3.2646 \(\int \frac{x^{-1-\frac{3 n}{2}}}{a+b x^n} \, dx\)

Optimal. Leaf size=68 \[ -\frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )}{a^{5/2} n}+\frac{2 b x^{-n/2}}{a^2 n}-\frac{2 x^{-3 n/2}}{3 a n} \]

[Out]

-2/(3*a*n*x^((3*n)/2)) + (2*b)/(a^2*n*x^(n/2)) - (2*b^(3/2)*ArcTan[Sqrt[a]/(Sqrt[b]*x^(n/2))])/(a^(5/2)*n)

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Rubi [A]  time = 0.0360525, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {362, 345, 193, 321, 205} \[ -\frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )}{a^{5/2} n}+\frac{2 b x^{-n/2}}{a^2 n}-\frac{2 x^{-3 n/2}}{3 a n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - (3*n)/2)/(a + b*x^n),x]

[Out]

-2/(3*a*n*x^((3*n)/2)) + (2*b)/(a^2*n*x^(n/2)) - (2*b^(3/2)*ArcTan[Sqrt[a]/(Sqrt[b]*x^(n/2))])/(a^(5/2)*n)

Rule 362

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[x^(m + 1)/(a*(m + 1)), x] - Dist[b/a, Int[x^Simplify
[m + n]/(a + b*x^n), x], x] /; FreeQ[{a, b, m, n}, x] && FractionQ[Simplify[(m + 1)/n]] && SumSimplerQ[m, 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{x^{-1-\frac{3 n}{2}}}{a+b x^n} \, dx &=-\frac{2 x^{-3 n/2}}{3 a n}-\frac{b \int \frac{x^{-1-\frac{n}{2}}}{a+b x^n} \, dx}{a}\\ &=-\frac{2 x^{-3 n/2}}{3 a n}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{a+\frac{b}{x^2}} \, dx,x,x^{-n/2}\right )}{a n}\\ &=-\frac{2 x^{-3 n/2}}{3 a n}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{x^2}{b+a x^2} \, dx,x,x^{-n/2}\right )}{a n}\\ &=-\frac{2 x^{-3 n/2}}{3 a n}+\frac{2 b x^{-n/2}}{a^2 n}-\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,x^{-n/2}\right )}{a^2 n}\\ &=-\frac{2 x^{-3 n/2}}{3 a n}+\frac{2 b x^{-n/2}}{a^2 n}-\frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )}{a^{5/2} n}\\ \end{align*}

Mathematica [C]  time = 0.0071368, size = 34, normalized size = 0.5 \[ -\frac{2 x^{-3 n/2} \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-\frac{b x^n}{a}\right )}{3 a n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - (3*n)/2)/(a + b*x^n),x]

[Out]

(-2*Hypergeometric2F1[-3/2, 1, -1/2, -((b*x^n)/a)])/(3*a*n*x^((3*n)/2))

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Maple [A]  time = 0.072, size = 97, normalized size = 1.4 \begin{align*} 2\,{\frac{b}{{a}^{2}n{x}^{n/2}}}-{\frac{2}{3\,an} \left ({x}^{{\frac{n}{2}}} \right ) ^{-3}}+{\frac{b}{{a}^{3}n}\sqrt{-ab}\ln \left ({x}^{{\frac{n}{2}}}+{\frac{1}{b}\sqrt{-ab}} \right ) }-{\frac{b}{{a}^{3}n}\sqrt{-ab}\ln \left ({x}^{{\frac{n}{2}}}-{\frac{1}{b}\sqrt{-ab}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-3/2*n)/(a+b*x^n),x)

[Out]

2*b/a^2/n/(x^(1/2*n))-2/3/a/n/(x^(1/2*n))^3+(-a*b)^(1/2)/a^3*b/n*ln(x^(1/2*n)+1/b*(-a*b)^(1/2))-(-a*b)^(1/2)/a
^3*b/n*ln(x^(1/2*n)-1/b*(-a*b)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} b^{2} \int \frac{x^{\frac{1}{2} \, n}}{a^{2} b x x^{n} + a^{3} x}\,{d x} + \frac{2 \,{\left (3 \, b x^{n} - a\right )}}{3 \, a^{2} n x^{\frac{3}{2} \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-3/2*n)/(a+b*x^n),x, algorithm="maxima")

