3.31.3 \(\int (a+b (c x^n)^{\frac {1}{n}}) \, dx\) [3003]

Optimal. Leaf size=19 \[ a x+\frac {1}{2} b x \left (c x^n\right )^{\frac {1}{n}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 30} \begin {gather*} a x+\frac {1}{2} b x \left (c x^n\right )^{\frac {1}{n}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[a + b*(c*x^n)^n^(-1),x]

[Out]

a*x + (b*x*(c*x^n)^n^(-1))/2

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 )^{\frac {1}{n}}\right ) \, dx &=a x+b \int \left (c x^n\right )^{\frac {1}{n}} \, dx\\ &=a x+\frac {\left (b \left (c x^n\right )^{\frac {1}{n}}\right ) \int x \, dx}{x}\\ &=a x+\frac {1}{2} b x \left (c x^n\right )^{\frac {1}{n}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 19, normalized size = 1.00 \begin {gather*} a x+\frac {1}{2} b x \left (c x^n\right )^{\frac {1}{n}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a + b*(c*x^n)^n^(-1),x]

[Out]

a*x + (b*x*(c*x^n)^n^(-1))/2

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Maple [A]
time = 0.06, size = 22, normalized size = 1.16

method	result	size
default	\(a x +\frac {x b \,{\mathrm e}^{\frac {\ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )}{n}}}{2}\)	\(22\)
norman	\(a x +\frac {x b \,{\mathrm e}^{\frac {\ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )}{n}}}{2}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.37, size = 15, normalized size = 0.79 \begin {gather*} \frac {1}{2} \, b c^{\left (\frac {1}{n}\right )} x^{2} + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*c^(1/n)*x^2 + a*x

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Sympy [A]
time = 0.09, size = 15, normalized size = 0.79 \begin {gather*} a x + \frac {b x \left (c x^{n}\right )^{\frac {1}{n}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*x*(c*x**n)**(1/n)/2

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Giac [A]
time = 3.53, size = 15, normalized size = 0.79 \begin {gather*} \frac {1}{2} \, b c^{\left (\frac {1}{n}\right )} x^{2} + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*c^(1/n)*x^2 + a*x

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Mupad [B]
time = 1.20, size = 17, normalized size = 0.89 \begin {gather*} a\,x+\frac {b\,x\,{\left (c\,x^n\right )}^{1/n}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*(c*x^n)^(1/n),x)

[Out]

a*x + (b*x*(c*x^n)^(1/n))/2

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