\(\int (a+b (c x^n)^{\frac {1}{n}})^2 \, dx\) [3004]

Optimal result

Integrand size = 15, antiderivative size = 34 \[ \int \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2 \, dx=\frac {x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^3}{3 b} \]

[Out]

1/3*x*(a+b*(c*x^n)^(1/n))^3/b/((c*x^n)^(1/n))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 34, 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} \[ \int \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2 \, dx=\frac {x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^3}{3 b} \]

[In]

Int[(a + b*(c*x^n)^n^(-1))^2,x]

[Out]

(x*(a + b*(c*x^n)^n^(-1))^3)/(3*b*(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)^2 \, 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 )^3}{3 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.12 \[ \int \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2 \, dx=a^2 x+a b x \left (c x^n\right )^{\frac {1}{n}}+\frac {1}{3} b^2 x \left (c x^n\right )^{2/n} \]

[In]

Integrate[(a + b*(c*x^n)^n^(-1))^2,x]

[Out]

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

Maple [A] (verified)

Time = 4.11 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.44

method	result	size
parallelrisch	\(\frac {x^{2} \left (c \,x^{n}\right )^{\frac {2}{n}} b^{2}+3 x^{2} \left (c \,x^{n}\right )^{\frac {1}{n}} a b +3 a^{2} x^{2}}{3 x}\)	\(49\)

[In]

int((a+b*(c*x^n)^(1/n))^2,x,method=_RETURNVERBOSE)

[Out]

1/3*(x^2*((c*x^n)^(1/n))^2*b^2+3*x^2*(c*x^n)^(1/n)*a*b+3*a^2*x^2)/x

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2 \, dx=\frac {1}{3} \, b^{2} c^{\frac {2}{n}} x^{3} + a b c^{\left (\frac {1}{n}\right )} x^{2} + a^{2} x \]

[In]

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

[Out]

1/3*b^2*c^(2/n)*x^3 + a*b*c^(1/n)*x^2 + a^2*x

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2 \, dx=a^{2} x + a b x \left (c x^{n}\right )^{\frac {1}{n}} + \frac {b^{2} x \left (c x^{n}\right )^{\frac {2}{n}}}{3} \]

[In]

integrate((a+b*(c*x**n)**(1/n))**2,x)

[Out]

a**2*x + a*b*x*(c*x**n)**(1/n) + b**2*x*(c*x**n)**(2/n)/3

Maxima [F]

\[ \int \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2 \, dx=\int { {\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{2} \,d x } \]

[In]

integrate((a+b*(c*x^n)^(1/n))^2,x, algorithm="maxima")

[Out]

b^2*c^(2/n)*integrate((x^n)^(2/n), x) + 2*a*b*c^(1/n)*integrate((x^n)^(1/n), x) + a^2*x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2 \, dx=\frac {1}{3} \, b^{2} c^{\frac {2}{n}} x^{3} + a b c^{\left (\frac {1}{n}\right )} x^{2} + a^{2} x \]

[In]

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

[Out]

1/3*b^2*c^(2/n)*x^3 + a*b*c^(1/n)*x^2 + a^2*x

Mupad [B] (verification not implemented)

Time = 5.48 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2 \, dx=a^2\,x+\frac {b^2\,x\,{\left (c\,x^n\right )}^{2/n}}{3}+a\,b\,x\,{\left (c\,x^n\right )}^{1/n} \]

[In]

int((a + b*(c*x^n)^(1/n))^2,x)

[Out]

a^2*x + (b^2*x*(c*x^n)^(2/n))/3 + a*b*x*(c*x^n)^(1/n)