3.2562 \(\int x^{-1-7 n} (a+b x^n)^5 \, dx\)

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 = 0.0166591, 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, Rules used = {266, 45, 37} \[ \frac{b x^{-6 n} \left (a+b x^n\right )^6}{42 a^2 n}-\frac{x^{-7 n} \left (a+b x^n\right )^6}{7 a n} \]

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))

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int x^{-1-7 n} \left (a+b x^n\right )^5 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^8} \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-7 n} \left (a+b x^n\right )^6}{7 a n}-\frac{b \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^7} \, dx,x,x^n\right )}{7 a n}\\ &=-\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}\\ \end{align*}

Mathematica [A]  time = 0.0114364, size = 33, normalized size = 0.66 \[ \frac{x^{-7 n} \left (b x^n-6 a\right ) \left (a+b x^n\right )^6}{42 a^2 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.019, size = 88, normalized size = 1.8 \begin{align*} -{\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}}} \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-7*n)*(a+b*x^n)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.40352, size = 163, normalized size = 3.26 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-7*n)*(a+b*x^n)^5,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 [A]  time = 174.642, size = 95, normalized size = 1.9 \begin{align*} \begin{cases} - \frac{a^{5} x^{- 7 n}}{7 n} - \frac{5 a^{4} b x^{- 6 n}}{6 n} - \frac{2 a^{3} b^{2} x^{- 5 n}}{n} - \frac{5 a^{2} b^{3} x^{- 4 n}}{2 n} - \frac{5 a b^{4} x^{- 3 n}}{3 n} - \frac{b^{5} x^{- 2 n}}{2 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{5} \log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1-7*n)*(a+b*x**n)**5,x)

[Out]

Piecewise((-a**5*x**(-7*n)/(7*n) - 5*a**4*b*x**(-6*n)/(6*n) - 2*a**3*b**2*x**(-5*n)/n - 5*a**2*b**3*x**(-4*n)/
(2*n) - 5*a*b**4*x**(-3*n)/(3*n) - b**5*x**(-2*n)/(2*n), Ne(n, 0)), ((a + b)**5*log(x), True))

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Giac [A]  time = 1.24349, size = 100, normalized size = 2. \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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))