\(\int (a+b (c x^n)^{\frac {1}{n}})^p \, dx\) [3025]

Optimal result

Integrand size = 15, antiderivative size = 38 \[ \int \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\frac {x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{1+p}}{b (1+p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {260, 32} \[ \int \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\frac {x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+1}}{b (p+1)} \]

[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 260

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*} \text {integral}& = \left (x \left (c x^n\right )^{-1/n}\right ) \text {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] (verified)

Time = 0.16 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.68 \[ \int \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\frac {x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \left (1+\frac {a \left (c x^n\right )^{-1/n} \left (1-\left (1+\frac {b \left (c x^n\right )^{\frac {1}{n}}}{a}\right )^{-p}\right )}{b}\right )}{1+p} \]

[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)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 6.29 (sec) , antiderivative size = 147, normalized size of antiderivative = 3.87

method	result	size
risch	\(\frac {{\left (b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right )}^{1+p} c^{-\frac {1}{n}} x \left (x^{n}\right )^{-\frac {1}{n}} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}}{b \left (1+p \right )}\)	\(147\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97 \[ \int \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\frac {{\left (b c^{\left (\frac {1}{n}\right )} x + a\right )} {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p}}{{\left (b p + b\right )} c^{\left (\frac {1}{n}\right )}} \]

[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))

Sympy [F]

\[ \int \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\int \left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )^{p}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\int { {\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{p} \,d x } \]

[In]

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

[Out]

integrate(((c*x^n)^(1/n)*b + a)^p, x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.42 \[ \int \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\frac {{\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} b c^{\left (\frac {1}{n}\right )} x + {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} a}{b c^{\left (\frac {1}{n}\right )} p + b c^{\left (\frac {1}{n}\right )}} \]

[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))

Mupad [F(-1)]

Timed out. \[ \int \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\int {\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^p \,d x \]

[In]

int((a + b*(c*x^n)^(1/n))^p,x)

[Out]

int((a + b*(c*x^n)^(1/n))^p, x)