3.558 \(\int \frac {x^{-1+\frac {n}{2}}}{a+b x^n+c x^{2 n}} \, dx\)

Optimal. Leaf size=169 \[ \frac {2 \sqrt {2} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x^{n/2}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{n \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {2 \sqrt {2} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x^{n/2}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{n \sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}} \]

[Out]

2*arctan(x^(1/2*n)*2^(1/2)*c^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)*c^(1/2)/n/(-4*a*c+b^2)^(1/2)/(b-(-4*a
*c+b^2)^(1/2))^(1/2)-2*arctan(x^(1/2*n)*2^(1/2)*c^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)*c^(1/2)/n/(-4*a*
c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)

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Rubi [A]  time = 0.19, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1381, 1093, 205} \[ \frac {2 \sqrt {2} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x^{n/2}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{n \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {2 \sqrt {2} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x^{n/2}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{n \sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[2]*Sqrt[c]*ArcTan[(Sqrt[2]*Sqrt[c]*x^(n/2))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b -
Sqrt[b^2 - 4*a*c]]*n) - (2*Sqrt[2]*Sqrt[c]*ArcTan[(Sqrt[2]*Sqrt[c]*x^(n/2))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqr
t[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*n)

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 1093

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/
2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && PosQ[b^2 - 4*a*c]

Rule 1381

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/(m + 1), Subst[Int[(a +
b*x^Simplify[n/(m + 1)] + c*x^Simplify[(2*n)/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, c, m, n, p}, x
] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[n/(m + 1)]] &&  !IntegerQ[n]

Rubi steps

\begin {align*} \int \frac {x^{-1+\frac {n}{2}}}{a+b x^n+c x^{2 n}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{a+b x^2+c x^4} \, dx,x,x^{n/2}\right )}{n}\\ &=\frac {(2 c) \operatorname {Subst}\left (\int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,x^{n/2}\right )}{\sqrt {b^2-4 a c} n}-\frac {(2 c) \operatorname {Subst}\left (\int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,x^{n/2}\right )}{\sqrt {b^2-4 a c} n}\\ &=\frac {2 \sqrt {2} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x^{n/2}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}} n}-\frac {2 \sqrt {2} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x^{n/2}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}} n}\\ \end {align*}

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Mathematica [A]  time = 0.26, size = 145, normalized size = 0.86 \[ \frac {2 \sqrt {2} \sqrt {c} \left (\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x^{n/2}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x^{n/2}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{n \sqrt {b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[2]*Sqrt[c]*(ArcTan[(Sqrt[2]*Sqrt[c]*x^(n/2))/Sqrt[b - Sqrt[b^2 - 4*a*c]]]/Sqrt[b - Sqrt[b^2 - 4*a*c]]
- ArcTan[(Sqrt[2]*Sqrt[c]*x^(n/2))/Sqrt[b + Sqrt[b^2 - 4*a*c]]]/Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(Sqrt[b^2 - 4*a*
c]*n)

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fricas [B]  time = 0.71, size = 801, normalized size = 4.74 \[ \frac {1}{2} \, \sqrt {2} \sqrt {-\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} + b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}} \log \left (\frac {4 \, c x x^{\frac {1}{2} \, n - 1} + \sqrt {2} {\left ({\left (a b^{3} - 4 \, a^{2} b c\right )} n^{3} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} - {\left (b^{2} - 4 \, a c\right )} n\right )} \sqrt {-\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} + b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}}}{x}\right ) - \frac {1}{2} \, \sqrt {2} \sqrt {-\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} + b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}} \log \left (\frac {4 \, c x x^{\frac {1}{2} \, n - 1} - \sqrt {2} {\left ({\left (a b^{3} - 4 \, a^{2} b c\right )} n^{3} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} - {\left (b^{2} - 4 \, a c\right )} n\right )} \sqrt {-\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} + b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}}}{x}\right ) - \frac {1}{2} \, \sqrt {2} \sqrt {\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} - b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}} \log \left (\frac {4 \, c x x^{\frac {1}{2} \, n - 1} + \sqrt {2} {\left ({\left (a b^{3} - 4 \, a^{2} b c\right )} n^{3} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} + {\left (b^{2} - 4 \, a c\right )} n\right )} \sqrt {\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} - b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}}}{x}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} - b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}} \log \left (\frac {4 \, c x x^{\frac {1}{2} \, n - 1} - \sqrt {2} {\left ({\left (a b^{3} - 4 \, a^{2} b c\right )} n^{3} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} + {\left (b^{2} - 4 \, a c\right )} n\right )} \sqrt {\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} - b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+1/2*n)/(a+b*x^n+c*x^(2*n)),x, algorithm="fricas")

[Out]

