3.2545 \(\int x^{-1+3 n} \left (a+b x^n\right )^5 \, dx\)

Optimal. Leaf size=62 \[ \frac{a^2 \left (a+b x^n\right )^6}{6 b^3 n}+\frac{\left (a+b x^n\right )^8}{8 b^3 n}-\frac{2 a \left (a+b x^n\right )^7}{7 b^3 n} \]

[Out]

(a^2*(a + b*x^n)^6)/(6*b^3*n) - (2*a*(a + b*x^n)^7)/(7*b^3*n) + (a + b*x^n)^8/(8
*b^3*n)

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Rubi [A]  time = 0.0944203, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{a^2 \left (a+b x^n\right )^6}{6 b^3 n}+\frac{\left (a+b x^n\right )^8}{8 b^3 n}-\frac{2 a \left (a+b x^n\right )^7}{7 b^3 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 3*n)*(a + b*x^n)^5,x]

[Out]

(a^2*(a + b*x^n)^6)/(6*b^3*n) - (2*a*(a + b*x^n)^7)/(7*b^3*n) + (a + b*x^n)^8/(8
*b^3*n)

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Rubi in Sympy [A]  time = 7.55437, size = 51, normalized size = 0.82 \[ \frac{a^{2} \left (a + b x^{n}\right )^{6}}{6 b^{3} n} - \frac{2 a \left (a + b x^{n}\right )^{7}}{7 b^{3} n} + \frac{\left (a + b x^{n}\right )^{8}}{8 b^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+3*n)*(a+b*x**n)**5,x)

[Out]

a**2*(a + b*x**n)**6/(6*b**3*n) - 2*a*(a + b*x**n)**7/(7*b**3*n) + (a + b*x**n)*
*8/(8*b**3*n)

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Mathematica [A]  time = 0.0322421, size = 74, normalized size = 1.19 \[ \frac{x^{3 n} \left (56 a^5+210 a^4 b x^n+336 a^3 b^2 x^{2 n}+280 a^2 b^3 x^{3 n}+120 a b^4 x^{4 n}+21 b^5 x^{5 n}\right )}{168 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 3*n)*(a + b*x^n)^5,x]

[Out]

(x^(3*n)*(56*a^5 + 210*a^4*b*x^n + 336*a^3*b^2*x^(2*n) + 280*a^2*b^3*x^(3*n) + 1
20*a*b^4*x^(4*n) + 21*b^5*x^(5*n)))/(168*n)

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Maple [A]  time = 0.034, size = 88, normalized size = 1.4 \[{\frac{{b}^{5} \left ({x}^{n} \right ) ^{8}}{8\,n}}+{\frac{5\,a{b}^{4} \left ({x}^{n} \right ) ^{7}}{7\,n}}+{\frac{5\,{a}^{2}{b}^{3} \left ({x}^{n} \right ) ^{6}}{3\,n}}+2\,{\frac{{a}^{3}{b}^{2} \left ({x}^{n} \right ) ^{5}}{n}}+{\frac{5\,{a}^{4}b \left ({x}^{n} \right ) ^{4}}{4\,n}}+{\frac{{a}^{5} \left ({x}^{n} \right ) ^{3}}{3\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+3*n)*(a+b*x^n)^5,x)

[Out]

1/8*b^5/n*(x^n)^8+5/7*a*b^4/n*(x^n)^7+5/3*a^2*b^3/n*(x^n)^6+2*a^3*b^2/n*(x^n)^5+
5/4*a^4*b/n*(x^n)^4+1/3*a^5/n*(x^n)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(3*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224709, size = 100, normalized size = 1.61 \[ \frac{21 \, b^{5} x^{8 \, n} + 120 \, a b^{4} x^{7 \, n} + 280 \, a^{2} b^{3} x^{6 \, n} + 336 \, a^{3} b^{2} x^{5 \, n} + 210 \, a^{4} b x^{4 \, n} + 56 \, a^{5} x^{3 \, n}}{168 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(3*n - 1),x, algorithm="fricas")

[Out]

1/168*(21*b^5*x^(8*n) + 120*a*b^4*x^(7*n) + 280*a^2*b^3*x^(6*n) + 336*a^3*b^2*x^
(5*n) + 210*a^4*b*x^(4*n) + 56*a^5*x^(3*n))/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+3*n)*(a+b*x**n)**5,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{5} x^{3 \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(3*n - 1),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^5*x^(3*n - 1), x)