3.2601 \(\int \frac{x^{2+3 (-1+n)}}{a+b x^n} \, dx\)

Optimal. Leaf size=46 \[ \frac{a^2 \log \left (a+b x^n\right )}{b^3 n}-\frac{a x^n}{b^2 n}+\frac{x^{2 n}}{2 b n} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0658019, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{a^2 \log \left (a+b x^n\right )}{b^3 n}-\frac{a x^n}{b^2 n}+\frac{x^{2 n}}{2 b n} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*log(a + b*x**n)/(b**3*n) + Integral(x, (x, x**n))/(b*n) - Integral(a, (x, x
**n))/(b**2*n)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0252838, size = 38, normalized size = 0.83 \[ \frac{2 a^2 \log \left (a+b x^n\right )+b x^n \left (b x^n-2 a\right )}{2 b^3 n} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 51, normalized size = 1.1 \[{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,bn}}-{\frac{a{{\rm e}^{n\ln \left ( x \right ) }}}{{b}^{2}n}}+{\frac{{a}^{2}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{3}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+3*n)/(a+b*x^n),x)

[Out]

1/2/b/n*exp(n*ln(x))^2-a/b^2/n*exp(n*ln(x))+a^2/b^3/n*ln(a+b*exp(n*ln(x)))

_______________________________________________________________________________________

Maxima [A]  time = 1.45392, size = 61, normalized size = 1.33 \[ \frac{a^{2} \log \left (\frac{b x^{n} + a}{b}\right )}{b^{3} n} + \frac{b x^{2 \, n} - 2 \, a x^{n}}{2 \, b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^2*log((b*x^n + a)/b)/(b^3*n) + 1/2*(b*x^(2*n) - 2*a*x^n)/(b^2*n)

_______________________________________________________________________________________

Fricas [A]  time = 0.226673, size = 51, normalized size = 1.11 \[ \frac{b^{2} x^{2 \, n} - 2 \, a b x^{n} + 2 \, a^{2} \log \left (b x^{n} + a\right )}{2 \, b^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^(3*n - 1)/(b*x^n + a), x)