1/2*sqrt(2)*sqrt(-((a*b^2 - 4*a^2*c)*n^2*sqrt(1/((a^2*b^2 - 4*a^3*c)*n^4)) + b)/((a*b^2 - 4*a^2*c)*n^2))*log((
4*c*x*x^(1/2*n - 1) + sqrt(2)*((a*b^3 - 4*a^2*b*c)*n^3*sqrt(1/((a^2*b^2 - 4*a^3*c)*n^4)) - (b^2 - 4*a*c)*n)*sq
rt(-((a*b^2 - 4*a^2*c)*n^2*sqrt(1/((a^2*b^2 - 4*a^3*c)*n^4)) + b)/((a*b^2 - 4*a^2*c)*n^2)))/x) - 1/2*sqrt(2)*s
qrt(-((a*b^2 - 4*a^2*c)*n^2*sqrt(1/((a^2*b^2 - 4*a^3*c)*n^4)) + b)/((a*b^2 - 4*a^2*c)*n^2))*log((4*c*x*x^(1/2*
n - 1) - sqrt(2)*((a*b^3 - 4*a^2*b*c)*n^3*sqrt(1/((a^2*b^2 - 4*a^3*c)*n^4)) - (b^2 - 4*a*c)*n)*sqrt(-((a*b^2 -
4*a^2*c)*n^2*sqrt(1/((a^2*b^2 - 4*a^3*c)*n^4)) + b)/((a*b^2 - 4*a^2*c)*n^2)))/x) - 1/2*sqrt(2)*sqrt(((a*b^2 -
4*a^2*c)*n^2*sqrt(1/((a^2*b^2 - 4*a^3*c)*n^4)) - b)/((a*b^2 - 4*a^2*c)*n^2))*log((4*c*x*x^(1/2*n - 1) + sqrt(
2)*((a*b^3 - 4*a^2*b*c)*n^3*sqrt(1/((a^2*b^2 - 4*a^3*c)*n^4)) + (b^2 - 4*a*c)*n)*sqrt(((a*b^2 - 4*a^2*c)*n^2*s
qrt(1/((a^2*b^2 - 4*a^3*c)*n^4)) - b)/((a*b^2 - 4*a^2*c)*n^2)))/x) + 1/2*sqrt(2)*sqrt(((a*b^2 - 4*a^2*c)*n^2*s
qrt(1/((a^2*b^2 - 4*a^3*c)*n^4)) - b)/((a*b^2 - 4*a^2*c)*n^2))*log((4*c*x*x^(1/2*n - 1) - sqrt(2)*((a*b^3 - 4*
a^2*b*c)*n^3*sqrt(1/((a^2*b^2 - 4*a^3*c)*n^4)) + (b^2 - 4*a*c)*n)*sqrt(((a*b^2 - 4*a^2*c)*n^2*sqrt(1/((a^2*b^2
- 4*a^3*c)*n^4)) - b)/((a*b^2 - 4*a^2*c)*n^2)))/x)

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giac [B]  time = 1.85, size = 1035, normalized size = 6.12 \[ \frac {\frac {{\left (\sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b^{4} - 8 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a b^{2} c - 2 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b^{3} c - 2 \, b^{4} c + 16 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a^{2} c^{2} + 8 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a b c^{2} + \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b^{2} c^{2} + 16 \, a b^{2} c^{2} - 2 \, b^{3} c^{2} - 4 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a c^{3} - 32 \, a^{2} c^{3} + 8 \, a b c^{3} + \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b^{3} - 4 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a b c - 2 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b^{2} c + \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b c^{2} + 2 \, {\left (b^{2} - 4 \, a c\right )} b^{2} c - 8 \, {\left (b^{2} - 4 \, a c\right )} a c^{2} + 2 \, {\left (b^{2} - 4 \, a c\right )} b c^{2}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{n}}}{\sqrt {\frac {b + \sqrt {b^{2} - 4 \, a c}}{c}}}\right )}{{\left (a b^{4} - 8 \, a^{2} b^{2} c - 2 \, a b^{3} c + 16 \, a^{3} c^{2} + 8 \, a^{2} b c^{2} + a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} {\left | c \right |}} + \frac {{\left (\sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b^{4} - 8 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a b^{2} c - 2 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b^{3} c + 2 \, b^{4} c + 16 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a^{2} c^{2} + 8 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a b c^{2} + \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b^{2} c^{2} - 16 \, a b^{2} c^{2} - 2 \, b^{3} c^{2} - 4 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a c^{3} + 32 \, a^{2} c^{3} + 8 \, a b c^{3} + \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b^{3} - 4 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a b c - 2 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b^{2} c + \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b c^{2} - 2 \, {\left (b^{2} - 4 \, a c\right )} b^{2} c + 8 \, {\left (b^{2} - 4 \, a c\right )} a c^{2} + 2 \, {\left (b^{2} - 4 \, a c\right )} b c^{2}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{n}}}{\sqrt {\frac {b - \sqrt {b^{2} - 4 \, a c}}{c}}}\right )}{{\left (a b^{4} - 8 \, a^{2} b^{2} c - 2 \, a b^{3} c + 16 \, a^{3} c^{2} + 8 \, a^{2} b c^{2} + a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} {\left | c \right |}}}{2 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+1/2*n)/(a+b*x^n+c*x^(2*n)),x, algorithm="giac")

