3.2641 \(\int \frac{x^{-1-\frac{5 n}{4}}}{a+b x^n} \, dx\)

Optimal. Leaf size=252 \[ \frac{b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{9/4} n}-\frac{b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{9/4} n}+\frac{\sqrt{2} b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{9/4} n}-\frac{\sqrt{2} b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}+1\right )}{a^{9/4} n}+\frac{4 b x^{-n/4}}{a^2 n}-\frac{4 x^{-5 n/4}}{5 a n} \]

[Out]

-4/(5*a*n*x^((5*n)/4)) + (4*b)/(a^2*n*x^(n/4)) + (Sqrt[2]*b^(5/4)*ArcTan[1 - (Sq
rt[2]*a^(1/4))/(b^(1/4)*x^(n/4))])/(a^(9/4)*n) - (Sqrt[2]*b^(5/4)*ArcTan[1 + (Sq
rt[2]*a^(1/4))/(b^(1/4)*x^(n/4))])/(a^(9/4)*n) + (b^(5/4)*Log[Sqrt[b] + Sqrt[a]/
x^(n/2) - (Sqrt[2]*a^(1/4)*b^(1/4))/x^(n/4)])/(Sqrt[2]*a^(9/4)*n) - (b^(5/4)*Log
[Sqrt[b] + Sqrt[a]/x^(n/2) + (Sqrt[2]*a^(1/4)*b^(1/4))/x^(n/4)])/(Sqrt[2]*a^(9/4
)*n)

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Rubi [A]  time = 0.446151, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ \frac{b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{9/4} n}-\frac{b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{9/4} n}+\frac{\sqrt{2} b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{9/4} n}-\frac{\sqrt{2} b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}+1\right )}{a^{9/4} n}+\frac{4 b x^{-n/4}}{a^2 n}-\frac{4 x^{-5 n/4}}{5 a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - (5*n)/4)/(a + b*x^n),x]

[Out]

-4/(5*a*n*x^((5*n)/4)) + (4*b)/(a^2*n*x^(n/4)) + (Sqrt[2]*b^(5/4)*ArcTan[1 - (Sq
rt[2]*a^(1/4))/(b^(1/4)*x^(n/4))])/(a^(9/4)*n) - (Sqrt[2]*b^(5/4)*ArcTan[1 + (Sq
rt[2]*a^(1/4))/(b^(1/4)*x^(n/4))])/(a^(9/4)*n) + (b^(5/4)*Log[Sqrt[b] + Sqrt[a]/
x^(n/2) - (Sqrt[2]*a^(1/4)*b^(1/4))/x^(n/4)])/(Sqrt[2]*a^(9/4)*n) - (b^(5/4)*Log
[Sqrt[b] + Sqrt[a]/x^(n/2) + (Sqrt[2]*a^(1/4)*b^(1/4))/x^(n/4)])/(Sqrt[2]*a^(9/4
)*n)

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Rubi in Sympy [A]  time = 74.4801, size = 219, normalized size = 0.87 \[ - \frac{4 x^{- \frac{5 n}{4}}}{5 a n} + \frac{4 b x^{- \frac{n}{4}}}{a^{2} n} + \frac{\sqrt{2} b^{\frac{5}{4}} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{- \frac{n}{4}} + \sqrt{a} x^{- \frac{n}{2}} + \sqrt{b} \right )}}{2 a^{\frac{9}{4}} n} - \frac{\sqrt{2} b^{\frac{5}{4}} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{- \frac{n}{4}} + \sqrt{a} x^{- \frac{n}{2}} + \sqrt{b} \right )}}{2 a^{\frac{9}{4}} n} - \frac{\sqrt{2} b^{\frac{5}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{a} x^{- \frac{n}{4}}}{\sqrt [4]{b}} - 1 \right )}}{a^{\frac{9}{4}} n} - \frac{\sqrt{2} b^{\frac{5}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{a} x^{- \frac{n}{4}}}{\sqrt [4]{b}} + 1 \right )}}{a^{\frac{9}{4}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-5/4*n)/(a+b*x**n),x)

[Out]

-4*x**(-5*n/4)/(5*a*n) + 4*b*x**(-n/4)/(a**2*n) + sqrt(2)*b**(5/4)*log(-sqrt(2)*
a**(1/4)*b**(1/4)*x**(-n/4) + sqrt(a)*x**(-n/2) + sqrt(b))/(2*a**(9/4)*n) - sqrt
(2)*b**(5/4)*log(sqrt(2)*a**(1/4)*b**(1/4)*x**(-n/4) + sqrt(a)*x**(-n/2) + sqrt(
b))/(2*a**(9/4)*n) - sqrt(2)*b**(5/4)*atan(sqrt(2)*a**(1/4)*x**(-n/4)/b**(1/4) -
1)/(a**(9/4)*n) - sqrt(2)*b**(5/4)*atan(sqrt(2)*a**(1/4)*x**(-n/4)/b**(1/4) + 1
)/(a**(9/4)*n)

