3.2640 \(\int \frac{x^{-1-\frac{4 n}{3}}}{a+b x^n} \, dx\)

Optimal. Leaf size=176 \[ -\frac{b^{4/3} \log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{a^{7/3} n}+\frac{b^{4/3} \log \left (a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}\right )}{2 a^{7/3} n}+\frac{\sqrt{3} b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt{3} \sqrt [3]{b}}\right )}{a^{7/3} n}+\frac{3 b x^{-n/3}}{a^2 n}-\frac{3 x^{-4 n/3}}{4 a n} \]

[Out]

-3/(4*a*n*x^((4*n)/3)) + (3*b)/(a^2*n*x^(n/3)) + (Sqrt[3]*b^(4/3)*ArcTan[(b^(1/3
) - (2*a^(1/3))/x^(n/3))/(Sqrt[3]*b^(1/3))])/(a^(7/3)*n) - (b^(4/3)*Log[b^(1/3)
+ a^(1/3)/x^(n/3)])/(a^(7/3)*n) + (b^(4/3)*Log[b^(2/3) + a^(2/3)/x^((2*n)/3) - (
a^(1/3)*b^(1/3))/x^(n/3)])/(2*a^(7/3)*n)

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Rubi [A]  time = 0.293486, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ -\frac{b^{4/3} \log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{a^{7/3} n}+\frac{b^{4/3} \log \left (a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}\right )}{2 a^{7/3} n}+\frac{\sqrt{3} b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt{3} \sqrt [3]{b}}\right )}{a^{7/3} n}+\frac{3 b x^{-n/3}}{a^2 n}-\frac{3 x^{-4 n/3}}{4 a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - (4*n)/3)/(a + b*x^n),x]

[Out]

-3/(4*a*n*x^((4*n)/3)) + (3*b)/(a^2*n*x^(n/3)) + (Sqrt[3]*b^(4/3)*ArcTan[(b^(1/3
) - (2*a^(1/3))/x^(n/3))/(Sqrt[3]*b^(1/3))])/(a^(7/3)*n) - (b^(4/3)*Log[b^(1/3)
+ a^(1/3)/x^(n/3)])/(a^(7/3)*n) + (b^(4/3)*Log[b^(2/3) + a^(2/3)/x^((2*n)/3) - (
a^(1/3)*b^(1/3))/x^(n/3)])/(2*a^(7/3)*n)

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Rubi in Sympy [A]  time = 45.3485, size = 151, normalized size = 0.86 \[ - \frac{3 x^{- \frac{4 n}{3}}}{4 a n} + \frac{3 b x^{- \frac{n}{3}}}{a^{2} n} - \frac{b^{\frac{4}{3}} \log{\left (\sqrt [3]{a} x^{- \frac{n}{3}} + \sqrt [3]{b} \right )}}{a^{\frac{7}{3}} n} + \frac{b^{\frac{4}{3}} \log{\left (a^{\frac{2}{3}} x^{- \frac{2 n}{3}} - \sqrt [3]{a} \sqrt [3]{b} x^{- \frac{n}{3}} + b^{\frac{2}{3}} \right )}}{2 a^{\frac{7}{3}} n} + \frac{\sqrt{3} b^{\frac{4}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- \frac{2 \sqrt [3]{a} x^{- \frac{n}{3}}}{3} + \frac{\sqrt [3]{b}}{3}\right )}{\sqrt [3]{b}} \right )}}{a^{\frac{7}{3}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-4/3*n)/(a+b*x**n),x)

[Out]

-3*x**(-4*n/3)/(4*a*n) + 3*b*x**(-n/3)/(a**2*n) - b**(4/3)*log(a**(1/3)*x**(-n/3
) + b**(1/3))/(a**(7/3)*n) + b**(4/3)*log(a**(2/3)*x**(-2*n/3) - a**(1/3)*b**(1/
3)*x**(-n/3) + b**(2/3))/(2*a**(7/3)*n) + sqrt(3)*b**(4/3)*atan(sqrt(3)*(-2*a**(
1/3)*x**(-n/3)/3 + b**(1/3)/3)/b**(1/3))/(a**(7/3)*n)

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Mathematica [C]  time = 0.0799826, size = 70, normalized size = 0.4 \[ -\frac{4 b^2 \text{RootSum}\left [\text{$\#$1}^3 a+b\&,\frac{3 \log \left (x^{-n/3}-\text{$\#$1}\right )+n \log (x)}{\text{$\#$1}^2}\&\right ]+9 a x^{-4 n/3} \left (a-4 b x^n\right )}{12 a^3 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - (4*n)/3)/(a + b*x^n),x]

[Out]

-((9*a*(a - 4*b*x^n))/x^((4*n)/3) + 4*b^2*RootSum[b + a*#1^3 & , (n*Log[x] + 3*L
og[x^(-n/3) - #1])/#1^2 & ])/(12*a^3*n)

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Maple [C]  time = 0.102, size = 73, normalized size = 0.4 \[ 3\,{\frac{b}{{a}^{2}n{x}^{n/3}}}-{\frac{3}{4\,an} \left ({x}^{{\frac{n}{3}}} \right ) ^{-4}}+\sum _{{\it \_R}={\it RootOf} \left ({a}^{7}{n}^{3}{{\it \_Z}}^{3}+{b}^{4} \right ) }{\it \_R}\,\ln \left ({x}^{{\frac{n}{3}}}+{\frac{{a}^{5}{n}^{2}{{\it \_R}}^{2}}{{b}^{3}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-4/3*n)/(a+b*x^n),x)

[Out]

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

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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^(-4/3*n - 1)/(b*x^n + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.267212, size = 236, normalized size = 1.34 \[ -\frac{3 \, a x x^{-\frac{4}{3} \, n - 1} + 4 \, \sqrt{3} b \left (-\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{4}} x^{-\frac{1}{3} \, n - \frac{1}{4}} + \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right ) - 4 \, b \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (\frac{x^{\frac{1}{4}} x^{-\frac{1}{3} \, n - \frac{1}{4}} - \left (-\frac{b}{a}\right )^{\frac{1}{3}}}{x^{\frac{1}{4}}}\right ) + 2 \, b \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (\frac{x^{\frac{1}{4}} x^{-\frac{1}{3} \, n - \frac{1}{4}} \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \sqrt{x} x^{-\frac{2}{3} \, n - \frac{1}{2}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}}{\sqrt{x}}\right ) - 12 \, b x^{\frac{1}{4}} x^{-\frac{1}{3} \, n - \frac{1}{4}}}{4 \, a^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-4/3*n - 1)/(b*x^n + a),x, algorithm="fricas")

[Out]

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-\frac{4}{3} \, n - 1}}{b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-4/3*n - 1)/(b*x^n + a),x, algorithm="giac")

[Out]

integrate(x^(-4/3*n - 1)/(b*x^n + a), x)