3.2541 \(\int x^{-1+2 n} (a+b x^n)^3 \, dx\)

Optimal. Leaf size=40 \[ \frac{\left (a+b x^n\right )^5}{5 b^2 n}-\frac{a \left (a+b x^n\right )^4}{4 b^2 n} \]

[Out]

-(a*(a + b*x^n)^4)/(4*b^2*n) + (a + b*x^n)^5/(5*b^2*n)

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Rubi [A]  time = 0.0184661, antiderivative size = 40, 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} \[ \frac{\left (a+b x^n\right )^5}{5 b^2 n}-\frac{a \left (a+b x^n\right )^4}{4 b^2 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + 2*n)*(a + b*x^n)^3,x]

[Out]

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

Mathematica [A]  time = 0.0186436, size = 27, normalized size = 0.68 \[ -\frac{\left (a-4 b x^n\right ) \left (a+b x^n\right )^4}{20 b^2 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + 2*n)*(a + b*x^n)^3,x]

[Out]

-((a - 4*b*x^n)*(a + b*x^n)^4)/(20*b^2*n)

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Maple [A]  time = 0.016, size = 63, normalized size = 1.6 \begin{align*}{\frac{b{a}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{n}}+{\frac{{a}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,n}}+{\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}}{5\,n}}+{\frac{3\,{b}^{2}a \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{4\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+2*n)*(a+b*x^n)^3,x)

[Out]

b*a^2/n*exp(n*ln(x))^3+1/2*a^3/n*exp(n*ln(x))^2+1/5*b^3/n*exp(n*ln(x))^5+3/4*b^2*a/n*exp(n*ln(x))^4

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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+2*n)*(a+b*x^n)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.2857, size = 107, normalized size = 2.68 \begin{align*} \frac{4 \, b^{3} x^{5 \, n} + 15 \, a b^{2} x^{4 \, n} + 20 \, a^{2} b x^{3 \, n} + 10 \, a^{3} x^{2 \, n}}{20 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+2*n)*(a+b*x^n)^3,x, algorithm="fricas")

[Out]

1/20*(4*b^3*x^(5*n) + 15*a*b^2*x^(4*n) + 20*a^2*b*x^(3*n) + 10*a^3*x^(2*n))/n

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Sympy [A]  time = 29.2719, size = 58, normalized size = 1.45 \begin{align*} \begin{cases} \frac{a^{3} x^{2 n}}{2 n} + \frac{a^{2} b x^{3 n}}{n} + \frac{3 a b^{2} x^{4 n}}{4 n} + \frac{b^{3} x^{5 n}}{5 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+2*n)*(a+b*x**n)**3,x)

[Out]

Piecewise((a**3*x**(2*n)/(2*n) + a**2*b*x**(3*n)/n + 3*a*b**2*x**(4*n)/(4*n) + b**3*x**(5*n)/(5*n), Ne(n, 0)),
((a + b)**3*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 )}^{3} x^{2 \, n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+2*n)*(a+b*x^n)^3,x, algorithm="giac")

[Out]

integrate((b*x^n + a)^3*x^(2*n - 1), x)