[Out]

b^2*integrate(x^(1/2*n)/(a^2*b*x*x^n + a^3*x), x) + 2/3*(3*b*x^n - a)/(a^2*n*x^(3/2*n))

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Fricas [A]  time = 1.26689, size = 637, normalized size = 9.37 \begin{align*} \left [-\frac{2 \, a x x^{-\frac{3}{2} \, n - 1} - 3 \, b \sqrt{-\frac{b}{a}} \log \left (-\frac{2 \, a^{3} x^{\frac{5}{3}} x^{-\frac{5}{2} \, n - \frac{5}{3}} \sqrt{-\frac{b}{a}} - a^{3} x^{2} x^{-3 \, n - 2} - 2 \, a^{2} b x x^{-\frac{3}{2} \, n - 1} \sqrt{-\frac{b}{a}} + 2 \, a^{2} b x^{\frac{4}{3}} x^{-2 \, n - \frac{4}{3}} + 2 \, a b^{2} x^{\frac{1}{3}} x^{-\frac{1}{2} \, n - \frac{1}{3}} \sqrt{-\frac{b}{a}} - 2 \, a b^{2} x^{\frac{2}{3}} x^{-n - \frac{2}{3}} + b^{3}}{a^{3} x^{2} x^{-3 \, n - 2} + b^{3}}\right ) - 6 \, b x^{\frac{1}{3}} x^{-\frac{1}{2} \, n - \frac{1}{3}}}{3 \, a^{2} n}, -\frac{2 \,{\left (a x x^{-\frac{3}{2} \, n - 1} - 3 \, b \sqrt{\frac{b}{a}} \arctan \left (\frac{\sqrt{\frac{b}{a}}}{x^{\frac{1}{3}} x^{-\frac{1}{2} \, n - \frac{1}{3}}}\right ) - 3 \, b x^{\frac{1}{3}} x^{-\frac{1}{2} \, n - \frac{1}{3}}\right )}}{3 \, a^{2} n}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-3/2*n)/(a+b*x^n),x, algorithm="fricas")

[Out]

[-1/3*(2*a*x*x^(-3/2*n - 1) - 3*b*sqrt(-b/a)*log(-(2*a^3*x^(5/3)*x^(-5/2*n - 5/3)*sqrt(-b/a) - a^3*x^2*x^(-3*n
- 2) - 2*a^2*b*x*x^(-3/2*n - 1)*sqrt(-b/a) + 2*a^2*b*x^(4/3)*x^(-2*n - 4/3) + 2*a*b^2*x^(1/3)*x^(-1/2*n - 1/3
)*sqrt(-b/a) - 2*a*b^2*x^(2/3)*x^(-n - 2/3) + b^3)/(a^3*x^2*x^(-3*n - 2) + b^3)) - 6*b*x^(1/3)*x^(-1/2*n - 1/3
))/(a^2*n), -2/3*(a*x*x^(-3/2*n - 1) - 3*b*sqrt(b/a)*arctan(sqrt(b/a)/(x^(1/3)*x^(-1/2*n - 1/3))) - 3*b*x^(1/3
)*x^(-1/2*n - 1/3))/(a^2*n)]

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Sympy [A]  time = 5.41361, size = 56, normalized size = 0.82 \begin{align*} - \frac{2 x^{- \frac{3 n}{2}}}{3 a n} + \frac{2 b x^{- \frac{n}{2}}}{a^{2} n} + \frac{2 b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x^{\frac{n}{2}}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1-3/2*n)/(a+b*x**n),x)

[Out]

-2*x**(-3*n/2)/(3*a*n) + 2*b*x**(-n/2)/(a**2*n) + 2*b**(3/2)*atan(sqrt(b)*x**(n/2)/sqrt(a))/(a**(5/2)*n)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{-\frac{3}{2} \, n - 1}}{b x^{n} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-3/2*n)/(a+b*x^n),x, algorithm="giac")

[Out]

integrate(x^(-3/2*n - 1)/(b*x^n + a), x)