3.2638 \(\int \frac{x^{-1-\frac{n}{4}}}{a+b x^n} \, dx\)

Optimal. Leaf size=234 \[ -\frac{\sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{5/4} n}+\frac{\sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{5/4} n}-\frac{\sqrt{2} \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{5/4} n}+\frac{\sqrt{2} \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}+1\right )}{a^{5/4} n}-\frac{4 x^{-n/4}}{a n} \]

[Out]

-4/(a*n*x^(n/4)) - (Sqrt[2]*b^(1/4)*ArcTan[1 - (Sqrt[2]*a^(1/4))/(b^(1/4)*x^(n/4
))])/(a^(5/4)*n) + (Sqrt[2]*b^(1/4)*ArcTan[1 + (Sqrt[2]*a^(1/4))/(b^(1/4)*x^(n/4
))])/(a^(5/4)*n) - (b^(1/4)*Log[Sqrt[b] + Sqrt[a]/x^(n/2) - (Sqrt[2]*a^(1/4)*b^(
1/4))/x^(n/4)])/(Sqrt[2]*a^(5/4)*n) + (b^(1/4)*Log[Sqrt[b] + Sqrt[a]/x^(n/2) + (
Sqrt[2]*a^(1/4)*b^(1/4))/x^(n/4)])/(Sqrt[2]*a^(5/4)*n)

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Rubi [A]  time = 0.408694, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ -\frac{\sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{5/4} n}+\frac{\sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{5/4} n}-\frac{\sqrt{2} \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{5/4} n}+\frac{\sqrt{2} \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}+1\right )}{a^{5/4} n}-\frac{4 x^{-n/4}}{a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - n/4)/(a + b*x^n),x]

[Out]

-4/(a*n*x^(n/4)) - (Sqrt[2]*b^(1/4)*ArcTan[1 - (Sqrt[2]*a^(1/4))/(b^(1/4)*x^(n/4
))])/(a^(5/4)*n) + (Sqrt[2]*b^(1/4)*ArcTan[1 + (Sqrt[2]*a^(1/4))/(b^(1/4)*x^(n/4
))])/(a^(5/4)*n) - (b^(1/4)*Log[Sqrt[b] + Sqrt[a]/x^(n/2) - (Sqrt[2]*a^(1/4)*b^(
1/4))/x^(n/4)])/(Sqrt[2]*a^(5/4)*n) + (b^(1/4)*Log[Sqrt[b] + Sqrt[a]/x^(n/2) + (
Sqrt[2]*a^(1/4)*b^(1/4))/x^(n/4)])/(Sqrt[2]*a^(5/4)*n)

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Rubi in Sympy [A]  time = 62.902, size = 202, normalized size = 0.86 \[ - \frac{4 x^{- \frac{n}{4}}}{a n} - \frac{\sqrt{2} \sqrt [4]{b} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{- \frac{n}{4}} + \sqrt{a} x^{- \frac{n}{2}} + \sqrt{b} \right )}}{2 a^{\frac{5}{4}} n} + \frac{\sqrt{2} \sqrt [4]{b} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{- \frac{n}{4}} + \sqrt{a} x^{- \frac{n}{2}} + \sqrt{b} \right )}}{2 a^{\frac{5}{4}} n} + \frac{\sqrt{2} \sqrt [4]{b} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{a} x^{- \frac{n}{4}}}{\sqrt [4]{b}} - 1 \right )}}{a^{\frac{5}{4}} n} + \frac{\sqrt{2} \sqrt [4]{b} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{a} x^{- \frac{n}{4}}}{\sqrt [4]{b}} + 1 \right )}}{a^{\frac{5}{4}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-1/4*n)/(a+b*x**n),x)

[Out]

-4*x**(-n/4)/(a*n) - sqrt(2)*b**(1/4)*log(-sqrt(2)*a**(1/4)*b**(1/4)*x**(-n/4) +
sqrt(a)*x**(-n/2) + sqrt(b))/(2*a**(5/4)*n) + sqrt(2)*b**(1/4)*log(sqrt(2)*a**(
1/4)*b**(1/4)*x**(-n/4) + sqrt(a)*x**(-n/2) + sqrt(b))/(2*a**(5/4)*n) + sqrt(2)*
b**(1/4)*atan(sqrt(2)*a**(1/4)*x**(-n/4)/b**(1/4) - 1)/(a**(5/4)*n) + sqrt(2)*b*
*(1/4)*atan(sqrt(2)*a**(1/4)*x**(-n/4)/b**(1/4) + 1)/(a**(5/4)*n)

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Mathematica [C]  time = 0.0378012, size = 59, normalized size = 0.25 \[ \frac{b \text{RootSum}\left [\text{$\#$1}^4 a+b\&,\frac{4 \log \left (x^{-n/4}-\text{$\#$1}\right )+n \log (x)}{\text{$\#$1}^3}\&\right ]-16 a x^{-n/4}}{4 a^2 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - n/4)/(a + b*x^n),x]

[Out]

((-16*a)/x^(n/4) + b*RootSum[b + a*#1^4 & , (n*Log[x] + 4*Log[x^(-n/4) - #1])/#1
^3 & ])/(4*a^2*n)

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Maple [C]  time = 0.085, size = 56, normalized size = 0.2 \[ -4\,{\frac{1}{an{x}^{n/4}}}+\sum _{{\it \_R}={\it RootOf} \left ({a}^{5}{n}^{4}{{\it \_Z}}^{4}+b \right ) }{\it \_R}\,\ln \left ({x}^{{\frac{n}{4}}}-{\frac{{a}^{4}{n}^{3}{{\it \_R}}^{3}}{b}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-1/4*n)/(a+b*x^n),x)

[Out]

-4/a/n/(x^(1/4*n))+sum(_R*ln(x^(1/4*n)-a^4*n^3/b*_R^3),_R=RootOf(_Z^4*a^5*n^4+b)
)

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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^(-1/4*n - 1)/(b*x^n + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246396, size = 254, normalized size = 1.09 \[ -\frac{4 \, a n \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{1}{4}} \arctan \left (\frac{a n \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{1}{4}}}{x x^{-\frac{1}{4} \, n - 1} + x \sqrt{\frac{a^{2} n^{2} \sqrt{-\frac{b}{a^{5} n^{4}}} + x^{2} x^{-\frac{1}{2} \, n - 2}}{x^{2}}}}\right ) - a n \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{1}{4}} \log \left (\frac{a n \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{1}{4}} + x x^{-\frac{1}{4} \, n - 1}}{x}\right ) + a n \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{1}{4}} \log \left (-\frac{a n \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{1}{4}} - x x^{-\frac{1}{4} \, n - 1}}{x}\right ) + 4 \, x x^{-\frac{1}{4} \, n - 1}}{a n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-1/4*n - 1)/(b*x^n + a),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-\frac{1}{4} \, n - 1}}{b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-1/4*n - 1)/(b*x^n + a),x, algorithm="giac")

[Out]

integrate(x^(-1/4*n - 1)/(b*x^n + a), x)