3.2547 \(\int x^{-1+n} \left (a+b x^n\right )^5 \, dx\)

Optimal. Leaf size=19 \[ \frac{\left (a+b x^n\right )^6}{6 b n} \]

[Out]

(a + b*x^n)^6/(6*b*n)

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Rubi [A]  time = 0.0187513, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{\left (a+b x^n\right )^6}{6 b n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + n)*(a + b*x^n)^5,x]

[Out]

(a + b*x^n)^6/(6*b*n)

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Rubi in Sympy [A]  time = 1.22625, size = 12, normalized size = 0.63 \[ \frac{\left (a + b x^{n}\right )^{6}}{6 b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x**n)**6/(6*b*n)

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Mathematica [A]  time = 0.0149867, size = 19, normalized size = 1. \[ \frac{\left (a+b x^n\right )^6}{6 b n} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x^n)^6/(6*b*n)

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Maple [B]  time = 0.036, size = 84, normalized size = 4.4 \[{\frac{{b}^{5} \left ({x}^{n} \right ) ^{6}}{6\,n}}+{\frac{a{b}^{4} \left ({x}^{n} \right ) ^{5}}{n}}+{\frac{5\,{a}^{2}{b}^{3} \left ({x}^{n} \right ) ^{4}}{2\,n}}+{\frac{10\,{a}^{3}{b}^{2} \left ({x}^{n} \right ) ^{3}}{3\,n}}+{\frac{5\,{a}^{4}b \left ({x}^{n} \right ) ^{2}}{2\,n}}+{\frac{{a}^{5}{x}^{n}}{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+n)*(a+b*x^n)^5,x)

[Out]

1/6*b^5/n*(x^n)^6+a*b^4/n*(x^n)^5+5/2*a^2*b^3/n*(x^n)^4+10/3*a^3*b^2/n*(x^n)^3+5
/2*a^4*b/n*(x^n)^2+a^5/n*x^n

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Maxima [A]  time = 1.43421, size = 23, normalized size = 1.21 \[ \frac{{\left (b x^{n} + a\right )}^{6}}{6 \, b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(b*x^n + a)^6/(b*n)

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Fricas [A]  time = 0.224737, size = 96, normalized size = 5.05 \[ \frac{b^{5} x^{6 \, n} + 6 \, a b^{4} x^{5 \, n} + 15 \, a^{2} b^{3} x^{4 \, n} + 20 \, a^{3} b^{2} x^{3 \, n} + 15 \, a^{4} b x^{2 \, n} + 6 \, a^{5} x^{n}}{6 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(b^5*x^(6*n) + 6*a*b^4*x^(5*n) + 15*a^2*b^3*x^(4*n) + 20*a^3*b^2*x^(3*n) + 1
5*a^4*b*x^(2*n) + 6*a^5*x^n)/n

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Sympy [A]  time = 49.307, size = 88, normalized size = 4.63 \[ \begin{cases} \frac{a^{5} x^{n}}{n} + \frac{5 a^{4} b x^{2 n}}{2 n} + \frac{10 a^{3} b^{2} x^{3 n}}{3 n} + \frac{5 a^{2} b^{3} x^{4 n}}{2 n} + \frac{a b^{4} x^{5 n}}{n} + \frac{b^{5} x^{6 n}}{6 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{5} \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((a**5*x**n/n + 5*a**4*b*x**(2*n)/(2*n) + 10*a**3*b**2*x**(3*n)/(3*n) +
5*a**2*b**3*x**(4*n)/(2*n) + a*b**4*x**(5*n)/n + b**5*x**(6*n)/(6*n), Ne(n, 0))
, ((a + b)**5*log(x), True))

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GIAC/XCAS [A]  time = 0.215021, size = 96, normalized size = 5.05 \[ \frac{b^{5} x^{6 \, n} + 6 \, a b^{4} x^{5 \, n} + 15 \, a^{2} b^{3} x^{4 \, n} + 20 \, a^{3} b^{2} x^{3 \, n} + 15 \, a^{4} b x^{2 \, n} + 6 \, a^{5} x^{n}}{6 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(b^5*x^(6*n) + 6*a*b^4*x^(5*n) + 15*a^2*b^3*x^(4*n) + 20*a^3*b^2*x^(3*n) + 1
5*a^4*b*x^(2*n) + 6*a^5*x^n)/n