\(\int \frac {x^2}{a+b x^n+c x^{2 n}} \, dx\) [562]

Optimal result

Integrand size = 20, antiderivative size = 140 \[ \int \frac {x^2}{a+b x^n+c x^{2 n}} \, dx=-\frac {2 c x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {3+n}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{3 \left (b^2-4 a c-b \sqrt {b^2-4 a c}\right )}-\frac {2 c x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {3+n}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{3 \left (b^2-4 a c+b \sqrt {b^2-4 a c}\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1397, 371} \[ \int \frac {x^2}{a+b x^n+c x^{2 n}} \, dx=-\frac {2 c x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {n+3}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{3 \left (-b \sqrt {b^2-4 a c}-4 a c+b^2\right )}-\frac {2 c x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {n+3}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{3 \left (b \sqrt {b^2-4 a c}-4 a c+b^2\right )} \]

[In]

Int[x^2/(a + b*x^n + c*x^(2*n)),x]

[Out]

(-2*c*x^3*Hypergeometric2F1[1, 3/n, (3 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(3*(b^2 - 4*a*c - b*Sqrt[b
^2 - 4*a*c])) - (2*c*x^3*Hypergeometric2F1[1, 3/n, (3 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(3*(b^2 - 4
*a*c + b*Sqrt[b^2 - 4*a*c]))

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 1397

Int[((d_.)*(x_))^(m_.)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]
}, Dist[2*(c/q), Int[(d*x)^m/(b - q + 2*c*x^n), x], x] - Dist[2*(c/q), Int[(d*x)^m/(b + q + 2*c*x^n), x], x]]
/; FreeQ[{a, b, c, d, m, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(2 c) \int \frac {x^2}{b-\sqrt {b^2-4 a c}+2 c x^n} \, dx}{\sqrt {b^2-4 a c}}-\frac {(2 c) \int \frac {x^2}{b+\sqrt {b^2-4 a c}+2 c x^n} \, dx}{\sqrt {b^2-4 a c}} \\ & = -\frac {2 c x^3 \, _2F_1\left (1,\frac {3}{n};\frac {3+n}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{3 \left (b^2-4 a c-b \sqrt {b^2-4 a c}\right )}-\frac {2 c x^3 \, _2F_1\left (1,\frac {3}{n};\frac {3+n}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{3 \left (b^2-4 a c+b \sqrt {b^2-4 a c}\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.89 \[ \int \frac {x^2}{a+b x^n+c x^{2 n}} \, dx=-\frac {2}{3} c x^3 \left (\frac {1-\left (\frac {x^n}{-\frac {-b+\sqrt {b^2-4 a c}}{2 c}+x^n}\right )^{-3/n} \operatorname {Hypergeometric2F1}\left (-\frac {3}{n},-\frac {3}{n},\frac {-3+n}{n},\frac {b-\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}+2 c x^n}\right )}{b^2-4 a c-b \sqrt {b^2-4 a c}}+\frac {1-8^{-1/n} \left (\frac {c x^n}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )^{-3/n} \operatorname {Hypergeometric2F1}\left (-\frac {3}{n},-\frac {3}{n},\frac {-3+n}{n},\frac {b+\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}+2 c x^n}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )}\right ) \]

[In]

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

[Out]

(-2*c*x^3*((1 - Hypergeometric2F1[-3/n, -3/n, (-3 + n)/n, (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c] + 2*c
*x^n)]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(3/n))/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) + (1 - Hyperge
ometric2F1[-3/n, -3/n, (-3 + n)/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(8^n^(-1)*((c*x^
n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^(3/n)))/(Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c]))))/3

Maple [F]

\[\int \frac {x^{2}}{a +b \,x^{n}+c \,x^{2 n}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {x^2}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x^{2}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x^2}{a+b x^n+c x^{2 n}} \, dx=\int \frac {x^{2}}{a + b x^{n} + c x^{2 n}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {x^2}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x^{2}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {x^2}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x^{2}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{a+b x^n+c x^{2 n}} \, dx=\int \frac {x^2}{a+b\,x^n+c\,x^{2\,n}} \,d x \]

[In]

int(x^2/(a + b*x^n + c*x^(2*n)),x)

[Out]

int(x^2/(a + b*x^n + c*x^(2*n)), x)