3.25.3 \(\int (a+b (c x^n)^{\frac {1}{n}})^3 \, dx\)

Optimal. Leaf size=34 \[ \frac {x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^4}{4 b} \]

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Rubi [A]  time = 0.01, antiderivative size = 34, 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 )^4}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*(c*x^n)^n^(-1))^3,x]

[Out]

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

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Mathematica [A]  time = 0.04, size = 60, normalized size = 1.76 \begin {gather*} a^3 x+\frac {3}{2} a^2 b x \left (c x^n\right )^{\frac {1}{n}}+a b^2 x \left (c x^n\right )^{2/n}+\frac {1}{4} b^3 x \left (c x^n\right )^{3/n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*(c*x^n)^n^(-1))^3,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.50, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*(c*x^n)^n^(-1))^3,x]

[Out]

a^3*x + Defer[IntegrateAlgebraic][b*(c*x^n)^n^(-1)*(3*a^2 + 3*a*b*(c*x^n)^n^(-1) + b^2*(c*x^n)^(2/n)), x]

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fricas [A]  time = 1.44, size = 50, normalized size = 1.47 \begin {gather*} \frac {1}{4} \, b^{3} c^{\frac {3}{n}} x^{4} + a b^{2} c^{\frac {2}{n}} x^{3} + \frac {3}{2} \, a^{2} b c^{\left (\frac {1}{n}\right )} x^{2} + a^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x^n)^(1/n))^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.21, size = 50, normalized size = 1.47 \begin {gather*} \frac {1}{4} \, b^{3} c^{\frac {3}{n}} x^{4} + a b^{2} c^{\frac {2}{n}} x^{3} + \frac {3}{2} \, a^{2} b c^{\left (\frac {1}{n}\right )} x^{2} + a^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x^n)^(1/n))^3,x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b \left (c \,x^{n}\right )^{\frac {1}{n}}+a \right )^{3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} b^{3} c^{\frac {3}{n}} \int {\left (x^{n}\right )}^{\frac {3}{n}}\,{d x} + 3 \, a b^{2} c^{\frac {2}{n}} \int {\left (x^{n}\right )}^{\frac {2}{n}}\,{d x} + 3 \, a^{2} b c^{\left (\frac {1}{n}\right )} \int {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )}\,{d x} + a^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x^n)^(1/n))^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.27, size = 56, normalized size = 1.65 \begin {gather*} a^3\,x+\frac {b^3\,x\,{\left (c\,x^n\right )}^{3/n}}{4}+\frac {3\,a^2\,b\,x\,{\left (c\,x^n\right )}^{1/n}}{2}+a\,b^2\,x\,{\left (c\,x^n\right )}^{2/n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*(c*x^n)^(1/n))^3,x)

[Out]

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

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sympy [B]  time = 0.78, size = 63, normalized size = 1.85 \begin {gather*} a^{3} x + \frac {3 a^{2} b c^{\frac {1}{n}} x \left (x^{n}\right )^{\frac {1}{n}}}{2} + a b^{2} c^{\frac {2}{n}} x \left (x^{n}\right )^{\frac {2}{n}} + \frac {b^{3} c^{\frac {3}{n}} x \left (x^{n}\right )^{\frac {3}{n}}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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