3.2639 \(\int \frac{x^{-1-\frac{3 n}{2}}}{a+b x^n} \, dx\)

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 = 0.0975609, 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 \[ -\frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )}{a^{5/2} n}+\frac{2 b x^{-n/2}}{a^2 n}-\frac{2 x^{-3 n/2}}{3 a n} \]

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)

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Rubi in Sympy [A]  time = 15.5261, size = 56, normalized size = 0.82 \[ - \frac{2 x^{- \frac{3 n}{2}}}{3 a n} + \frac{2 b x^{- \frac{n}{2}}}{a^{2} n} - \frac{2 b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} x^{- \frac{n}{2}}}{\sqrt{b}} \right )}}{a^{\frac{5}{2}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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(a)*x**(-n
/2)/sqrt(b))/(a**(5/2)*n)

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Mathematica [A]  time = 0.0911584, size = 62, normalized size = 0.91 \[ \frac{2 \left (\sqrt{a} x^{-3 n/2} \left (3 b x^n-a\right )-3 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )\right )}{3 a^{5/2} n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.12, size = 97, normalized size = 1.4 \[ 2\,{\frac{b}{{a}^{2}n{x}^{n/2}}}-{\frac{2}{3\,an} \left ({x}^{{\frac{n}{2}}} \right ) ^{-3}}+{\frac{b}{{a}^{3}n}\sqrt{-ab}\ln \left ({x}^{{\frac{n}{2}}}+{\frac{1}{b}\sqrt{-ab}} \right ) }-{\frac{b}{{a}^{3}n}\sqrt{-ab}\ln \left ({x}^{{\frac{n}{2}}}-{\frac{1}{b}\sqrt{-ab}} \right ) } \]

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. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.261498, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, a x x^{-\frac{3}{2} \, n - 1} - 3 \, b \sqrt{-\frac{b}{a}} \log \left (-\frac{2 \, a x^{\frac{1}{3}} x^{-\frac{1}{2} \, n - \frac{1}{3}} \sqrt{-\frac{b}{a}} - a x^{\frac{2}{3}} x^{-n - \frac{2}{3}} + b}{a x^{\frac{2}{3}} x^{-n - \frac{2}{3}} + b}\right ) - 6 \, b x^{\frac{1}{3}} x^{-\frac{1}{2} \, n - \frac{1}{3}}}{3 \, a^{2} n}, -\frac{2 \,{\left (a x x^{-\frac{3}{2} \, n - 1} - 3 \, b \sqrt{\frac{b}{a}} \arctan \left (\frac{\sqrt{\frac{b}{a}}}{x^{\frac{1}{3}} x^{-\frac{1}{2} \, n - \frac{1}{3}}}\right ) - 3 \, b x^{\frac{1}{3}} x^{-\frac{1}{2} \, n - \frac{1}{3}}\right )}}{3 \, a^{2} n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/3*(2*a*x*x^(-3/2*n - 1) - 3*b*sqrt(-b/a)*log(-(2*a*x^(1/3)*x^(-1/2*n - 1/3)*
sqrt(-b/a) - a*x^(2/3)*x^(-n - 2/3) + b)/(a*x^(2/3)*x^(-n - 2/3) + b)) - 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 [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-\frac{3}{2} \, n - 1}}{b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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