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

Optimal. Leaf size=205 \[ \frac{\sqrt{2} \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{a} x^{-n/2}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{a^{3/2} n \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{a} x^{-n/2}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{a^{3/2} n \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{2 x^{-n/2}}{a n} \]

[Out]

-2/(a*n*x^(n/2)) + (Sqrt[2]*(b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2
]*Sqrt[a])/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*x^(n/2))])/(a^(3/2)*Sqrt[b - Sqrt[b^2 -
4*a*c]]*n) + (Sqrt[2]*(b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt
[a])/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*x^(n/2))])/(a^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]
]*n)

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Rubi [A]  time = 0.794926, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\sqrt{2} \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{a} x^{-n/2}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{a^{3/2} n \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{a} x^{-n/2}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{a^{3/2} n \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{2 x^{-n/2}}{a n} \]

Antiderivative was successfully verified.

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

[Out]

-2/(a*n*x^(n/2)) + (Sqrt[2]*(b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2
]*Sqrt[a])/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*x^(n/2))])/(a^(3/2)*Sqrt[b - Sqrt[b^2 -
4*a*c]]*n) + (Sqrt[2]*(b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt
[a])/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*x^(n/2))])/(a^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]
]*n)

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Rubi in Sympy [A]  time = 77.2929, size = 202, normalized size = 0.99 \[ - \frac{2 x^{- \frac{n}{2}}}{a n} + \frac{\sqrt{2} \left (- 2 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{a} x^{- \frac{n}{2}}}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{a^{\frac{3}{2}} n \sqrt{b + \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} - \frac{\sqrt{2} \left (- 2 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{a} x^{- \frac{n}{2}}}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{a^{\frac{3}{2}} n \sqrt{b - \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**(-n/2)/(a*n) + sqrt(2)*(-2*a*c + b**2 + b*sqrt(-4*a*c + b**2))*atan(sqrt(2
)*sqrt(a)*x**(-n/2)/sqrt(b + sqrt(-4*a*c + b**2)))/(a**(3/2)*n*sqrt(b + sqrt(-4*
a*c + b**2))*sqrt(-4*a*c + b**2)) - sqrt(2)*(-2*a*c + b**2 - b*sqrt(-4*a*c + b**
2))*atan(sqrt(2)*sqrt(a)*x**(-n/2)/sqrt(b - sqrt(-4*a*c + b**2)))/(a**(3/2)*n*sq
rt(b - sqrt(-4*a*c + b**2))*sqrt(-4*a*c + b**2))

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Mathematica [C]  time = 0.0975331, size = 105, normalized size = 0.51 \[ -\frac{4 x^{-n/2}-\text{RootSum}\left [\text{$\#$1}^4 a+\text{$\#$1}^2 b+c\&,\frac{2 \text{$\#$1}^2 b \log \left (x^{-n/2}-\text{$\#$1}\right )+\text{$\#$1}^2 b n \log (x)+2 c \log \left (x^{-n/2}-\text{$\#$1}\right )+c n \log (x)}{2 \text{$\#$1}^3 a+\text{$\#$1} b}\&\right ]}{2 a n} \]

Antiderivative was successfully verified.

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

[Out]

-(4/x^(n/2) - RootSum[c + b*#1^2 + a*#1^4 & , (c*n*Log[x] + 2*c*Log[x^(-n/2) - #
1] + b*n*Log[x]*#1^2 + 2*b*Log[x^(-n/2) - #1]*#1^2)/(b*#1 + 2*a*#1^3) & ])/(2*a*
n)

