3.2634 \(\int \frac{x^{-1-\frac{3 n}{4}}}{a+b x^n} \, dx\)

Optimal. Leaf size=236 \[ \frac{b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt{a}+\sqrt{b} x^{n/2}\right )}{\sqrt{2} a^{7/4} n}-\frac{b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt{a}+\sqrt{b} x^{n/2}\right )}{\sqrt{2} a^{7/4} n}+\frac{\sqrt{2} b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{a^{7/4} n}-\frac{\sqrt{2} b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}+1\right )}{a^{7/4} n}-\frac{4 x^{-3 n/4}}{3 a n} \]

[Out]

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

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Rubi [A]  time = 0.440814, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ \frac{b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt{a}+\sqrt{b} x^{n/2}\right )}{\sqrt{2} a^{7/4} n}-\frac{b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt{a}+\sqrt{b} x^{n/2}\right )}{\sqrt{2} a^{7/4} n}+\frac{\sqrt{2} b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{a^{7/4} n}-\frac{\sqrt{2} b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}+1\right )}{a^{7/4} n}-\frac{4 x^{-3 n/4}}{3 a n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 65.3219, size = 206, normalized size = 0.87 \[ - \frac{4 x^{- \frac{3 n}{4}}}{3 a n} + \frac{\sqrt{2} b^{\frac{3}{4}} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{\frac{n}{4}} + \sqrt{a} + \sqrt{b} x^{\frac{n}{2}} \right )}}{2 a^{\frac{7}{4}} n} - \frac{\sqrt{2} b^{\frac{3}{4}} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{\frac{n}{4}} + \sqrt{a} + \sqrt{b} x^{\frac{n}{2}} \right )}}{2 a^{\frac{7}{4}} n} + \frac{\sqrt{2} b^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x^{\frac{n}{4}}}{\sqrt [4]{a}} \right )}}{a^{\frac{7}{4}} n} - \frac{\sqrt{2} b^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x^{\frac{n}{4}}}{\sqrt [4]{a}} \right )}}{a^{\frac{7}{4}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0438319, size = 60, normalized size = 0.25 \[ \frac{3 b \text{RootSum}\left [\text{$\#$1}^4 a+b\&,\frac{4 \log \left (x^{-n/4}-\text{$\#$1}\right )+n \log (x)}{\text{$\#$1}}\&\right ]-16 a x^{-3 n/4}}{12 a^2 n} \]

Antiderivative was successfully verified.

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

[Out]

((-16*a)/x^((3*n)/4) + 3*b*RootSum[b + a*#1^4 & , (n*Log[x] + 4*Log[x^(-n/4) - #
1])/#1 & ])/(12*a^2*n)

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Maple [C]  time = 0.089, size = 54, normalized size = 0.2 \[ -{\frac{4}{3\,an} \left ({x}^{{\frac{n}{4}}} \right ) ^{-3}}+\sum _{{\it \_R}={\it RootOf} \left ({a}^{7}{n}^{4}{{\it \_Z}}^{4}+{b}^{3} \right ) }{\it \_R}\,\ln \left ({x}^{{\frac{n}{4}}}-{\frac{{a}^{2}n{\it \_R}}{b}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-3/4*n)/(a+b*x^n),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.342121, size = 324, normalized size = 1.37 \[ \frac{12 \, a n \left (-\frac{b^{3}}{a^{7} n^{4}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{5} n^{3} \left (-\frac{b^{3}}{a^{7} n^{4}}\right )^{\frac{3}{4}}}{b^{2} x^{\frac{1}{3}} x^{-\frac{1}{4} \, n - \frac{1}{3}} + x^{\frac{1}{3}} \sqrt{-\frac{a^{3} b^{3} n^{2} \sqrt{-\frac{b^{3}}{a^{7} n^{4}}} - b^{4} x^{\frac{2}{3}} x^{-\frac{1}{2} \, n - \frac{2}{3}}}{x^{\frac{2}{3}}}}}\right ) + 3 \, a n \left (-\frac{b^{3}}{a^{7} n^{4}}\right )^{\frac{1}{4}} \log \left (\frac{a^{5} n^{3} \left (-\frac{b^{3}}{a^{7} n^{4}}\right )^{\frac{3}{4}} + b^{2} x^{\frac{1}{3}} x^{-\frac{1}{4} \, n - \frac{1}{3}}}{x^{\frac{1}{3}}}\right ) - 3 \, a n \left (-\frac{b^{3}}{a^{7} n^{4}}\right )^{\frac{1}{4}} \log \left (-\frac{a^{5} n^{3} \left (-\frac{b^{3}}{a^{7} n^{4}}\right )^{\frac{3}{4}} - b^{2} x^{\frac{1}{3}} x^{-\frac{1}{4} \, n - \frac{1}{3}}}{x^{\frac{1}{3}}}\right ) - 4 \, x x^{-\frac{3}{4} \, n - 1}}{3 \, a n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(12*a*n*(-b^3/(a^7*n^4))^(1/4)*arctan(a^5*n^3*(-b^3/(a^7*n^4))^(3/4)/(b^2*x^
(1/3)*x^(-1/4*n - 1/3) + x^(1/3)*sqrt(-(a^3*b^3*n^2*sqrt(-b^3/(a^7*n^4)) - b^4*x
^(2/3)*x^(-1/2*n - 2/3))/x^(2/3)))) + 3*a*n*(-b^3/(a^7*n^4))^(1/4)*log((a^5*n^3*
(-b^3/(a^7*n^4))^(3/4) + b^2*x^(1/3)*x^(-1/4*n - 1/3))/x^(1/3)) - 3*a*n*(-b^3/(a
^7*n^4))^(1/4)*log(-(a^5*n^3*(-b^3/(a^7*n^4))^(3/4) - b^2*x^(1/3)*x^(-1/4*n - 1/
3))/x^(1/3)) - 4*x*x^(-3/4*n - 1))/(a*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**(-1-3/4*n)/(a+b*x**n),x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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