3.549 \(\int \frac{x^{-1+4 n}}{a+b x^n+c x^{2 n}} \, dx\)

Optimal. Leaf size=111 \[ \frac{b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right )}{c^3 n \sqrt{b^2-4 a c}}+\frac{\left (b^2-a c\right ) \log \left (a+b x^n+c x^{2 n}\right )}{2 c^3 n}-\frac{b x^n}{c^2 n}+\frac{x^{2 n}}{2 c n} \]

[Out]

-((b*x^n)/(c^2*n)) + x^(2*n)/(2*c*n) + (b*(b^2 - 3*a*c)*ArcTanh[(b + 2*c*x^n)/Sq
rt[b^2 - 4*a*c]])/(c^3*Sqrt[b^2 - 4*a*c]*n) + ((b^2 - a*c)*Log[a + b*x^n + c*x^(
2*n)])/(2*c^3*n)

_______________________________________________________________________________________

Rubi [A]  time = 0.239469, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right )}{c^3 n \sqrt{b^2-4 a c}}+\frac{\left (b^2-a c\right ) \log \left (a+b x^n+c x^{2 n}\right )}{2 c^3 n}-\frac{b x^n}{c^2 n}+\frac{x^{2 n}}{2 c n} \]

Antiderivative was successfully verified.

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

[Out]

-((b*x^n)/(c^2*n)) + x^(2*n)/(2*c*n) + (b*(b^2 - 3*a*c)*ArcTanh[(b + 2*c*x^n)/Sq
rt[b^2 - 4*a*c]])/(c^3*Sqrt[b^2 - 4*a*c]*n) + ((b^2 - a*c)*Log[a + b*x^n + c*x^(
2*n)])/(2*c^3*n)

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b \left (- 3 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{n}}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{3} n \sqrt{- 4 a c + b^{2}}} + \frac{\int ^{x^{n}} x\, dx}{c n} - \frac{\int ^{x^{n}} b\, dx}{c^{2} n} + \frac{\left (- a c + b^{2}\right ) \log{\left (a + b x^{n} + c x^{2 n} \right )}}{2 c^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+4*n)/(a+b*x**n+c*x**(2*n)),x)

[Out]

b*(-3*a*c + b**2)*atanh((b + 2*c*x**n)/sqrt(-4*a*c + b**2))/(c**3*n*sqrt(-4*a*c
+ b**2)) + Integral(x, (x, x**n))/(c*n) - Integral(b, (x, x**n))/(c**2*n) + (-a*
c + b**2)*log(a + b*x**n + c*x**(2*n))/(2*c**3*n)

_______________________________________________________________________________________

Mathematica [A]  time = 0.260004, size = 97, normalized size = 0.87 \[ \frac{\left (b^2-a c\right ) \log \left (a+x^n \left (b+c x^n\right )\right )-\frac{2 b \left (b^2-3 a c\right ) \tan ^{-1}\left (\frac{b+2 c x^n}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+c x^n \left (c x^n-2 b\right )}{2 c^3 n} \]

Antiderivative was successfully verified.

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

[Out]

(c*x^n*(-2*b + c*x^n) - (2*b*(b^2 - 3*a*c)*ArcTan[(b + 2*c*x^n)/Sqrt[-b^2 + 4*a*
c]])/Sqrt[-b^2 + 4*a*c] + (b^2 - a*c)*Log[a + x^n*(b + c*x^n)])/(2*c^3*n)

