3.3040 \(\int (a+b (c x^n)^{3/n})^3 \, dx\)

Optimal. Leaf size=65 \[ \frac{3}{4} a^2 b x \left (c x^n\right )^{3/n}+a^3 x+\frac{3}{7} a b^2 x \left (c x^n\right )^{6/n}+\frac{1}{10} b^3 x \left (c x^n\right )^{9/n} \]

[Out]

a^3*x + (3*a^2*b*x*(c*x^n)^(3/n))/4 + (3*a*b^2*x*(c*x^n)^(6/n))/7 + (b^3*x*(c*x^n)^(9/n))/10

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Rubi [A]  time = 0.0227796, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {254, 194} \[ \frac{3}{4} a^2 b x \left (c x^n\right )^{3/n}+a^3 x+\frac{3}{7} a b^2 x \left (c x^n\right )^{6/n}+\frac{1}{10} b^3 x \left (c x^n\right )^{9/n} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*(c*x^n)^(3/n))^3,x]

[Out]

a^3*x + (3*a^2*b*x*(c*x^n)^(3/n))/4 + (3*a*b^2*x*(c*x^n)^(6/n))/7 + (b^3*x*(c*x^n)^(9/n))/10

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 194

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]

Rubi steps

\begin{align*} \int \left (a+b \left (c x^n\right )^{3/n}\right )^3 \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \left (a+b x^3\right )^3 \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \left (a^3+3 a^2 b x^3+3 a b^2 x^6+b^3 x^9\right ) \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=a^3 x+\frac{3}{4} a^2 b x \left (c x^n\right )^{3/n}+\frac{3}{7} a b^2 x \left (c x^n\right )^{6/n}+\frac{1}{10} b^3 x \left (c x^n\right )^{9/n}\\ \end{align*}

Mathematica [A]  time = 0.0221851, size = 65, normalized size = 1. \[ \frac{3}{4} a^2 b x \left (c x^n\right )^{3/n}+a^3 x+\frac{3}{7} a b^2 x \left (c x^n\right )^{6/n}+\frac{1}{10} b^3 x \left (c x^n\right )^{9/n} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*(c*x^n)^(3/n))^3,x]

[Out]

a^3*x + (3*a^2*b*x*(c*x^n)^(3/n))/4 + (3*a*b^2*x*(c*x^n)^(6/n))/7 + (b^3*x*(c*x^n)^(9/n))/10

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Maple [F]  time = 0.178, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ( c{x}^{n} \right ) ^{3\,{n}^{-1}} \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} b^{3} c^{\frac{9}{n}} \int{\left (x^{n}\right )}^{\frac{9}{n}}\,{d x} + 3 \, a b^{2} c^{\frac{6}{n}} \int{\left (x^{n}\right )}^{\frac{6}{n}}\,{d x} + 3 \, a^{2} b c^{\frac{3}{n}} \int{\left (x^{n}\right )}^{\frac{3}{n}}\,{d x} + a^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^3*c^(9/n)*integrate((x^n)^(9/n), x) + 3*a*b^2*c^(6/n)*integrate((x^n)^(6/n), x) + 3*a^2*b*c^(3/n)*integrate(
(x^n)^(3/n), x) + a^3*x

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Fricas [A]  time = 1.28372, size = 107, normalized size = 1.65 \begin{align*} \frac{1}{10} \, b^{3} c^{\frac{9}{n}} x^{10} + \frac{3}{7} \, a b^{2} c^{\frac{6}{n}} x^{7} + \frac{3}{4} \, a^{2} b c^{\frac{3}{n}} x^{4} + a^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^3*c^(9/n)*x^10 + 3/7*a*b^2*c^(6/n)*x^7 + 3/4*a^2*b*c^(3/n)*x^4 + a^3*x

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Sympy [A]  time = 0.76088, size = 66, normalized size = 1.02 \begin{align*} a^{3} x + \frac{3 a^{2} b c^{\frac{3}{n}} x \left (x^{n}\right )^{\frac{3}{n}}}{4} + \frac{3 a b^{2} c^{\frac{6}{n}} x \left (x^{n}\right )^{\frac{6}{n}}}{7} + \frac{b^{3} c^{\frac{9}{n}} x \left (x^{n}\right )^{\frac{9}{n}}}{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x + 3*a**2*b*c**(3/n)*x*(x**n)**(3/n)/4 + 3*a*b**2*c**(6/n)*x*(x**n)**(6/n)/7 + b**3*c**(9/n)*x*(x**n)**(
9/n)/10

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Giac [A]  time = 1.59795, size = 72, normalized size = 1.11 \begin{align*} \frac{1}{10} \, b^{3} c^{\frac{9}{n}} x^{10} + \frac{3}{7} \, a b^{2} c^{\frac{6}{n}} x^{7} + \frac{3}{4} \, a^{2} b c^{\frac{3}{n}} x^{4} + a^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^3*c^(9/n)*x^10 + 3/7*a*b^2*c^(6/n)*x^7 + 3/4*a^2*b*c^(3/n)*x^4 + a^3*x