sqrt{e^2-4 d f} e-2 a^2 d f^5+4 a c d^2 f^4-2 c^2 d^3 f^3\right ) \log \left (e+2 f x+\sqrt{e^2-4 d f}\right )}{\sqrt{2} f^5 \sqrt{e^2-4 d f} \sqrt{c e^2+c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f}}+\frac{\left (8 c^2 e^4+12 a c f^2 e^2-24 c^2 d f e^2+3 a^2 f^4-12 a c d f^3+8 c^2 d^2 f^2\right ) \log \left (c x+\sqrt{c} \sqrt{c x^2+a}\right )}{8 \sqrt{c} f^5}+\frac{\left (-c^2 e^6+c^2 \sqrt{e^2-4 d f} e^5-2 a c f^2 e^4+6 c^2 d f e^4+2 a c f^2 \sqrt{e^2-4 d f} e^3-4 c^2 d f \sqrt{e^2-4 d f} e^3-a^2 f^4 e^2+8 a c d f^3 e^2-9 c^2 d^2 f^2 e^2+a^2 f^4 \sqrt{e^2-4 d f} e-4 a c d f^3 \sqrt{e^2-4 d f} e+3 c^2 d^2 f^2 \sqrt{e^2-4 d f} e+2 a^2 d f^5-4 a c d^2 f^4+2 c^2 d^3 f^3\right ) \log \left (c x e^2-c \sqrt{e^2-4 d f} x e-4 c d f x+2 a f \sqrt{e^2-4 d f}+\sqrt{2} \sqrt{e^2-4 d f} \sqrt{c e^2-c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f} \sqrt{c x^2+a}\right )}{\sqrt{2} f^5 \sqrt{e^2-4 d f} \sqrt{c e^2-c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f}}+\frac{\left (c^2 e^6+c^2 \sqrt{e^2-4 d f} e^5+2 a c f^2 e^4-6 c^2 d f e^4+2 a c f^2 \sqrt{e^2-4 d f} e^3-4 c^2 d f \sqrt{e^2-4 d f} e^3+a^2 f^4 e^2-8 a c d f^3 e^2+9 c^2 d^2 f^2 e^2+a^2 f^4 \sqrt{e^2-4 d f} e-4 a c d f^3 \sqrt{e^2-4 d f} e+3 c^2 d^2 f^2 \sqrt{e^2-4 d f} e-2 a^2 d f^5+4 a c d^2 f^4-2 c^2 d^3 f^3\right ) \log \left (-c x e^2-c \sqrt{e^2-4 d f} x e+4 c d f x+2 a f \sqrt{e^2-4 d f}+\sqrt{2} \sqrt{e^2-4 d f} \sqrt{c e^2+c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f} \sqrt{c x^2+a}\right )}{\sqrt{2} f^5 \sqrt{e^2-4 d f} \sqrt{c e^2+c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + c*x^2]*(-(e*(3*c*e^2 - 6*c*d*f + 4*a*f^2))/(3*f^4) + ((4*c*e^2 - 4*c*d*
f + 5*a*f^2)*x)/(8*f^3) - (c*e*x^2)/(3*f^2) + (c*x^3)/(4*f)) - ((-(c^2*e^6) + 6*
c^2*d*e^4*f - 9*c^2*d^2*e^2*f^2 - 2*a*c*e^4*f^2 + 2*c^2*d^3*f^3 + 8*a*c*d*e^2*f^
3 - 4*a*c*d^2*f^4 - a^2*e^2*f^4 + 2*a^2*d*f^5 + c^2*e^5*Sqrt[e^2 - 4*d*f] - 4*c^
2*d*e^3*f*Sqrt[e^2 - 4*d*f] + 3*c^2*d^2*e*f^2*Sqrt[e^2 - 4*d*f] + 2*a*c*e^3*f^2*
Sqrt[e^2 - 4*d*f] - 4*a*c*d*e*f^3*Sqrt[e^2 - 4*d*f] + a^2*e*f^4*Sqrt[e^2 - 4*d*f
])*Log[-e + Sqrt[e^2 - 4*d*f] - 2*f*x])/(Sqrt[2]*f^5*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^
2 - 2*c*d*f + 2*a*f^2 - c*e*Sqrt[e^2 - 4*d*f]]) - ((c^2*e^6 - 6*c^2*d*e^4*f + 9*