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Mathematica [C]  time = 0.0803548, size = 70, normalized size = 0.28 \[ -\frac{5 b^2 \text{RootSum}\left [\text{$\#$1}^4 a+b\&,\frac{4 \log \left (x^{-n/4}-\text{$\#$1}\right )+n \log (x)}{\text{$\#$1}^3}\&\right ]+16 a x^{-5 n/4} \left (a-5 b x^n\right )}{20 a^3 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - (5*n)/4)/(a + b*x^n),x]

[Out]

-((16*a*(a - 5*b*x^n))/x^((5*n)/4) + 5*b^2*RootSum[b + a*#1^4 & , (n*Log[x] + 4*
Log[x^(-n/4) - #1])/#1^3 & ])/(20*a^3*n)

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Maple [C]  time = 0.115, size = 73, normalized size = 0.3 \[ 4\,{\frac{b}{{a}^{2}n{x}^{n/4}}}-{\frac{4}{5\,an} \left ({x}^{{\frac{n}{4}}} \right ) ^{-5}}+\sum _{{\it \_R}={\it RootOf} \left ({a}^{9}{n}^{4}{{\it \_Z}}^{4}+{b}^{5} \right ) }{\it \_R}\,\ln \left ({x}^{{\frac{n}{4}}}+{\frac{{a}^{7}{n}^{3}{{\it \_R}}^{3}}{{b}^{4}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-5/4*n)/(a+b*x^n),x)

[Out]

4*b/a^2/n/(x^(1/4*n))-4/5/a/n/(x^(1/4*n))^5+sum(_R*ln(x^(1/4*n)+a^7*n^3/b^4*_R^3
),_R=RootOf(_Z^4*a^9*n^4+b^5))

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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^(-5/4*n - 1)/(b*x^n + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.28204, size = 331, normalized size = 1.31 \[ \frac{20 \, a^{2} n \left (-\frac{b^{5}}{a^{9} n^{4}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{2} n x^{\frac{4}{5}} \left (-\frac{b^{5}}{a^{9} n^{4}}\right )^{\frac{1}{4}}}{b x x^{-\frac{1}{4} \, n - \frac{1}{5}} + x \sqrt{\frac{a^{4} n^{2} x^{\frac{3}{5}} \sqrt{-\frac{b^{5}}{a^{9} n^{4}}} + b^{2} x x^{-\frac{1}{2} \, n - \frac{2}{5}}}{x}}}\right ) - 5 \, a^{2} n \left (-\frac{b^{5}}{a^{9} n^{4}}\right )^{\frac{1}{4}} \log \left (\frac{a^{2} n x^{\frac{4}{5}} \left (-\frac{b^{5}}{a^{9} n^{4}}\right )^{\frac{1}{4}} + b x x^{-\frac{1}{4} \, n - \frac{1}{5}}}{x}\right ) + 5 \, a^{2} n \left (-\frac{b^{5}}{a^{9} n^{4}}\right )^{\frac{1}{4}} \log \left (-\frac{a^{2} n x^{\frac{4}{5}} \left (-\frac{b^{5}}{a^{9} n^{4}}\right )^{\frac{1}{4}} - b x x^{-\frac{1}{4} \, n - \frac{1}{5}}}{x}\right ) - 4 \, a x x^{-\frac{5}{4} \, n - 1} + 20 \, b x^{\frac{1}{5}} x^{-\frac{1}{4} \, n - \frac{1}{5}}}{5 \, a^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-5/4*n - 1)/(b*x^n + a),x, algorithm="fricas")

[Out]

1/5*(20*a^2*n*(-b^5/(a^9*n^4))^(1/4)*arctan(a^2*n*x^(4/5)*(-b^5/(a^9*n^4))^(1/4)
/(b*x*x^(-1/4*n - 1/5) + x*sqrt((a^4*n^2*x^(3/5)*sqrt(-b^5/(a^9*n^4)) + b^2*x*x^
(-1/2*n - 2/5))/x))) - 5*a^2*n*(-b^5/(a^9*n^4))^(1/4)*log((a^2*n*x^(4/5)*(-b^5/(
a^9*n^4))^(1/4) + b*x*x^(-1/4*n - 1/5))/x) + 5*a^2*n*(-b^5/(a^9*n^4))^(1/4)*log(
-(a^2*n*x^(4/5)*(-b^5/(a^9*n^4))^(1/4) - b*x*x^(-1/4*n - 1/5))/x) - 4*a*x*x^(-5/
4*n - 1) + 20*b*x^(1/5)*x^(-1/4*n - 1/5))/(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**(-1-5/4*n)/(a+b*x**n),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-\frac{5}{4} \, n - 1}}{b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-5/4*n - 1)/(b*x^n + a),x, algorithm="giac")

[Out]

integrate(x^(-5/4*n - 1)/(b*x^n + a), x)