3.499 \(\int \frac{x^{-1-3 n}}{b x^n+c x^{2 n}} \, dx\)

Optimal. Leaf size=93 \[ -\frac{c^4 \log \left (b+c x^n\right )}{b^5 n}+\frac{c^4 \log (x)}{b^5}+\frac{c^3 x^{-n}}{b^4 n}-\frac{c^2 x^{-2 n}}{2 b^3 n}+\frac{c x^{-3 n}}{3 b^2 n}-\frac{x^{-4 n}}{4 b n} \]

[Out]

-1/(4*b*n*x^(4*n)) + c/(3*b^2*n*x^(3*n)) - c^2/(2*b^3*n*x^(2*n)) + c^3/(b^4*n*x^
n) + (c^4*Log[x])/b^5 - (c^4*Log[b + c*x^n])/(b^5*n)

_______________________________________________________________________________________

Rubi [A]  time = 0.108819, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ -\frac{c^4 \log \left (b+c x^n\right )}{b^5 n}+\frac{c^4 \log (x)}{b^5}+\frac{c^3 x^{-n}}{b^4 n}-\frac{c^2 x^{-2 n}}{2 b^3 n}+\frac{c x^{-3 n}}{3 b^2 n}-\frac{x^{-4 n}}{4 b n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 3*n)/(b*x^n + c*x^(2*n)),x]

[Out]

-1/(4*b*n*x^(4*n)) + c/(3*b^2*n*x^(3*n)) - c^2/(2*b^3*n*x^(2*n)) + c^3/(b^4*n*x^
n) + (c^4*Log[x])/b^5 - (c^4*Log[b + c*x^n])/(b^5*n)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 20.2316, size = 82, normalized size = 0.88 \[ - \frac{x^{- 4 n}}{4 b n} + \frac{c x^{- 3 n}}{3 b^{2} n} - \frac{c^{2} x^{- 2 n}}{2 b^{3} n} + \frac{c^{3} x^{- n}}{b^{4} n} + \frac{c^{4} \log{\left (x^{n} \right )}}{b^{5} n} - \frac{c^{4} \log{\left (b + c x^{n} \right )}}{b^{5} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-3*n)/(b*x**n+c*x**(2*n)),x)

[Out]

-x**(-4*n)/(4*b*n) + c*x**(-3*n)/(3*b**2*n) - c**2*x**(-2*n)/(2*b**3*n) + c**3*x
**(-n)/(b**4*n) + c**4*log(x**n)/(b**5*n) - c**4*log(b + c*x**n)/(b**5*n)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0588833, size = 74, normalized size = 0.8 \[ -\frac{x^{-4 n} \left (b \left (3 b^3-4 b^2 c x^n+6 b c^2 x^{2 n}-12 c^3 x^{3 n}\right )+12 c^4 x^{4 n} \log \left (b x^{-n}+c\right )\right )}{12 b^5 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 3*n)/(b*x^n + c*x^(2*n)),x]

[Out]

-(b*(3*b^3 - 4*b^2*c*x^n + 6*b*c^2*x^(2*n) - 12*c^3*x^(3*n)) + 12*c^4*x^(4*n)*Lo
g[c + b/x^n])/(12*b^5*n*x^(4*n))

_______________________________________________________________________________________

Maple [A]  time = 0.045, size = 105, normalized size = 1.1 \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}} \left ({\frac{{c}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{{b}^{4}n}}-{\frac{1}{4\,bn}}+{\frac{c{{\rm e}^{n\ln \left ( x \right ) }}}{3\,{b}^{2}n}}-{\frac{{c}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,{b}^{3}n}}+{\frac{{c}^{4}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{{b}^{5}}} \right ) }-{\frac{{c}^{4}\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }}+b \right ) }{{b}^{5}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-3*n)/(b*x^n+c*x^(2*n)),x)

[Out]

(c^3/b^4/n*exp(n*ln(x))^3-1/4/b/n+1/3*c/b^2/n*exp(n*ln(x))-1/2*c^2/b^3/n*exp(n*l
n(x))^2+c^4/b^5*ln(x)*exp(n*ln(x))^4)/exp(n*ln(x))^4-c^4/b^5/n*ln(c*exp(n*ln(x))
+b)

_______________________________________________________________________________________

Maxima [A]  time = 0.749778, size = 111, normalized size = 1.19 \[ \frac{c^{4} \log \left (x\right )}{b^{5}} - \frac{c^{4} \log \left (\frac{c x^{n} + b}{c}\right )}{b^{5} n} + \frac{{\left (12 \, c^{3} x^{3 \, n} - 6 \, b c^{2} x^{2 \, n} + 4 \, b^{2} c x^{n} - 3 \, b^{3}\right )} x^{-4 \, n}}{12 \, b^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-3*n - 1)/(c*x^(2*n) + b*x^n),x, algorithm="maxima")

[Out]

c^4*log(x)/b^5 - c^4*log((c*x^n + b)/c)/(b^5*n) + 1/12*(12*c^3*x^(3*n) - 6*b*c^2
*x^(2*n) + 4*b^2*c*x^n - 3*b^3)*x^(-4*n)/(b^4*n)

_______________________________________________________________________________________

Fricas [A]  time = 0.291579, size = 115, normalized size = 1.24 \[ \frac{12 \, c^{4} n x^{4 \, n} \log \left (x\right ) - 12 \, c^{4} x^{4 \, n} \log \left (c x^{n} + b\right ) + 12 \, b c^{3} x^{3 \, n} - 6 \, b^{2} c^{2} x^{2 \, n} + 4 \, b^{3} c x^{n} - 3 \, b^{4}}{12 \, b^{5} n x^{4 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-3*n - 1)/(c*x^(2*n) + b*x^n),x, algorithm="fricas")

[Out]

1/12*(12*c^4*n*x^(4*n)*log(x) - 12*c^4*x^(4*n)*log(c*x^n + b) + 12*b*c^3*x^(3*n)
- 6*b^2*c^2*x^(2*n) + 4*b^3*c*x^n - 3*b^4)/(b^5*n*x^(4*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**(-1-3*n)/(b*x**n+c*x**(2*n)),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-3 \, n - 1}}{c x^{2 \, n} + b x^{n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-3*n - 1)/(c*x^(2*n) + b*x^n),x, algorithm="giac")

[Out]

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