3.233 \(\int \left (c+d x^n\right )^{-2-\frac{1}{n}} \, dx\)

Optimal. Leaf size=50 \[ \frac{n x \left (c+d x^n\right )^{-1/n}}{c^2 (n+1)}+\frac{x \left (c+d x^n\right )^{-\frac{1}{n}-1}}{c (n+1)} \]

[Out]

(x*(c + d*x^n)^(-1 - n^(-1)))/(c*(1 + n)) + (n*x)/(c^2*(1 + n)*(c + d*x^n)^n^(-1
))

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Rubi [A]  time = 0.0396865, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{n x \left (c+d x^n\right )^{-1/n}}{c^2 (n+1)}+\frac{x \left (c+d x^n\right )^{-\frac{1}{n}-1}}{c (n+1)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^n)^(-2 - n^(-1)),x]

[Out]

(x*(c + d*x^n)^(-1 - n^(-1)))/(c*(1 + n)) + (n*x)/(c^2*(1 + n)*(c + d*x^n)^n^(-1
))

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Rubi in Sympy [A]  time = 3.68181, size = 39, normalized size = 0.78 \[ \frac{x \left (c + d x^{n}\right )^{-1 - \frac{1}{n}}}{c \left (n + 1\right )} + \frac{n x \left (c + d x^{n}\right )^{- \frac{1}{n}}}{c^{2} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c+d*x**n)**(-2-1/n),x)

[Out]

x*(c + d*x**n)**(-1 - 1/n)/(c*(n + 1)) + n*x*(c + d*x**n)**(-1/n)/(c**2*(n + 1))

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Mathematica [C]  time = 0.0370937, size = 55, normalized size = 1.1 \[ \frac{x \left (c+d x^n\right )^{-1/n} \left (\frac{d x^n}{c}+1\right )^{\frac{1}{n}} \, _2F_1\left (2+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^n)^(-2 - n^(-1)),x]

[Out]

(x*(1 + (d*x^n)/c)^n^(-1)*Hypergeometric2F1[2 + n^(-1), n^(-1), 1 + n^(-1), -((d
*x^n)/c)])/(c^2*(c + d*x^n)^n^(-1))

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Maple [F]  time = 0.131, size = 0, normalized size = 0. \[ \int \left ( c+d{x}^{n} \right ) ^{-2-{n}^{-1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d*x^n)^(-2-1/n),x)

[Out]

int((c+d*x^n)^(-2-1/n),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (d x^{n} + c\right )}^{-\frac{1}{n} - 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^n + c)^(-1/n - 2),x, algorithm="maxima")

[Out]

integrate((d*x^n + c)^(-1/n - 2), x)

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Fricas [A]  time = 0.256259, size = 92, normalized size = 1.84 \[ \frac{d^{2} n x x^{2 \, n} +{\left (2 \, c d n + c d\right )} x x^{n} +{\left (c^{2} n + c^{2}\right )} x}{{\left (c^{2} n + c^{2}\right )}{\left (d x^{n} + c\right )}^{\frac{2 \, n + 1}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^n + c)^(-1/n - 2),x, algorithm="fricas")

[Out]

(d^2*n*x*x^(2*n) + (2*c*d*n + c*d)*x*x^n + (c^2*n + c^2)*x)/((c^2*n + c^2)*(d*x^
n + c)^((2*n + 1)/n))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c+d*x**n)**(-2-1/n),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (d x^{n} + c\right )}^{-\frac{1}{n} - 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^n + c)^(-1/n - 2),x, algorithm="giac")

[Out]

integrate((d*x^n + c)^(-1/n - 2), x)