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Maple [C]  time = 0.307, size = 268, normalized size = 1.3 \[ -2\,{\frac{1}{an{x}^{n/2}}}+\sum _{{\it \_R}={\it RootOf} \left ( \left ( 16\,{a}^{5}{c}^{2}{n}^{4}-8\,{a}^{4}{b}^{2}c{n}^{4}+{a}^{3}{b}^{4}{n}^{4} \right ){{\it \_Z}}^{4}+ \left ( 12\,{a}^{2}b{c}^{2}{n}^{2}-7\,a{b}^{3}c{n}^{2}+{b}^{5}{n}^{2} \right ){{\it \_Z}}^{2}+{c}^{3} \right ) }{\it \_R}\,\ln \left ({x}^{{\frac{n}{2}}}+ \left ( -8\,{\frac{{a}^{5}{n}^{3}{c}^{2}}{a{c}^{3}-{b}^{2}{c}^{2}}}+6\,{\frac{{n}^{3}{b}^{2}{a}^{4}c}{a{c}^{3}-{b}^{2}{c}^{2}}}-{\frac{{n}^{3}{b}^{4}{a}^{3}}{a{c}^{3}-{b}^{2}{c}^{2}}} \right ){{\it \_R}}^{3}+ \left ( -5\,{\frac{{a}^{2}b{c}^{2}n}{a{c}^{3}-{b}^{2}{c}^{2}}}+5\,{\frac{a{b}^{3}cn}{a{c}^{3}-{b}^{2}{c}^{2}}}-{\frac{{b}^{5}n}{a{c}^{3}-{b}^{2}{c}^{2}}} \right ){\it \_R} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/a/n/(x^(1/2*n))+sum(_R*ln(x^(1/2*n)+(-8/(a*c^3-b^2*c^2)*n^3*a^5*c^2+6/(a*c^3-
b^2*c^2)*n^3*b^2*a^4*c-1/(a*c^3-b^2*c^2)*n^3*b^4*a^3)*_R^3+(-5/(a*c^3-b^2*c^2)*n
*b*a^2*c^2+5/(a*c^3-b^2*c^2)*n*b^3*a*c-1/(a*c^3-b^2*c^2)*n*b^5)*_R),_R=RootOf((1
6*a^5*c^2*n^4-8*a^4*b^2*c*n^4+a^3*b^4*n^4)*_Z^4+(12*a^2*b*c^2*n^2-7*a*b^3*c*n^2+
b^5*n^2)*_Z^2+c^3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.311095, size = 1659, normalized size = 8.09 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(sqrt(2)*a*n*sqrt(-((a^3*b^2 - 4*a^4*c)*n^2*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)
/((a^6*b^2 - 4*a^7*c)*n^4)) + b^3 - 3*a*b*c)/((a^3*b^2 - 4*a^4*c)*n^2))*log(-(4*
(b^2*c - a*c^2)*x*x^(-1/2*n - 1) + sqrt(2)*((a^3*b^3 - 4*a^4*b*c)*n^3*sqrt((b^4
- 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*n^4)) - (b^4 - 5*a*b^2*c + 4*a^2*c^2
)*n)*sqrt(-((a^3*b^2 - 4*a^4*c)*n^2*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 -
4*a^7*c)*n^4)) + b^3 - 3*a*b*c)/((a^3*b^2 - 4*a^4*c)*n^2)))/x) - sqrt(2)*a*n*sq
rt(-((a^3*b^2 - 4*a^4*c)*n^2*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*
c)*n^4)) + b^3 - 3*a*b*c)/((a^3*b^2 - 4*a^4*c)*n^2))*log(-(4*(b^2*c - a*c^2)*x*x
^(-1/2*n - 1) - sqrt(2)*((a^3*b^3 - 4*a^4*b*c)*n^3*sqrt((b^4 - 2*a*b^2*c + a^2*c
^2)/((a^6*b^2 - 4*a^7*c)*n^4)) - (b^4 - 5*a*b^2*c + 4*a^2*c^2)*n)*sqrt(-((a^3*b^
2 - 4*a^4*c)*n^2*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*n^4)) + b
^3 - 3*a*b*c)/((a^3*b^2 - 4*a^4*c)*n^2)))/x) - sqrt(2)*a*n*sqrt(((a^3*b^2 - 4*a^
4*c)*n^2*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*n^4)) - b^3 + 3*a
*b*c)/((a^3*b^2 - 4*a^4*c)*n^2))*log(-(4*(b^2*c - a*c^2)*x*x^(-1/2*n - 1) + sqrt
(2)*((a^3*b^3 - 4*a^4*b*c)*n^3*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^
7*c)*n^4)) + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*n)*sqrt(((a^3*b^2 - 4*a^4*c)*n^2*sqrt
((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*n^4)) - b^3 + 3*a*b*c)/((a^3*b
^2 - 4*a^4*c)*n^2)))/x) + sqrt(2)*a*n*sqrt(((a^3*b^2 - 4*a^4*c)*n^2*sqrt((b^4 -
2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*n^4)) - b^3 + 3*a*b*c)/((a^3*b^2 - 4*a
^4*c)*n^2))*log(-(4*(b^2*c - a*c^2)*x*x^(-1/2*n - 1) - sqrt(2)*((a^3*b^3 - 4*a^4
*b*c)*n^3*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*n^4)) + (b^4 - 5
*a*b^2*c + 4*a^2*c^2)*n)*sqrt(((a^3*b^2 - 4*a^4*c)*n^2*sqrt((b^4 - 2*a*b^2*c + a
^2*c^2)/((a^6*b^2 - 4*a^7*c)*n^4)) - b^3 + 3*a*b*c)/((a^3*b^2 - 4*a^4*c)*n^2)))/
x) - 4*x*x^(-1/2*n - 1))/(a*n)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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