3.2723 \(\int x^{-1+4 n} (a+b x^n)^p \, dx\)

Optimal. Leaf size=103 \[ -\frac{a^3 \left (a+b x^n\right )^{p+1}}{b^4 n (p+1)}+\frac{3 a^2 \left (a+b x^n\right )^{p+2}}{b^4 n (p+2)}-\frac{3 a \left (a+b x^n\right )^{p+3}}{b^4 n (p+3)}+\frac{\left (a+b x^n\right )^{p+4}}{b^4 n (p+4)} \]

[Out]

-((a^3*(a + b*x^n)^(1 + p))/(b^4*n*(1 + p))) + (3*a^2*(a + b*x^n)^(2 + p))/(b^4*n*(2 + p)) - (3*a*(a + b*x^n)^
(3 + p))/(b^4*n*(3 + p)) + (a + b*x^n)^(4 + p)/(b^4*n*(4 + p))

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Rubi [A]  time = 0.0607738, antiderivative size = 103, 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} \[ -\frac{a^3 \left (a+b x^n\right )^{p+1}}{b^4 n (p+1)}+\frac{3 a^2 \left (a+b x^n\right )^{p+2}}{b^4 n (p+2)}-\frac{3 a \left (a+b x^n\right )^{p+3}}{b^4 n (p+3)}+\frac{\left (a+b x^n\right )^{p+4}}{b^4 n (p+4)} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + 4*n)*(a + b*x^n)^p,x]

[Out]

-((a^3*(a + b*x^n)^(1 + p))/(b^4*n*(1 + p))) + (3*a^2*(a + b*x^n)^(2 + p))/(b^4*n*(2 + p)) - (3*a*(a + b*x^n)^
(3 + p))/(b^4*n*(3 + p)) + (a + b*x^n)^(4 + p)/(b^4*n*(4 + p))

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 )^p \, dx &=\frac{\operatorname{Subst}\left (\int x^3 (a+b x)^p \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^p}{b^3}+\frac{3 a^2 (a+b x)^{1+p}}{b^3}-\frac{3 a (a+b x)^{2+p}}{b^3}+\frac{(a+b x)^{3+p}}{b^3}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^3 \left (a+b x^n\right )^{1+p}}{b^4 n (1+p)}+\frac{3 a^2 \left (a+b x^n\right )^{2+p}}{b^4 n (2+p)}-\frac{3 a \left (a+b x^n\right )^{3+p}}{b^4 n (3+p)}+\frac{\left (a+b x^n\right )^{4+p}}{b^4 n (4+p)}\\ \end{align*}

Mathematica [A]  time = 0.0722685, size = 78, normalized size = 0.76 \[ \frac{\left (a+b x^n\right )^{p+1} \left (\frac{3 a^2 \left (a+b x^n\right )}{p+2}-\frac{a^3}{p+1}-\frac{3 a \left (a+b x^n\right )^2}{p+3}+\frac{\left (a+b x^n\right )^3}{p+4}\right )}{b^4 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x^n)^(1 + p)*(-(a^3/(1 + p)) + (3*a^2*(a + b*x^n))/(2 + p) - (3*a*(a + b*x^n)^2)/(3 + p) + (a + b*x^n)
^3/(4 + p)))/(b^4*n)

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Maple [A]  time = 0.067, size = 171, normalized size = 1.7 \begin{align*} -{\frac{ \left ( -{b}^{4}{p}^{3} \left ({x}^{n} \right ) ^{4}-a{b}^{3}{p}^{3} \left ({x}^{n} \right ) ^{3}-6\,{b}^{4}{p}^{2} \left ({x}^{n} \right ) ^{4}-3\,a{b}^{3}{p}^{2} \left ({x}^{n} \right ) ^{3}-11\,{b}^{4}p \left ({x}^{n} \right ) ^{4}+3\,{a}^{2}{b}^{2}{p}^{2} \left ({x}^{n} \right ) ^{2}-2\,ap \left ({x}^{n} \right ) ^{3}{b}^{3}-6\, \left ({x}^{n} \right ) ^{4}{b}^{4}+3\,{a}^{2}p \left ({x}^{n} \right ) ^{2}{b}^{2}-6\,{a}^{3}p{x}^{n}b+6\,{a}^{4} \right ) \left ( a+b{x}^{n} \right ) ^{p}}{ \left ( 3+p \right ) \left ( 4+p \right ) \left ( 2+p \right ) \left ( 1+p \right ) n{b}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-b^4*p^3*(x^n)^4-a*b^3*p^3*(x^n)^3-6*b^4*p^2*(x^n)^4-3*a*b^3*p^2*(x^n)^3-11*b^4*p*(x^n)^4+3*a^2*b^2*p^2*(x^n
)^2-2*a*p*(x^n)^3*b^3-6*(x^n)^4*b^4+3*a^2*p*(x^n)^2*b^2-6*a^3*p*x^n*b+6*a^4)/(3+p)/(4+p)/(2+p)/(1+p)/n/b^4*(a+
b*x^n)^p

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Maxima [A]  time = 1.00344, size = 154, normalized size = 1.5 \begin{align*} \frac{{\left ({\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} b^{4} x^{4 \, n} +{\left (p^{3} + 3 \, p^{2} + 2 \, p\right )} a b^{3} x^{3 \, n} - 3 \,{\left (p^{2} + p\right )} a^{2} b^{2} x^{2 \, n} + 6 \, a^{3} b p x^{n} - 6 \, a^{4}\right )}{\left (b x^{n} + a\right )}^{p}}{{\left (p^{4} + 10 \, p^{3} + 35 \, p^{2} + 50 \, p + 24\right )} b^{4} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((p^3 + 6*p^2 + 11*p + 6)*b^4*x^(4*n) + (p^3 + 3*p^2 + 2*p)*a*b^3*x^(3*n) - 3*(p^2 + p)*a^2*b^2*x^(2*n) + 6*a^
3*b*p*x^n - 6*a^4)*(b*x^n + a)^p/((p^4 + 10*p^3 + 35*p^2 + 50*p + 24)*b^4*n)

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Fricas [A]  time = 1.42323, size = 327, normalized size = 3.17 \begin{align*} \frac{{\left (6 \, a^{3} b p x^{n} - 6 \, a^{4} +{\left (b^{4} p^{3} + 6 \, b^{4} p^{2} + 11 \, b^{4} p + 6 \, b^{4}\right )} x^{4 \, n} +{\left (a b^{3} p^{3} + 3 \, a b^{3} p^{2} + 2 \, a b^{3} p\right )} x^{3 \, n} - 3 \,{\left (a^{2} b^{2} p^{2} + a^{2} b^{2} p\right )} x^{2 \, n}\right )}{\left (b x^{n} + a\right )}^{p}}{b^{4} n p^{4} + 10 \, b^{4} n p^{3} + 35 \, b^{4} n p^{2} + 50 \, b^{4} n p + 24 \, b^{4} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6*a^3*b*p*x^n - 6*a^4 + (b^4*p^3 + 6*b^4*p^2 + 11*b^4*p + 6*b^4)*x^(4*n) + (a*b^3*p^3 + 3*a*b^3*p^2 + 2*a*b^3
*p)*x^(3*n) - 3*(a^2*b^2*p^2 + a^2*b^2*p)*x^(2*n))*(b*x^n + a)^p/(b^4*n*p^4 + 10*b^4*n*p^3 + 35*b^4*n*p^2 + 50
*b^4*n*p + 24*b^4*n)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{n} + a\right )}^{p} 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)^p,x, algorithm="giac")

[Out]

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