\(\int (a+b (c x^n)^{3/n})^2 \, dx\) [3041]

Optimal result

Integrand size = 17, antiderivative size = 43 \[ \int \left (a+b \left (c x^n\right )^{3/n}\right )^2 \, dx=a^2 x+\frac {1}{2} a b x \left (c x^n\right )^{3/n}+\frac {1}{7} b^2 x \left (c x^n\right )^{6/n} \]

[Out]

a^2*x+1/2*a*b*x*(c*x^n)^(3/n)+1/7*b^2*x*(c*x^n)^(6/n)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {260, 200} \[ \int \left (a+b \left (c x^n\right )^{3/n}\right )^2 \, dx=a^2 x+\frac {1}{2} a b x \left (c x^n\right )^{3/n}+\frac {1}{7} b^2 x \left (c x^n\right )^{6/n} \]

[In]

Int[(a + b*(c*x^n)^(3/n))^2,x]

[Out]

a^2*x + (a*b*x*(c*x^n)^(3/n))/2 + (b^2*x*(c*x^n)^(6/n))/7

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

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 \left (a+b x^3\right )^2 \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = \left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \left (a^2+2 a b x^3+b^2 x^6\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = a^2 x+\frac {1}{2} a b x \left (c x^n\right )^{3/n}+\frac {1}{7} b^2 x \left (c x^n\right )^{6/n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \left (a+b \left (c x^n\right )^{3/n}\right )^2 \, dx=a^2 x+\frac {1}{2} a b x \left (c x^n\right )^{3/n}+\frac {1}{7} b^2 x \left (c x^n\right )^{6/n} \]

[In]

Integrate[(a + b*(c*x^n)^(3/n))^2,x]

[Out]

a^2*x + (a*b*x*(c*x^n)^(3/n))/2 + (b^2*x*(c*x^n)^(6/n))/7

Maple [A] (verified)

Time = 5.88 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.98

method	result	size
parallelrisch	\(a^{2} x +\frac {a b x \left (c \,x^{n}\right )^{\frac {3}{n}}}{2}+\frac {b^{2} x \left (c \,x^{n}\right )^{\frac {6}{n}}}{7}\)	\(42\)

[In]

int((a+b*(c*x^n)^(3/n))^2,x,method=_RETURNVERBOSE)

[Out]

a^2*x+1/2*a*b*x*(c*x^n)^(3/n)+1/7*x*((c*x^n)^(3/n))^2*b^2

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \left (a+b \left (c x^n\right )^{3/n}\right )^2 \, dx=\frac {1}{7} \, b^{2} c^{\frac {6}{n}} x^{7} + \frac {1}{2} \, a b c^{\frac {3}{n}} x^{4} + a^{2} x \]

[In]

integrate((a+b*(c*x^n)^(3/n))^2,x, algorithm="fricas")

[Out]

1/7*b^2*c^(6/n)*x^7 + 1/2*a*b*c^(3/n)*x^4 + a^2*x

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.79 \[ \int \left (a+b \left (c x^n\right )^{3/n}\right )^2 \, dx=a^{2} x + \frac {a b x \left (c x^{n}\right )^{\frac {3}{n}}}{2} + \frac {b^{2} x \left (c x^{n}\right )^{\frac {6}{n}}}{7} \]

[In]

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

[Out]

a**2*x + a*b*x*(c*x**n)**(3/n)/2 + b**2*x*(c*x**n)**(6/n)/7

Maxima [F]

\[ \int \left (a+b \left (c x^n\right )^{3/n}\right )^2 \, dx=\int { {\left (\left (c x^{n}\right )^{\frac {3}{n}} b + a\right )}^{2} \,d x } \]

[In]

integrate((a+b*(c*x^n)^(3/n))^2,x, algorithm="maxima")

[Out]

b^2*c^(6/n)*integrate((x^n)^(6/n), x) + 2*a*b*c^(3/n)*integrate((x^n)^(3/n), x) + a^2*x

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \left (a+b \left (c x^n\right )^{3/n}\right )^2 \, dx=\frac {1}{7} \, b^{2} c^{\frac {6}{n}} x^{7} + \frac {1}{2} \, a b c^{\frac {3}{n}} x^{4} + a^{2} x \]

[In]

integrate((a+b*(c*x^n)^(3/n))^2,x, algorithm="giac")

[Out]

1/7*b^2*c^(6/n)*x^7 + 1/2*a*b*c^(3/n)*x^4 + a^2*x

Mupad [B] (verification not implemented)

Time = 5.34 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91 \[ \int \left (a+b \left (c x^n\right )^{3/n}\right )^2 \, dx=a^2\,x+\frac {b^2\,x\,{\left (c\,x^n\right )}^{6/n}}{7}+\frac {a\,b\,x\,{\left (c\,x^n\right )}^{3/n}}{2} \]

[In]

int((a + b*(c*x^n)^(3/n))^2,x)

[Out]

a^2*x + (b^2*x*(c*x^n)^(6/n))/7 + (a*b*x*(c*x^n)^(3/n))/2