c^2*d^2*e^2*f^2 + 2*a*c*e^4*f^2 - 2*c^2*d^3*f^3 - 8*a*c*d*e^2*f^3 + 4*a*c*d^2*f^
4 + a^2*e^2*f^4 - 2*a^2*d*f^5 + c^2*e^5*Sqrt[e^2 - 4*d*f] - 4*c^2*d*e^3*f*Sqrt[e
^2 - 4*d*f] + 3*c^2*d^2*e*f^2*Sqrt[e^2 - 4*d*f] + 2*a*c*e^3*f^2*Sqrt[e^2 - 4*d*f
] - 4*a*c*d*e*f^3*Sqrt[e^2 - 4*d*f] + a^2*e*f^4*Sqrt[e^2 - 4*d*f])*Log[e + Sqrt[
e^2 - 4*d*f] + 2*f*x])/(Sqrt[2]*f^5*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f + 2*a
*f^2 + c*e*Sqrt[e^2 - 4*d*f]]) + ((8*c^2*e^4 - 24*c^2*d*e^2*f + 8*c^2*d^2*f^2 +
12*a*c*e^2*f^2 - 12*a*c*d*f^3 + 3*a^2*f^4)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(
8*Sqrt[c]*f^5) + ((-(c^2*e^6) + 6*c^2*d*e^4*f - 9*c^2*d^2*e^2*f^2 - 2*a*c*e^4*f^
2 + 2*c^2*d^3*f^3 + 8*a*c*d*e^2*f^3 - 4*a*c*d^2*f^4 - a^2*e^2*f^4 + 2*a^2*d*f^5
+ c^2*e^5*Sqrt[e^2 - 4*d*f] - 4*c^2*d*e^3*f*Sqrt[e^2 - 4*d*f] + 3*c^2*d^2*e*f^2*
Sqrt[e^2 - 4*d*f] + 2*a*c*e^3*f^2*Sqrt[e^2 - 4*d*f] - 4*a*c*d*e*f^3*Sqrt[e^2 - 4
*d*f] + a^2*e*f^4*Sqrt[e^2 - 4*d*f])*Log[2*a*f*Sqrt[e^2 - 4*d*f] + c*e^2*x - 4*c
*d*f*x - c*e*Sqrt[e^2 - 4*d*f]*x + Sqrt[2]*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*
f + 2*a*f^2 - c*e*Sqrt[e^2 - 4*d*f]]*Sqrt[a + c*x^2]])/(Sqrt[2]*f^5*Sqrt[e^2 - 4
*d*f]*Sqrt[c*e^2 - 2*c*d*f + 2*a*f^2 - c*e*Sqrt[e^2 - 4*d*f]]) + ((c^2*e^6 - 6*c
^2*d*e^4*f + 9*c^2*d^2*e^2*f^2 + 2*a*c*e^4*f^2 - 2*c^2*d^3*f^3 - 8*a*c*d*e^2*f^3
+ 4*a*c*d^2*f^4 + a^2*e^2*f^4 - 2*a^2*d*f^5 + c^2*e^5*Sqrt[e^2 - 4*d*f] - 4*c^2
*d*e^3*f*Sqrt[e^2 - 4*d*f] + 3*c^2*d^2*e*f^2*Sqrt[e^2 - 4*d*f] + 2*a*c*e^3*f^2*S
qrt[e^2 - 4*d*f] - 4*a*c*d*e*f^3*Sqrt[e^2 - 4*d*f] + a^2*e*f^4*Sqrt[e^2 - 4*d*f]
)*Log[2*a*f*Sqrt[e^2 - 4*d*f] - c*e^2*x + 4*c*d*f*x - c*e*Sqrt[e^2 - 4*d*f]*x +
Sqrt[2]*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f + 2*a*f^2 + c*e*Sqrt[e^2 - 4*d*f]
]*Sqrt[a + c*x^2]])/(Sqrt[2]*f^5*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f + 2*a*f^
2 + c*e*Sqrt[e^2 - 4*d*f]])

_______________________________________________________________________________________

Maple [B]  time = 0.043, size = 19148, normalized size = 24.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} \left (a + c x^{2}\right )^{\frac{3}{2}}}{d + e x + f x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.918341, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done