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

Optimal. Leaf size=21 \[ a x+\frac {1}{4} b x \left (c x^n\right )^{3/n} \]

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Rubi [A]  time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {15, 30} \begin {gather*} a x+\frac {1}{4} b x \left (c x^n\right )^{3/n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[a + b*(c*x^n)^(3/n),x]

[Out]

a*x + (b*x*(c*x^n)^(3/n))/4

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} a x+\frac {1}{4} b x \left (c x^n\right )^{3/n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a + b*(c*x^n)^(3/n),x]

[Out]

a*x + (b*x*(c*x^n)^(3/n))/4

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IntegrateAlgebraic [A]  time = 0.13, size = 21, normalized size = 1.00 \begin {gather*} a x+\frac {1}{4} b x \left (c x^n\right )^{3/n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a*x + (b*x*(c*x^n)^(3/n))/4

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fricas [A]  time = 0.83, size = 17, normalized size = 0.81 \begin {gather*} \frac {1}{4} \, b c^{\frac {3}{n}} x^{4} + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b*c^(3/n)*x^4 + a*x

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giac [A]  time = 0.17, size = 17, normalized size = 0.81 \begin {gather*} \frac {1}{4} \, b c^{\frac {3}{n}} x^{4} + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b*c^(3/n)*x^4 + a*x

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maple [A]  time = 0.05, size = 23, normalized size = 1.10 \begin {gather*} \frac {b x \,{\mathrm e}^{\frac {3 \ln \left (c \,{\mathrm e}^{n \ln \relax (x )}\right )}{n}}}{4}+a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.17, size = 19, normalized size = 0.90 \begin {gather*} a\,x+\frac {b\,x\,{\left (c\,x^n\right )}^{3/n}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*(c*x^n)^(3/n),x)

[Out]

a*x + (b*x*(c*x^n)^(3/n))/4

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sympy [A]  time = 0.30, size = 19, normalized size = 0.90 \begin {gather*} a x + \frac {b 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)**(3/n),x)

[Out]

a*x + b*c**(3/n)*x*(x**n)**(3/n)/4

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