3.2544 \(\int x^{-1+4 n} \left (a+b x^n\right )^5 \, dx\)

Optimal. Leaf size=84 \[ -\frac{a^3 \left (a+b x^n\right )^6}{6 b^4 n}+\frac{3 a^2 \left (a+b x^n\right )^7}{7 b^4 n}+\frac{\left (a+b x^n\right )^9}{9 b^4 n}-\frac{3 a \left (a+b x^n\right )^8}{8 b^4 n} \]

[Out]

-(a^3*(a + b*x^n)^6)/(6*b^4*n) + (3*a^2*(a + b*x^n)^7)/(7*b^4*n) - (3*a*(a + b*x
^n)^8)/(8*b^4*n) + (a + b*x^n)^9/(9*b^4*n)

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Rubi [A]  time = 0.114485, antiderivative size = 84, 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 \[ -\frac{a^3 \left (a+b x^n\right )^6}{6 b^4 n}+\frac{3 a^2 \left (a+b x^n\right )^7}{7 b^4 n}+\frac{\left (a+b x^n\right )^9}{9 b^4 n}-\frac{3 a \left (a+b x^n\right )^8}{8 b^4 n} \]

Antiderivative was successfully verified.

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

[Out]

-(a^3*(a + b*x^n)^6)/(6*b^4*n) + (3*a^2*(a + b*x^n)^7)/(7*b^4*n) - (3*a*(a + b*x
^n)^8)/(8*b^4*n) + (a + b*x^n)^9/(9*b^4*n)

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Rubi in Sympy [A]  time = 12.1463, size = 83, normalized size = 0.99 \[ \frac{a^{5} x^{4 n}}{4 n} + \frac{a^{4} b x^{5 n}}{n} + \frac{5 a^{3} b^{2} x^{6 n}}{3 n} + \frac{10 a^{2} b^{3} x^{7 n}}{7 n} + \frac{5 a b^{4} x^{8 n}}{8 n} + \frac{b^{5} x^{9 n}}{9 n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**5*x**(4*n)/(4*n) + a**4*b*x**(5*n)/n + 5*a**3*b**2*x**(6*n)/(3*n) + 10*a**2*b
**3*x**(7*n)/(7*n) + 5*a*b**4*x**(8*n)/(8*n) + b**5*x**(9*n)/(9*n)

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Mathematica [A]  time = 0.0359968, size = 74, normalized size = 0.88 \[ \frac{x^{4 n} \left (126 a^5+504 a^4 b x^n+840 a^3 b^2 x^{2 n}+720 a^2 b^3 x^{3 n}+315 a b^4 x^{4 n}+56 b^5 x^{5 n}\right )}{504 n} \]

Antiderivative was successfully verified.

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

[Out]

(x^(4*n)*(126*a^5 + 504*a^4*b*x^n + 840*a^3*b^2*x^(2*n) + 720*a^2*b^3*x^(3*n) +
315*a*b^4*x^(4*n) + 56*b^5*x^(5*n)))/(504*n)

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Maple [A]  time = 0.036, size = 87, normalized size = 1. \[{\frac{{b}^{5} \left ({x}^{n} \right ) ^{9}}{9\,n}}+{\frac{5\,a{b}^{4} \left ({x}^{n} \right ) ^{8}}{8\,n}}+{\frac{10\,{a}^{2}{b}^{3} \left ({x}^{n} \right ) ^{7}}{7\,n}}+{\frac{5\,{a}^{3}{b}^{2} \left ({x}^{n} \right ) ^{6}}{3\,n}}+{\frac{{a}^{4}b \left ({x}^{n} \right ) ^{5}}{n}}+{\frac{{a}^{5} \left ({x}^{n} \right ) ^{4}}{4\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+4*n)*(a+b*x^n)^5,x)

[Out]

1/9*b^5/n*(x^n)^9+5/8*a*b^4/n*(x^n)^8+10/7*a^2*b^3/n*(x^n)^7+5/3*a^3*b^2/n*(x^n)
^6+a^4*b/n*(x^n)^5+1/4*a^5/n*(x^n)^4

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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^(4*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225725, size = 100, normalized size = 1.19 \[ \frac{56 \, b^{5} x^{9 \, n} + 315 \, a b^{4} x^{8 \, n} + 720 \, a^{2} b^{3} x^{7 \, n} + 840 \, a^{3} b^{2} x^{6 \, n} + 504 \, a^{4} b x^{5 \, n} + 126 \, a^{5} x^{4 \, n}}{504 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(4*n - 1),x, algorithm="fricas")

[Out]

1/504*(56*b^5*x^(9*n) + 315*a*b^4*x^(8*n) + 720*a^2*b^3*x^(7*n) + 840*a^3*b^2*x^
(6*n) + 504*a^4*b*x^(5*n) + 126*a^5*x^(4*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+4*n)*(a+b*x**n)**5,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{5} x^{4 \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(4*n - 1),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^5*x^(4*n - 1), x)