_______________________________________________________________________________________

Maple [B]  time = 0.269, size = 973, normalized size = 8.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/c^2*ln(x)*a+1/c^3*ln(x)*b^2+1/2/c/n*(x^n)^2-b*x^n/c^2/n+4/(4*a*c^4*n^2-b^2*c^
3*n^2)*n^2*ln(x)*a^2*c^2-5/(4*a*c^4*n^2-b^2*c^3*n^2)*n^2*ln(x)*a*b^2*c+1/(4*a*c^
4*n^2-b^2*c^3*n^2)*n^2*ln(x)*b^4-2/c/(4*a*c-b^2)/n*ln(x^n+1/2*(3*a*b^2*c-b^4+(-3
6*a^3*b^2*c^3+33*a^2*b^4*c^2-10*a*b^6*c+b^8)^(1/2))/c/b/(3*a*c-b^2))*a^2+5/2/c^2
/(4*a*c-b^2)/n*ln(x^n+1/2*(3*a*b^2*c-b^4+(-36*a^3*b^2*c^3+33*a^2*b^4*c^2-10*a*b^
6*c+b^8)^(1/2))/c/b/(3*a*c-b^2))*a*b^2-1/2/c^3/(4*a*c-b^2)/n*ln(x^n+1/2*(3*a*b^2
*c-b^4+(-36*a^3*b^2*c^3+33*a^2*b^4*c^2-10*a*b^6*c+b^8)^(1/2))/c/b/(3*a*c-b^2))*b
^4+1/2/c^3/(4*a*c-b^2)/n*ln(x^n+1/2*(3*a*b^2*c-b^4+(-36*a^3*b^2*c^3+33*a^2*b^4*c
^2-10*a*b^6*c+b^8)^(1/2))/c/b/(3*a*c-b^2))*(-36*a^3*b^2*c^3+33*a^2*b^4*c^2-10*a*
b^6*c+b^8)^(1/2)-2/c/(4*a*c-b^2)/n*ln(x^n-1/2*(-3*a*b^2*c+b^4+(-36*a^3*b^2*c^3+3
3*a^2*b^4*c^2-10*a*b^6*c+b^8)^(1/2))/c/b/(3*a*c-b^2))*a^2+5/2/c^2/(4*a*c-b^2)/n*
ln(x^n-1/2*(-3*a*b^2*c+b^4+(-36*a^3*b^2*c^3+33*a^2*b^4*c^2-10*a*b^6*c+b^8)^(1/2)
)/c/b/(3*a*c-b^2))*a*b^2-1/2/c^3/(4*a*c-b^2)/n*ln(x^n-1/2*(-3*a*b^2*c+b^4+(-36*a
^3*b^2*c^3+33*a^2*b^4*c^2-10*a*b^6*c+b^8)^(1/2))/c/b/(3*a*c-b^2))*b^4-1/2/c^3/(4
*a*c-b^2)/n*ln(x^n-1/2*(-3*a*b^2*c+b^4+(-36*a^3*b^2*c^3+33*a^2*b^4*c^2-10*a*b^6*
c+b^8)^(1/2))/c/b/(3*a*c-b^2))*(-36*a^3*b^2*c^3+33*a^2*b^4*c^2-10*a*b^6*c+b^8)^(
1/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{{\left (b^{2} - a c\right )} \log \left (x\right )}{c^{3}} + \frac{c x^{2 \, n} - 2 \, b x^{n}}{2 \, c^{2} n} + \int -\frac{a b^{2} - a^{2} c +{\left (b^{3} - 2 \, a b c\right )} x^{n}}{c^{4} x x^{2 \, n} + b c^{3} x x^{n} + a c^{3} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(4*n - 1)/(c*x^(2*n) + b*x^n + a),x, algorithm="maxima")

[Out]

(b^2 - a*c)*log(x)/c^3 + 1/2*(c*x^(2*n) - 2*b*x^n)/(c^2*n) + integrate(-(a*b^2 -
a^2*c + (b^3 - 2*a*b*c)*x^n)/(c^4*x*x^(2*n) + b*c^3*x*x^n + a*c^3*x), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.295891, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{b^{2} - 4 \, a c} c^{2} x^{2 \, n} - 2 \, \sqrt{b^{2} - 4 \, a c} b c x^{n} +{\left (b^{2} - a c\right )} \sqrt{b^{2} - 4 \, a c} \log \left (c x^{2 \, n} + b x^{n} + a\right ) -{\left (b^{3} - 3 \, a b c\right )} \log \left (\frac{2 \, \sqrt{b^{2} - 4 \, a c} c^{2} x^{2 \, n} - b^{3} + 4 \, a b c - 2 \,{\left (b^{2} c - 4 \, a c^{2} - \sqrt{b^{2} - 4 \, a c} b c\right )} x^{n} +{\left (b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2 \, n} + b x^{n} + a}\right )}{2 \, \sqrt{b^{2} - 4 \, a c} c^{3} n}, \frac{\sqrt{-b^{2} + 4 \, a c} c^{2} x^{2 \, n} - 2 \, \sqrt{-b^{2} + 4 \, a c} b c x^{n} +{\left (b^{2} - a c\right )} \sqrt{-b^{2} + 4 \, a c} \log \left (c x^{2 \, n} + b x^{n} + a\right ) - 2 \,{\left (b^{3} - 3 \, a b c\right )} \arctan \left (-\frac{2 \, \sqrt{-b^{2} + 4 \, a c} c x^{n} + \sqrt{-b^{2} + 4 \, a c} b}{b^{2} - 4 \, a c}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} c^{3} n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(4*n - 1)/(c*x^(2*n) + b*x^n + a),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4 \, n - 1}}{c x^{2 \, n} + b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(4*n - 1)/(c*x^(2*n) + b*x^n + a),x, algorithm="giac")

[Out]

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