3.2552 \(\int x^{-1+3 n} (a+b x^n)^5 \, dx\)

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 = 0.0351237, 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, Rules used = {266, 43} \[ \frac{a^2 \left (a+b x^n\right )^6}{6 b^3 n}+\frac{\left (a+b x^n\right )^8}{8 b^3 n}-\frac{2 a \left (a+b x^n\right )^7}{7 b^3 n} \]

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)

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^{-1+3 n} \left (a+b x^n\right )^5 \, dx &=\frac{\operatorname{Subst}\left (\int x^2 (a+b x)^5 \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^2 (a+b x)^5}{b^2}-\frac{2 a (a+b x)^6}{b^2}+\frac{(a+b x)^7}{b^2}\right ) \, dx,x,x^n\right )}{n}\\ &=\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}\\ \end{align*}

Mathematica [A]  time = 0.0268074, size = 40, normalized size = 0.65 \[ \frac{\left (a+b x^n\right )^6 \left (a^2-6 a b x^n+21 b^2 x^{2 n}\right )}{168 b^3 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x^n)^6*(a^2 - 6*a*b*x^n + 21*b^2*x^(2*n)))/(168*b^3*n)

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Maple [A]  time = 0.02, size = 88, normalized size = 1.4 \begin{align*}{\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}} \end{align*}

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(-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+3*n)*(a+b*x^n)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.27435, size = 171, normalized size = 2.76 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+3*n)*(a+b*x^n)^5,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 [A]  time = 157.596, size = 94, normalized size = 1.52 \begin{align*} \begin{cases} \frac{a^{5} x^{3 n}}{3 n} + \frac{5 a^{4} b x^{4 n}}{4 n} + \frac{2 a^{3} b^{2} x^{5 n}}{n} + \frac{5 a^{2} b^{3} x^{6 n}}{3 n} + \frac{5 a b^{4} x^{7 n}}{7 n} + \frac{b^{5} x^{8 n}}{8 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+3*n)*(a+b*x**n)**5,x)

[Out]

Piecewise((a**5*x**(3*n)/(3*n) + 5*a**4*b*x**(4*n)/(4*n) + 2*a**3*b**2*x**(5*n)/n + 5*a**2*b**3*x**(6*n)/(3*n)
+ 5*a*b**4*x**(7*n)/(7*n) + b**5*x**(8*n)/(8*n), Ne(n, 0)), ((a + b)**5*log(x), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{n} + a\right )}^{5} x^{3 \, n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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