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)^(1/n))

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.04, size = 18, normalized size = 1.00 \[ \frac {x \left (c+d x^n\right )^{-1/n}}{c} \]

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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fricas [A]  time = 1.00, size = 31, normalized size = 1.72 \[ \frac {d x x^{n} + c x}{{\left (d x^{n} + c\right )}^{\frac {n + 1}{n}} c} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x^{n} + c\right )}^{-\frac {1}{n} - 1}\,{d x} \]

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)

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maple [B]  time = 0.07, size = 53, normalized size = 2.94 \[ \frac {d x \,{\mathrm e}^{n \ln \relax (x )} {\mathrm e}^{\left (-\frac {1}{n}-1\right ) \ln \left (d \,{\mathrm e}^{n \ln \relax (x )}+c \right )}}{c}+x \,{\mathrm e}^{\left (-\frac {1}{n}-1\right ) \ln \left (d \,{\mathrm e}^{n \ln \relax (x )}+c \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x^{n} + c\right )}^{-\frac {1}{n} - 1}\,{d x} \]

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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mupad [B]  time = 1.76, size = 75, normalized size = 4.17 \[ \frac {d\,x^{n+1}\,\left (\frac {c}{d\,x^n}-{\left (\frac {c}{d\,x^n}+1\right )}^{\frac {n+1}{n}}+1\right )}{c\,n\,\left (\frac {n+1}{n}-1\right )\,{\left (c+d\,x^n\right )}^{\frac {n+1}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c + d*x^n)^(1/n + 1),x)

[Out]

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

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sympy [A]  time = 33.05, size = 211, normalized size = 11.72 \[ \begin {cases} - \frac {d^{- \frac {1}{n}} x x^{- n} \left (x^{n}\right )^{- \frac {1}{n}}}{d n} & \text {for}\: c = 0 \\0^{-1 - \frac {1}{n}} x & \text {for}\: c = - d x^{n} \\x \left (0^{n}\right )^{-1 - \frac {1}{n}} & \text {for}\: c = 0^{n} - d x^{n} \\\frac {c^{2} x}{c^{3} \left (c + d x^{n}\right )^{\frac {1}{n}} + 2 c^{2} d x^{n} \left (c + d x^{n}\right )^{\frac {1}{n}} + c d^{2} x^{2 n} \left (c + d x^{n}\right )^{\frac {1}{n}}} + \frac {c d x x^{n}}{c^{3} \left (c + d x^{n}\right )^{\frac {1}{n}} + 2 c^{2} d x^{n} \left (c + d x^{n}\right )^{\frac {1}{n}} + c d^{2} x^{2 n} \left (c + d x^{n}\right )^{\frac {1}{n}}} + \frac {d x x^{n}}{c^{2} \left (c + d x^{n}\right )^{\frac {1}{n}} + c d x^{n} \left (c + d x^{n}\right )^{\frac {1}{n}}} & \text {otherwise} \end {cases} \]

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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