3.324 \(\int (c+d x^n)^{-1-\frac{1}{n}} \, dx\)

Optimal. Leaf size=18 \[ \frac{x \left (c+d x^n\right )^{-1/n}}{c} \]

[Out]

x/(c*(c + d*x^n)^n^(-1))

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Rubi [A]  time = 0.0026634, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {191} \[ \frac{x \left (c+d x^n\right )^{-1/n}}{c} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^n)^(-1 - n^(-1)),x]

[Out]

x/(c*(c + d*x^n)^n^(-1))

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

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

Mathematica [A]  time = 0.0267099, size = 18, normalized size = 1. \[ \frac{x \left (c+d x^n\right )^{-1/n}}{c} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^n)^(-1 - n^(-1)),x]

[Out]

x/(c*(c + d*x^n)^n^(-1))

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Maple [B]  time = 0.061, size = 53, normalized size = 2.9 \begin{align*} x{{\rm e}^{ \left ( -1-{n}^{-1} \right ) \ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) }}+{\frac{dx{{\rm e}^{n\ln \left ( x \right ) }}}{c}{{\rm e}^{ \left ( -1-{n}^{-1} \right ) \ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*x^n)^(-1-1/n),x)

[Out]

x*exp((-1-1/n)*ln(c+d*exp(n*ln(x))))+d/c*x*exp(n*ln(x))*exp((-1-1/n)*ln(c+d*exp(n*ln(x))))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x^{n} + c\right )}^{-\frac{1}{n} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x^n + c)^(-1/n - 1), x)

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Fricas [A]  time = 1.58121, size = 61, normalized size = 3.39 \begin{align*} \frac{d x x^{n} + c x}{{\left (d x^{n} + c\right )}^{\frac{n + 1}{n}} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x*x^n + c*x)/((d*x^n + c)^((n + 1)/n)*c)

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Sympy [A]  time = 35.6476, size = 211, normalized size = 11.72 \begin{align*} \begin{cases} - \frac{d^{- \frac{1}{n}} x x^{- n} \left (x^{n}\right )^{- \frac{1}{n}}}{d n} & \text{for}\: c = 0 \\0^{-1 - \frac{1}{n}} x & \text{for}\: c = - d x^{n} \\x \left (0^{n}\right )^{-1 - \frac{1}{n}} & \text{for}\: c = 0^{n} - d x^{n} \\\frac{c^{2} x}{c^{3} \left (c + d x^{n}\right )^{\frac{1}{n}} + 2 c^{2} d x^{n} \left (c + d x^{n}\right )^{\frac{1}{n}} + c d^{2} x^{2 n} \left (c + d x^{n}\right )^{\frac{1}{n}}} + \frac{c d x x^{n}}{c^{3} \left (c + d x^{n}\right )^{\frac{1}{n}} + 2 c^{2} d x^{n} \left (c + d x^{n}\right )^{\frac{1}{n}} + c d^{2} x^{2 n} \left (c + d x^{n}\right )^{\frac{1}{n}}} + \frac{d x x^{n}}{c^{2} \left (c + d x^{n}\right )^{\frac{1}{n}} + c d x^{n} \left (c + d x^{n}\right )^{\frac{1}{n}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*x**n)**(-1-1/n),x)

[Out]

Piecewise((-d**(-1/n)*x*x**(-n)*(x**n)**(-1/n)/(d*n), Eq(c, 0)), (0**(-1 - 1/n)*x, Eq(c, -d*x**n)), (x*(0**n)*
*(-1 - 1/n), Eq(c, 0**n - d*x**n)), (c**2*x/(c**3*(c + d*x**n)**(1/n) + 2*c**2*d*x**n*(c + d*x**n)**(1/n) + c*
d**2*x**(2*n)*(c + d*x**n)**(1/n)) + c*d*x*x**n/(c**3*(c + d*x**n)**(1/n) + 2*c**2*d*x**n*(c + d*x**n)**(1/n)
+ c*d**2*x**(2*n)*(c + d*x**n)**(1/n)) + d*x*x**n/(c**2*(c + d*x**n)**(1/n) + c*d*x**n*(c + d*x**n)**(1/n)), T
rue))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x^{n} + c\right )}^{-\frac{1}{n} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x^n + c)^(-1/n - 1), x)