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

Optimal. Leaf size=68 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right )}{c n \sqrt{b^2-4 a c}}+\frac{\log \left (a+b x^n+c x^{2 n}\right )}{2 c n} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.10408, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right )}{c n \sqrt{b^2-4 a c}}+\frac{\log \left (a+b x^n+c x^{2 n}\right )}{2 c n} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 19.8149, size = 58, normalized size = 0.85 \[ \frac{b \operatorname{atanh}{\left (\frac{b + 2 c x^{n}}{\sqrt{- 4 a c + b^{2}}} \right )}}{c n \sqrt{- 4 a c + b^{2}}} + \frac{\log{\left (a + b x^{n} + c x^{2 n} \right )}}{2 c n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*atanh((b + 2*c*x**n)/sqrt(-4*a*c + b**2))/(c*n*sqrt(-4*a*c + b**2)) + log(a +
b*x**n + c*x**(2*n))/(2*c*n)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0857897, size = 66, normalized size = 0.97 \[ \frac{\log \left (a+x^n \left (b+c x^n\right )\right )-\frac{2 b \tan ^{-1}\left (\frac{b+2 c x^n}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}}{2 c n} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.125, size = 402, normalized size = 5.9 \[{\frac{\ln \left ( x \right ) }{c}}-4\,{\frac{{n}^{2}\ln \left ( x \right ) ac}{4\,a{c}^{2}{n}^{2}-{b}^{2}c{n}^{2}}}+{\frac{{n}^{2}\ln \left ( x \right ){b}^{2}}{4\,a{c}^{2}{n}^{2}-{b}^{2}c{n}^{2}}}+2\,{\frac{a}{ \left ( 4\,ac-{b}^{2} \right ) n}\ln \left ({x}^{n}-1/2\,{\frac{-{b}^{2}+\sqrt{-4\,a{b}^{2}c+{b}^{4}}}{bc}} \right ) }-{\frac{{b}^{2}}{ \left ( 8\,ac-2\,{b}^{2} \right ) cn}\ln \left ({x}^{n}-{\frac{1}{2\,bc} \left ( -{b}^{2}+\sqrt{-4\,a{b}^{2}c+{b}^{4}} \right ) } \right ) }+{\frac{1}{ \left ( 8\,ac-2\,{b}^{2} \right ) cn}\ln \left ({x}^{n}-{\frac{1}{2\,bc} \left ( -{b}^{2}+\sqrt{-4\,a{b}^{2}c+{b}^{4}} \right ) } \right ) \sqrt{-4\,a{b}^{2}c+{b}^{4}}}+2\,{\frac{a}{ \left ( 4\,ac-{b}^{2} \right ) n}\ln \left ({x}^{n}+1/2\,{\frac{{b}^{2}+\sqrt{-4\,a{b}^{2}c+{b}^{4}}}{bc}} \right ) }-{\frac{{b}^{2}}{ \left ( 8\,ac-2\,{b}^{2} \right ) cn}\ln \left ({x}^{n}+{\frac{1}{2\,bc} \left ({b}^{2}+\sqrt{-4\,a{b}^{2}c+{b}^{4}} \right ) } \right ) }-{\frac{1}{ \left ( 8\,ac-2\,{b}^{2} \right ) cn}\ln \left ({x}^{n}+{\frac{1}{2\,bc} \left ({b}^{2}+\sqrt{-4\,a{b}^{2}c+{b}^{4}} \right ) } \right ) \sqrt{-4\,a{b}^{2}c+{b}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\log \left (x\right )}{c} - \int \frac{b x^{n} + a}{c^{2} x x^{2 \, n} + b c x x^{n} + a c x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.275514, size = 1, normalized size = 0.01 \[ \left [\frac{b \log \left (\frac{2 \, \sqrt{b^{2} - 4 \, a c} c^{2} x^{2 \, n} + b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2} + \sqrt{b^{2} - 4 \, a c} b c\right )} x^{n} +{\left (b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2 \, n} + b x^{n} + a}\right ) + \sqrt{b^{2} - 4 \, a c} \log \left (c x^{2 \, n} + b x^{n} + a\right )}{2 \, \sqrt{b^{2} - 4 \, a c} c n}, -\frac{2 \, b \arctan \left (-\frac{2 \, \sqrt{-b^{2} + 4 \, a c} c x^{n} + \sqrt{-b^{2} + 4 \, a c} b}{b^{2} - 4 \, a c}\right ) - \sqrt{-b^{2} + 4 \, a c} \log \left (c x^{2 \, n} + b x^{n} + a\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} c n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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