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

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

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Rubi [A]  time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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)} \end {gather*}

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 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]

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)]

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*}

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Mathematica [A]  time = 0.15, size = 64, normalized size = 1.68 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.02, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*(c*x^n)^n^(-1))^p,x]

[Out]

Defer[IntegrateAlgebraic][(a + b*(c*x^n)^n^(-1))^p, x]

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fricas [A]  time = 1.22, size = 37, normalized size = 0.97 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.31, size = 54, normalized size = 1.42 \begin {gather*} \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 {gather*}

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))

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maple [C]  time = 0.16, size = 147, normalized size = 3.87 \begin {gather*} \frac {x \,c^{-\frac {1}{n}} \left (x^{n}\right )^{-\frac {1}{n}} \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}+a \right )^{p +1} {\mathrm e}^{-\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}}{\left (p +1\right ) b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{p}\,{d x} \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int {\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*(c*x^n)^(1/n))^p,x)

[Out]

int((a + b*(c*x^n)^(1/n))^p, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )^{p}\, dx \end {gather*}

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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