3.2555 \(\int x^{-1-7 n} \left (a+b x^n\right )^5 \, dx\)

Optimal. Leaf size=50 \[ \frac{b x^{-6 n} \left (a+b x^n\right )^6}{42 a^2 n}-\frac{x^{-7 n} \left (a+b x^n\right )^6}{7 a n} \]

[Out]

-(a + b*x^n)^6/(7*a*n*x^(7*n)) + (b*(a + b*x^n)^6)/(42*a^2*n*x^(6*n))

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Rubi [A]  time = 0.055605, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{b x^{-6 n} \left (a+b x^n\right )^6}{42 a^2 n}-\frac{x^{-7 n} \left (a+b x^n\right )^6}{7 a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 7*n)*(a + b*x^n)^5,x]

[Out]

-(a + b*x^n)^6/(7*a*n*x^(7*n)) + (b*(a + b*x^n)^6)/(42*a^2*n*x^(6*n))

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Rubi in Sympy [A]  time = 6.85739, size = 39, normalized size = 0.78 \[ - \frac{x^{- 7 n} \left (a + b x^{n}\right )^{6}}{7 a n} + \frac{b x^{- 6 n} \left (a + b x^{n}\right )^{6}}{42 a^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-7*n)*(a+b*x**n)**5,x)

[Out]

-x**(-7*n)*(a + b*x**n)**6/(7*a*n) + b*x**(-6*n)*(a + b*x**n)**6/(42*a**2*n)

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Mathematica [A]  time = 0.0383964, size = 74, normalized size = 1.48 \[ -\frac{x^{-7 n} \left (6 a^5+35 a^4 b x^n+84 a^3 b^2 x^{2 n}+105 a^2 b^3 x^{3 n}+70 a b^4 x^{4 n}+21 b^5 x^{5 n}\right )}{42 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 7*n)*(a + b*x^n)^5,x]

[Out]

-(6*a^5 + 35*a^4*b*x^n + 84*a^3*b^2*x^(2*n) + 105*a^2*b^3*x^(3*n) + 70*a*b^4*x^(
4*n) + 21*b^5*x^(5*n))/(42*n*x^(7*n))

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Maple [A]  time = 0.036, size = 88, normalized size = 1.8 \[ -{\frac{{b}^{5}}{2\,n \left ({x}^{n} \right ) ^{2}}}-{\frac{5\,a{b}^{4}}{3\,n \left ({x}^{n} \right ) ^{3}}}-{\frac{5\,{a}^{2}{b}^{3}}{2\,n \left ({x}^{n} \right ) ^{4}}}-2\,{\frac{{a}^{3}{b}^{2}}{n \left ({x}^{n} \right ) ^{5}}}-{\frac{5\,{a}^{4}b}{6\,n \left ({x}^{n} \right ) ^{6}}}-{\frac{{a}^{5}}{7\,n \left ({x}^{n} \right ) ^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-7*n)*(a+b*x^n)^5,x)

[Out]

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

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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^(-7*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22702, size = 100, normalized size = 2. \[ -\frac{21 \, b^{5} x^{5 \, n} + 70 \, a b^{4} x^{4 \, n} + 105 \, a^{2} b^{3} x^{3 \, n} + 84 \, a^{3} b^{2} x^{2 \, n} + 35 \, a^{4} b x^{n} + 6 \, a^{5}}{42 \, n x^{7 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(-7*n - 1),x, algorithm="fricas")

[Out]

-1/42*(21*b^5*x^(5*n) + 70*a*b^4*x^(4*n) + 105*a^2*b^3*x^(3*n) + 84*a^3*b^2*x^(2
*n) + 35*a^4*b*x^n + 6*a^5)/(n*x^(7*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**(-1-7*n)*(a+b*x**n)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.227318, size = 107, normalized size = 2.14 \[ -\frac{{\left (21 \, b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} + 70 \, a b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} + 105 \, a^{2} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 84 \, a^{3} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 35 \, a^{4} b e^{\left (n{\rm ln}\left (x\right )\right )} + 6 \, a^{5}\right )} e^{\left (-7 \, n{\rm ln}\left (x\right )\right )}}{42 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(-7*n - 1),x, algorithm="giac")

[Out]

-1/42*(21*b^5*e^(5*n*ln(x)) + 70*a*b^4*e^(4*n*ln(x)) + 105*a^2*b^3*e^(3*n*ln(x))
+ 84*a^3*b^2*e^(2*n*ln(x)) + 35*a^4*b*e^(n*ln(x)) + 6*a^5)*e^(-7*n*ln(x))/n