3.3025 \(\int (a+b (c x^n)^{\frac{1}{n}})^p \, dx\)

Optimal. Leaf size=38 \[ \frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+1}}{b (p+1)} \]

[Out]

(x*(a + b*(c*x^n)^n^(-1))^(1 + p))/(b*(1 + p)*(c*x^n)^n^(-1))

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Rubi [A]  time = 0.0120628, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {254, 32} \[ \frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+1}}{b (p+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*(c*x^n)^n^(-1))^p,x]

[Out]

(x*(a + b*(c*x^n)^n^(-1))^(1 + p))/(b*(1 + p)*(c*x^n)^n^(-1))

Rule 254

Int[((a_) + (b_.)*((c_.)*(x_)^(q_.))^(n_))^(p_.), x_Symbol] :> Dist[x/(c*x^q)^(1/q), Subst[Int[(a + b*x^(n*q))
^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, n, p, q}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.131238, size = 64, normalized size = 1.68 \[ \frac{x \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^p \left (\frac{a \left (c x^n\right )^{-1/n} \left (1-\left (\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )^{-p}\right )}{b}+1\right )}{p+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*(c*x^n)^n^(-1))^p,x]

[Out]

(x*(a + b*(c*x^n)^n^(-1))^p*(1 + (a*(1 - (1 + (b*(c*x^n)^n^(-1))/a)^(-p)))/(b*(c*x^n)^n^(-1))))/(1 + p)

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Maple [C]  time = 0.23, size = 336, normalized size = 8.8 \begin{align*}{\frac{x}{1+p} \left ( b{{\rm e}^{-{\frac{i\pi \,{\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,n\ln \left ( x \right ) -2\,\ln \left ( c \right ) -2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}x+a \right ) ^{p}}+{\frac{a}{ \left ( 1+p \right ) \sqrt [n]{c}b} \left ( b{{\rm e}^{-{\frac{i\pi \,{\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,n\ln \left ( x \right ) -2\,\ln \left ( c \right ) -2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}x+a \right ) ^{p}{{\rm e}^{{\frac{i\pi \,{\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,n\ln \left ( x \right ) -2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*(c*x^n)^(1/n))^p,x)

[Out]

1/(1+p)*x*(b*exp(-1/2*(I*Pi*csgn(I*c*x^n)*csgn(I*c)*csgn(I*x^n)-I*Pi*csgn(I*c*x^n)^2*csgn(I*x^n)-I*Pi*csgn(I*c
*x^n)^2*csgn(I*c)+I*Pi*csgn(I*c*x^n)^3+2*n*ln(x)-2*ln(c)-2*ln(x^n))/n)*x+a)^p+(b*exp(-1/2*(I*Pi*csgn(I*c*x^n)*
csgn(I*c)*csgn(I*x^n)-I*Pi*csgn(I*c*x^n)^2*csgn(I*x^n)-I*Pi*csgn(I*c*x^n)^2*csgn(I*c)+I*Pi*csgn(I*c*x^n)^3+2*n
*ln(x)-2*ln(c)-2*ln(x^n))/n)*x+a)^p*a/(1+p)/(c^(1/n))/b*exp(1/2*(I*Pi*csgn(I*c*x^n)*csgn(I*c)*csgn(I*x^n)-I*Pi
*csgn(I*c*x^n)^2*csgn(I*x^n)-I*Pi*csgn(I*c*x^n)^2*csgn(I*c)+I*Pi*csgn(I*c*x^n)^3+2*n*ln(x)-2*ln(x^n))/n)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((c*x^n)^(1/n)*b + a)^p, x)

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Fricas [A]  time = 1.55481, size = 80, normalized size = 2.11 \begin{align*} \frac{{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}^{p}}{{\left (b p + b\right )} c^{\left (\frac{1}{n}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*c^(1/n)*x + a)*(b*c^(1/n)*x + a)^p/((b*p + b)*c^(1/n))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x**n)**(1/n))**p,x)

[Out]

Integral((a + b*(c*x**n)**(1/n))**p, x)

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Giac [A]  time = 1.2327, size = 73, normalized size = 1.92 \begin{align*} \frac{{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}^{p} b c^{\left (\frac{1}{n}\right )} x +{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}^{p} a}{b c^{\left (\frac{1}{n}\right )} p + b c^{\left (\frac{1}{n}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x^n)^(1/n))^p,x, algorithm="giac")

[Out]

((b*c^(1/n)*x + a)^p*b*c^(1/n)*x + (b*c^(1/n)*x + a)^p*a)/(b*c^(1/n)*p + b*c^(1/n))