3.583 \(\int \frac{x^2}{\sqrt{a+b x^n+c x^{2 n}}} \, dx\)

Optimal. Leaf size=148 \[ \frac{x^3 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{3}{n};\frac{1}{2},\frac{1}{2};\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{3 \sqrt{a+b x^n+c x^{2 n}}} \]

[Out]

(x^3*Sqrt[1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^n)/(b + Sqrt[b^
2 - 4*a*c])]*AppellF1[3/n, 1/2, 1/2, (3 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c
]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(3*Sqrt[a + b*x^n + c*x^(2*n)])

_______________________________________________________________________________________

Rubi [A]  time = 0.475795, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{x^3 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{3}{n};\frac{1}{2},\frac{1}{2};\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{3 \sqrt{a+b x^n+c x^{2 n}}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/Sqrt[a + b*x^n + c*x^(2*n)],x]

[Out]

(x^3*Sqrt[1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^n)/(b + Sqrt[b^
2 - 4*a*c])]*AppellF1[3/n, 1/2, 1/2, (3 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c
]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(3*Sqrt[a + b*x^n + c*x^(2*n)])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 33.1732, size = 126, normalized size = 0.85 \[ \frac{x^{3} \sqrt{a + b x^{n} + c x^{2 n}} \operatorname{appellf_{1}}{\left (\frac{3}{n},\frac{1}{2},\frac{1}{2},\frac{n + 3}{n},- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{3 a \sqrt{\frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(a+b*x**n+c*x**(2*n))**(1/2),x)

[Out]

x**3*sqrt(a + b*x**n + c*x**(2*n))*appellf1(3/n, 1/2, 1/2, (n + 3)/n, -2*c*x**n/
(b - sqrt(-4*a*c + b**2)), -2*c*x**n/(b + sqrt(-4*a*c + b**2)))/(3*a*sqrt(2*c*x*
*n/(b - sqrt(-4*a*c + b**2)) + 1)*sqrt(2*c*x**n/(b + sqrt(-4*a*c + b**2)) + 1))

_______________________________________________________________________________________

Mathematica [B]  time = 0.483157, size = 417, normalized size = 2.82 \[ -\frac{4 a^2 (n+3) x^3 \left (-\sqrt{b^2-4 a c}+b+2 c x^n\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^n\right ) F_1\left (\frac{3}{n};\frac{1}{2},\frac{1}{2};\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )}{3 \left (b-\sqrt{b^2-4 a c}\right ) \left (\sqrt{b^2-4 a c}+b\right ) \left (a+x^n \left (b+c x^n\right )\right )^{3/2} \left (n x^n \left (\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{n+3}{n};\frac{1}{2},\frac{3}{2};2+\frac{3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{n+3}{n};\frac{3}{2},\frac{1}{2};2+\frac{3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )-4 a (n+3) F_1\left (\frac{3}{n};\frac{1}{2},\frac{1}{2};\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^2/Sqrt[a + b*x^n + c*x^(2*n)],x]

[Out]

(-4*a^2*(3 + n)*x^3*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)*(b + Sqrt[b^2 - 4*a*c] + 2
*c*x^n)*AppellF1[3/n, 1/2, 1/2, (3 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (
2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/(3*(b - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4
*a*c])*(a + x^n*(b + c*x^n))^(3/2)*(-4*a*(3 + n)*AppellF1[3/n, 1/2, 1/2, (3 + n)
/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + n*
x^n*((b + Sqrt[b^2 - 4*a*c])*AppellF1[(3 + n)/n, 1/2, 3/2, 2 + 3/n, (-2*c*x^n)/(
b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*
a*c])*AppellF1[(3 + n)/n, 3/2, 1/2, 2 + 3/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]),
(2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])))

_______________________________________________________________________________________

Maple [F]  time = 0.029, size = 0, normalized size = 0. \[ \int{{x}^{2}{\frac{1}{\sqrt{a+b{x}^{n}+c{x}^{2\,n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b*x^n+c*x^(2*n))^(1/2),x)

[Out]

int(x^2/(a+b*x^n+c*x^(2*n))^(1/2),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{c x^{2 \, n} + b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(c*x^(2*n) + b*x^n + a),x, algorithm="maxima")

[Out]

integrate(x^2/sqrt(c*x^(2*n) + b*x^n + a), x)

_______________________________________________________________________________________

Fricas [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(x^2/sqrt(c*x^(2*n) + b*x^n + a),x, algorithm="fricas")

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{a + b x^{n} + c x^{2 n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(a+b*x**n+c*x**(2*n))**(1/2),x)

[Out]

Integral(x**2/sqrt(a + b*x**n + c*x**(2*n)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{c x^{2 \, n} + b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(c*x^(2*n) + b*x^n + a),x, algorithm="giac")

[Out]

integrate(x^2/sqrt(c*x^(2*n) + b*x^n + a), x)