3.2762 \(\int \frac{(c x)^{-1-\frac{n}{2}}}{a+b x^n} \, dx\)

Optimal. Leaf size=74 \[ \frac{2 \sqrt{b} x^{n/2} (c x)^{-n/2} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )}{a^{3/2} c n}-\frac{2 (c x)^{-n/2}}{a c n} \]

[Out]

-2/(a*c*n*(c*x)^(n/2)) + (2*Sqrt[b]*x^(n/2)*ArcTan[Sqrt[a]/(Sqrt[b]*x^(n/2))])/(a^(3/2)*c*n*(c*x)^(n/2))

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Rubi [A]  time = 0.0365788, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {347, 345, 193, 321, 205} \[ \frac{2 \sqrt{b} x^{n/2} (c x)^{-n/2} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )}{a^{3/2} c n}-\frac{2 (c x)^{-n/2}}{a c n} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^(-1 - n/2)/(a + b*x^n),x]

[Out]

-2/(a*c*n*(c*x)^(n/2)) + (2*Sqrt[b]*x^(n/2)*ArcTan[Sqrt[a]/(Sqrt[b]*x^(n/2))])/(a^(3/2)*c*n*(c*x)^(n/2))

Rule 347

Int[((c_)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracP
art[m], Int[x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && IntegerQ[Simplify[n/(m + 1)]] &&  !In
tegerQ[n]

Rule 345

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/(m + 1), Subst[Int[(a + b*x^Simplify[n/(m +
1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[n/(m + 1)]] &&  !IntegerQ[n]

Rule 193

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

Rule 321

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

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{(c x)^{-1-\frac{n}{2}}}{a+b x^n} \, dx &=\frac{\left (x^{n/2} (c x)^{-n/2}\right ) \int \frac{x^{-1-\frac{n}{2}}}{a+b x^n} \, dx}{c}\\ &=-\frac{\left (2 x^{n/2} (c x)^{-n/2}\right ) \operatorname{Subst}\left (\int \frac{1}{a+\frac{b}{x^2}} \, dx,x,x^{-n/2}\right )}{c n}\\ &=-\frac{\left (2 x^{n/2} (c x)^{-n/2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{b+a x^2} \, dx,x,x^{-n/2}\right )}{c n}\\ &=-\frac{2 (c x)^{-n/2}}{a c n}+\frac{\left (2 b x^{n/2} (c x)^{-n/2}\right ) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,x^{-n/2}\right )}{a c n}\\ &=-\frac{2 (c x)^{-n/2}}{a c n}+\frac{2 \sqrt{b} x^{n/2} (c x)^{-n/2} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )}{a^{3/2} c n}\\ \end{align*}

Mathematica [C]  time = 0.0109565, size = 37, normalized size = 0.5 \[ -\frac{2 x (c x)^{-\frac{n}{2}-1} \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{b x^n}{a}\right )}{a n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{a+b{x}^{n}} \left ( cx \right ) ^{-1-{\frac{n}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -b \int \frac{x^{\frac{1}{2} \, n}}{a b c^{\frac{1}{2} \, n + 1} x x^{n} + a^{2} c^{\frac{1}{2} \, n + 1} x}\,{d x} - \frac{2 \, c^{-\frac{1}{2} \, n - 1}}{a n x^{\frac{1}{2} \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.33034, size = 599, normalized size = 8.09 \begin{align*} \left [-\frac{2 \, x e^{\left (-\frac{1}{2} \,{\left (n + 2\right )} \log \left (c\right ) - \frac{1}{2} \,{\left (n + 2\right )} \log \left (x\right )\right )} - \sqrt{-\frac{b c^{-n - 2}}{a}} \log \left (\frac{a x^{2} e^{\left (-{\left (n + 2\right )} \log \left (c\right ) -{\left (n + 2\right )} \log \left (x\right )\right )} + 2 \, a \sqrt{-\frac{b c^{-n - 2}}{a}} x e^{\left (-\frac{1}{2} \,{\left (n + 2\right )} \log \left (c\right ) - \frac{1}{2} \,{\left (n + 2\right )} \log \left (x\right )\right )} - b c^{-n - 2}}{a x^{2} e^{\left (-{\left (n + 2\right )} \log \left (c\right ) -{\left (n + 2\right )} \log \left (x\right )\right )} + b c^{-n - 2}}\right )}{a n}, -\frac{2 \,{\left (x e^{\left (-\frac{1}{2} \,{\left (n + 2\right )} \log \left (c\right ) - \frac{1}{2} \,{\left (n + 2\right )} \log \left (x\right )\right )} + \sqrt{\frac{b c^{-n - 2}}{a}} \arctan \left (\frac{\sqrt{\frac{b c^{-n - 2}}{a}} e^{\left (\frac{1}{2} \,{\left (n + 2\right )} \log \left (c\right ) + \frac{1}{2} \,{\left (n + 2\right )} \log \left (x\right )\right )}}{x}\right )\right )}}{a n}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-(2*x*e^(-1/2*(n + 2)*log(c) - 1/2*(n + 2)*log(x)) - sqrt(-b*c^(-n - 2)/a)*log((a*x^2*e^(-(n + 2)*log(c) - (n
+ 2)*log(x)) + 2*a*sqrt(-b*c^(-n - 2)/a)*x*e^(-1/2*(n + 2)*log(c) - 1/2*(n + 2)*log(x)) - b*c^(-n - 2))/(a*x^
2*e^(-(n + 2)*log(c) - (n + 2)*log(x)) + b*c^(-n - 2))))/(a*n), -2*(x*e^(-1/2*(n + 2)*log(c) - 1/2*(n + 2)*log
(x)) + sqrt(b*c^(-n - 2)/a)*arctan(sqrt(b*c^(-n - 2)/a)*e^(1/2*(n + 2)*log(c) + 1/2*(n + 2)*log(x))/x))/(a*n)]

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Sympy [A]  time = 4.52979, size = 54, normalized size = 0.73 \begin{align*} - \frac{2 c^{- \frac{n}{2}} x^{- \frac{n}{2}}}{a c n} - \frac{2 \sqrt{b} c^{- \frac{n}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x^{\frac{n}{2}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} c n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*c**(-n/2)*x**(-n/2)/(a*c*n) - 2*sqrt(b)*c**(-n/2)*atan(sqrt(b)*x**(n/2)/sqrt(a))/(a**(3/2)*c*n)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{-\frac{1}{2} \, n - 1}}{b x^{n} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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