[Out]

1/2*((sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c - 2*sqrt
(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c - 2*b^4*c + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^2 + 8*s
qrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^2 + 16*a*b^2*c^
2 - 2*b^3*c^2 - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^3 - 32*a^2*c^3 + 8*a*b*c^3 + sqrt(2)*sqrt(b^2 -
4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c
- 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c + sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sq
rt(b^2 - 4*a*c)*c)*b*c^2 + 2*(b^2 - 4*a*c)*b^2*c - 8*(b^2 - 4*a*c)*a*c^2 + 2*(b^2 - 4*a*c)*b*c^2)*arctan(2*sqr
t(1/2)*sqrt(x^n)/sqrt((b + sqrt(b^2 - 4*a*c))/c))/((a*b^4 - 8*a^2*b^2*c - 2*a*b^3*c + 16*a^3*c^2 + 8*a^2*b*c^2
+ a*b^2*c^2 - 4*a^2*c^3)*abs(c)) + (sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^4 - 8*sqrt(2)*sqrt(b*c - sqrt(b
^2 - 4*a*c)*c)*a*b^2*c - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3*c + 2*b^4*c + 16*sqrt(2)*sqrt(b*c - sqr
t(b^2 - 4*a*c)*c)*a^2*c^2 + 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^2 + sqrt(2)*sqrt(b*c - sqrt(b^2 -
4*a*c)*c)*b^2*c^2 - 16*a*b^2*c^2 - 2*b^3*c^2 - 4*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*c^3 + 32*a^2*c^3 +
8*a*b*c^3 + sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b
*c - sqrt(b^2 - 4*a*c)*c)*a*b*c - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^2*c + sqrt(2)*
sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b*c^2 - 2*(b^2 - 4*a*c)*b^2*c + 8*(b^2 - 4*a*c)*a*c^2 + 2*(b
^2 - 4*a*c)*b*c^2)*arctan(2*sqrt(1/2)*sqrt(x^n)/sqrt((b - sqrt(b^2 - 4*a*c))/c))/((a*b^4 - 8*a^2*b^2*c - 2*a*b
^3*c + 16*a^3*c^2 + 8*a^2*b*c^2 + a*b^2*c^2 - 4*a^2*c^3)*abs(c)))/n

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maple [C]  time = 0.19, size = 114, normalized size = 0.67 \[ \RootOf \left (\left (16 a^{3} c^{2} n^{4}-8 a^{2} b^{2} c \,n^{4}+a \,b^{4} n^{4}\right ) \textit {\_Z}^{4}+\left (-4 a b c \,n^{2}+b^{3} n^{2}\right ) \textit {\_Z}^{2}+c \right ) \ln \left (\left (4 a^{2} b \,n^{3}-\frac {a \,b^{3} n^{3}}{c}\right ) \RootOf \left (\left (16 a^{3} c^{2} n^{4}-8 a^{2} b^{2} c \,n^{4}+a \,b^{4} n^{4}\right ) \textit {\_Z}^{4}+\left (-4 a b c \,n^{2}+b^{3} n^{2}\right ) \textit {\_Z}^{2}+c \right )^{3}+\left (2 a n -\frac {b^{2} n}{c}\right ) \RootOf \left (\left (16 a^{3} c^{2} n^{4}-8 a^{2} b^{2} c \,n^{4}+a \,b^{4} n^{4}\right ) \textit {\_Z}^{4}+\left (-4 a b c \,n^{2}+b^{3} n^{2}\right ) \textit {\_Z}^{2}+c \right )+x^{\frac {n}{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sum(_R*ln(x^(1/2*n)+(4*n^3*b*a^2-1/c*n^3*b^3*a)*_R^3+(2*a*n-1/c*n*b^2)*_R),_R=RootOf((16*a^3*c^2*n^4-8*a^2*b^2
*c*n^4+a*b^4*n^4)*_Z^4+(-4*a*b*c*n^2+b^3*n^2)*_Z^2+c))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{\frac {1}{2} \, n - 1}}{c x^{2 \, n} + b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+1/2*n)/(a+b*x^n+c*x^(2*n)),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^{\frac {n}{2}-1}}{a+b\,x^n+c\,x^{2\,n}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{\frac {n}{2} - 1}}{a + b x^{n} + c x^{2 n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+1/2*n)/(a+b*x**n+c*x**(2*n)),x)

[Out]

Integral(x**(n/2 - 1)/(a + b*x**n + c*x**(2*n)), x)

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