3.2599 \(\int \frac{x^{4+5 (-1+n)}}{a+b x^n} \, dx\)

Optimal. Leaf size=82 \[ \frac{a^4 \log \left (a+b x^n\right )}{b^5 n}-\frac{a^3 x^n}{b^4 n}+\frac{a^2 x^{2 n}}{2 b^3 n}-\frac{a x^{3 n}}{3 b^2 n}+\frac{x^{4 n}}{4 b n} \]

[Out]

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

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Rubi [A]  time = 0.105551, antiderivative size = 82, 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 \[ \frac{a^4 \log \left (a+b x^n\right )}{b^5 n}-\frac{a^3 x^n}{b^4 n}+\frac{a^2 x^{2 n}}{2 b^3 n}-\frac{a x^{3 n}}{3 b^2 n}+\frac{x^{4 n}}{4 b n} \]

Antiderivative was successfully verified.

[In]  Int[x^(4 + 5*(-1 + n))/(a + b*x^n),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{4} \log{\left (a + b x^{n} \right )}}{b^{5} n} + \frac{a^{2} \int ^{x^{n}} x\, dx}{b^{3} n} - \frac{a x^{3 n}}{3 b^{2} n} + \frac{x^{4 n}}{4 b n} - \frac{\int ^{x^{n}} a^{3}\, dx}{b^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+5*n)/(a+b*x**n),x)

[Out]

a**4*log(a + b*x**n)/(b**5*n) + a**2*Integral(x, (x, x**n))/(b**3*n) - a*x**(3*n
)/(3*b**2*n) + x**(4*n)/(4*b*n) - Integral(a**3, (x, x**n))/(b**4*n)

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Mathematica [A]  time = 0.0435782, size = 65, normalized size = 0.79 \[ \frac{12 a^4 \log \left (a+b x^n\right )+b x^n \left (-12 a^3+6 a^2 b x^n-4 a b^2 x^{2 n}+3 b^3 x^{3 n}\right )}{12 b^5 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(4 + 5*(-1 + n))/(a + b*x^n),x]

[Out]

(b*x^n*(-12*a^3 + 6*a^2*b*x^n - 4*a*b^2*x^(2*n) + 3*b^3*x^(3*n)) + 12*a^4*Log[a
+ b*x^n])/(12*b^5*n)

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Maple [A]  time = 0., size = 87, normalized size = 1.1 \[{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{4\,bn}}-{\frac{a \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{3\,{b}^{2}n}}+{\frac{{a}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,{b}^{3}n}}-{\frac{{a}^{3}{{\rm e}^{n\ln \left ( x \right ) }}}{{b}^{4}n}}+{\frac{{a}^{4}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{5}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+5*n)/(a+b*x^n),x)

[Out]

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

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Maxima [A]  time = 1.42853, size = 97, normalized size = 1.18 \[ \frac{a^{4} \log \left (\frac{b x^{n} + a}{b}\right )}{b^{5} n} + \frac{3 \, b^{3} x^{4 \, n} - 4 \, a b^{2} x^{3 \, n} + 6 \, a^{2} b x^{2 \, n} - 12 \, a^{3} x^{n}}{12 \, b^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5*n - 1)/(b*x^n + a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.230745, size = 88, normalized size = 1.07 \[ \frac{3 \, b^{4} x^{4 \, n} - 4 \, a b^{3} x^{3 \, n} + 6 \, a^{2} b^{2} x^{2 \, n} - 12 \, a^{3} b x^{n} + 12 \, a^{4} \log \left (b x^{n} + a\right )}{12 \, b^{5} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5*n - 1)/(b*x^n + a),x, algorithm="fricas")

[Out]

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

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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+5*n)/(a+b*x**n),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5 \, n - 1}}{b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5*n - 1)/(b*x^n + a),x, algorithm="giac")

[Out]

integrate(x^(5*n - 1)/(b*x^n + a), x)