3.25.78 \(\int \frac {1}{(a+b x^n)^2} \, dx\) [2478]

Optimal. Leaf size=24 \[ \frac {x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2} \]

[Out]

x*hypergeom([2, 1/n],[1+1/n],-b*x^n/a)/a^2

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Rubi [A]
time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {251} \begin {gather*} \frac {x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^(-2),x]

[Out]

(x*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/a^2

Rule 251

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b x^n\right )^2} \, dx &=\frac {x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 24, normalized size = 1.00 \begin {gather*} \frac {x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^n)^(-2),x]

[Out]

(x*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/a^2

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \,x^{n}\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(a+b*x^n)^2,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(1/(a+b*x^n)^2,x, algorithm="maxima")

[Out]

(n - 1)*integrate(1/(a*b*n*x^n + a^2*n), x) + x/(a*b*n*x^n + a^2*n)

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Fricas [F]
time = 0.39, size = 24, normalized size = 1.00 \begin {gather*} {\rm integral}\left (\frac {1}{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*x^n)^2,x, algorithm="fricas")

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.52, size = 257, normalized size = 10.71 \begin {gather*} \frac {n x \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} + \frac {n x \Gamma \left (\frac {1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} - \frac {x \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} + \frac {b n x x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a^{2} \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} - \frac {b x x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a^{2} \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*x**n)**2,x)

[Out]

n*x*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1/n)*gamma(1/n)/(a*(a*n**3*gamma(1 + 1/n) + b*n**3*x**n*gamma(1 + 1/
n))) + n*x*gamma(1/n)/(a*(a*n**3*gamma(1 + 1/n) + b*n**3*x**n*gamma(1 + 1/n))) - x*lerchphi(b*x**n*exp_polar(I
*pi)/a, 1, 1/n)*gamma(1/n)/(a*(a*n**3*gamma(1 + 1/n) + b*n**3*x**n*gamma(1 + 1/n))) + b*n*x*x**n*lerchphi(b*x*
*n*exp_polar(I*pi)/a, 1, 1/n)*gamma(1/n)/(a**2*(a*n**3*gamma(1 + 1/n) + b*n**3*x**n*gamma(1 + 1/n))) - b*x*x**
n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1/n)*gamma(1/n)/(a**2*(a*n**3*gamma(1 + 1/n) + b*n**3*x**n*gamma(1 + 1
/n)))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*x^n)^2,x, algorithm="giac")

[Out]

integrate((b*x^n + a)^(-2), x)

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Mupad [B]
time = 1.17, size = 25, normalized size = 1.04 \begin {gather*} \frac {x\,{{}}_2{\mathrm {F}}_1\left (2,\frac {1}{n};\ \frac {1}{n}+1;\ -\frac {b\,x^n}{a}\right )}{a^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x^n)^2,x)

[Out]

(x*hypergeom([2, 1/n], 1/n + 1, -(b*x^n)/a))/a^2

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