3.2539 \(\int x^{-1+4 n} (a+b x^n)^3 \, dx\)

Optimal. Leaf size=63 \[ \frac{3 a^2 b x^{5 n}}{5 n}+\frac{a^3 x^{4 n}}{4 n}+\frac{a b^2 x^{6 n}}{2 n}+\frac{b^3 x^{7 n}}{7 n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0271848, antiderivative size = 63, 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} \[ \frac{3 a^2 b x^{5 n}}{5 n}+\frac{a^3 x^{4 n}}{4 n}+\frac{a b^2 x^{6 n}}{2 n}+\frac{b^3 x^{7 n}}{7 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + 4*n)*(a + b*x^n)^3,x]

[Out]

(a^3*x^(4*n))/(4*n) + (3*a^2*b*x^(5*n))/(5*n) + (a*b^2*x^(6*n))/(2*n) + (b^3*x^(7*n))/(7*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+4 n} \left (a+b x^n\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int x^3 (a+b x)^3 \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^3 x^3+3 a^2 b x^4+3 a b^2 x^5+b^3 x^6\right ) \, dx,x,x^n\right )}{n}\\ &=\frac{a^3 x^{4 n}}{4 n}+\frac{3 a^2 b x^{5 n}}{5 n}+\frac{a b^2 x^{6 n}}{2 n}+\frac{b^3 x^{7 n}}{7 n}\\ \end{align*}

Mathematica [A]  time = 0.0213119, size = 48, normalized size = 0.76 \[ \frac{x^{4 n} \left (84 a^2 b x^n+35 a^3+70 a b^2 x^{2 n}+20 b^3 x^{3 n}\right )}{140 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + 4*n)*(a + b*x^n)^3,x]

[Out]

(x^(4*n)*(35*a^3 + 84*a^2*b*x^n + 70*a*b^2*x^(2*n) + 20*b^3*x^(3*n)))/(140*n)

________________________________________________________________________________________

Maple [A]  time = 0.017, size = 56, normalized size = 0.9 \begin{align*}{\frac{{b}^{3} \left ({x}^{n} \right ) ^{7}}{7\,n}}+{\frac{{b}^{2}a \left ({x}^{n} \right ) ^{6}}{2\,n}}+{\frac{3\,b{a}^{2} \left ({x}^{n} \right ) ^{5}}{5\,n}}+{\frac{{a}^{3} \left ({x}^{n} \right ) ^{4}}{4\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+4*n)*(a+b*x^n)^3,x)

[Out]

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

________________________________________________________________________________________

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+4*n)*(a+b*x^n)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.27533, size = 109, normalized size = 1.73 \begin{align*} \frac{20 \, b^{3} x^{7 \, n} + 70 \, a b^{2} x^{6 \, n} + 84 \, a^{2} b x^{5 \, n} + 35 \, a^{3} x^{4 \, n}}{140 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+4*n)*(a+b*x^n)^3,x, algorithm="fricas")

[Out]

1/140*(20*b^3*x^(7*n) + 70*a*b^2*x^(6*n) + 84*a^2*b*x^(5*n) + 35*a^3*x^(4*n))/n

________________________________________________________________________________________

Sympy [A]  time = 29.7732, size = 60, normalized size = 0.95 \begin{align*} \begin{cases} \frac{a^{3} x^{4 n}}{4 n} + \frac{3 a^{2} b x^{5 n}}{5 n} + \frac{a b^{2} x^{6 n}}{2 n} + \frac{b^{3} x^{7 n}}{7 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{3} \log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+4*n)*(a+b*x**n)**3,x)

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{n} + a\right )}^{3} x^{4 \, n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+4*n)*(a+b*x^n)^3,x, algorithm="giac")

[Out]

integrate((b*x^n + a)^3*x^(4*n - 1), x)