3.2780 \(\int \frac {(c x)^{-1+n}}{(a+b x^n)^2} \, dx\)

Optimal. Leaf size=24 \[ \frac {(c x)^n}{a c n \left (a+b x^n\right )} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {264} \[ \frac {(c x)^n}{a c n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^(-1 + n)/(a + b*x^n)^2,x]

[Out]

(c*x)^n/(a*c*n*(a + b*x^n))

Rule 264

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 31, normalized size = 1.29 \[ -\frac {x^{1-n} (c x)^{n-1}}{b n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.74, size = 22, normalized size = 0.92 \[ -\frac {c^{n - 1}}{b^{2} n x^{n} + a b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c^(n - 1)/(b^2*n*x^n + a*b*n)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x\right )^{n - 1}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.03, size = 99, normalized size = 4.12 \[ \frac {x \,{\mathrm e}^{\frac {\left (n -1\right ) \left (-i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i c x \right )+i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i c x \right )^{2}-i \pi \mathrm {csgn}\left (i c x \right )^{3}+2 \ln \relax (c )+2 \ln \relax (x )\right )}{2}}}{\left (b \,x^{n}+a \right ) a n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a/n*x/(b*x^n+a)*exp(1/2*(n-1)*(-I*Pi*csgn(I*c)*csgn(I*x)*csgn(I*c*x)+I*Pi*csgn(I*c)*csgn(I*c*x)^2+I*Pi*csgn(
I*x)*csgn(I*c*x)^2-I*Pi*csgn(I*c*x)^3+2*ln(c)+2*ln(x)))

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maxima [A]  time = 0.55, size = 22, normalized size = 0.92 \[ -\frac {c^{n}}{b^{2} c n x^{n} + a b c n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c^n/(b^2*c*n*x^n + a*b*c*n)

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mupad [B]  time = 1.16, size = 29, normalized size = 1.21 \[ \frac {x\,{\left (c\,x\right )}^{n-1}}{a\,b\,n\,\left (x^n+\frac {a}{b}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^(n - 1)/(a + b*x^n)^2,x)

[Out]

(x*(c*x)^(n - 1))/(a*b*n*(x^n + a/b))

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sympy [A]  time = 43.30, size = 330, normalized size = 13.75 \[ \begin {cases} \tilde {\infty } x & \text {for}\: a = 0 \wedge b = 0 \wedge c = 0 \wedge n = 0 \\- \frac {c^{n} x^{- n}}{b^{2} c n} & \text {for}\: a = 0 \\\frac {\tilde {\infty } c^{n} x^{n}}{c n} & \text {for}\: b = - a x^{- n} \\0^{n - 1} \left (\frac {n x \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} + \frac {n x \Gamma \left (\frac {1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} - \frac {x \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} + \frac {b n x x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a^{2} \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} - \frac {b x x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a^{2} \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )}\right ) & \text {for}\: c = 0 \\\frac {\log {\relax (x )}}{c \left (a + b\right )^{2}} & \text {for}\: n = 0 \\\frac {c^{n} x^{n}}{a^{2} c n + a b c n x^{n}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x, Eq(a, 0) & Eq(b, 0) & Eq(c, 0) & Eq(n, 0)), (-c**n*x**(-n)/(b**2*c*n), Eq(a, 0)), (zoo*c**n*
x**n/(c*n), Eq(b, -a*x**(-n))), (0**(n - 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))) + n*x*gamma(1/n)/(a*(a*n**3*gamma(1 + 1/n) + b*n**3*x**n*gam
ma(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)))), Eq(c, 0)), (log(x)/(c*(a + b)**2), Eq(n, 0)), (c**n*x**n/
(a**2*c*n + a*b*c*n*x**n), True))

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