3.500 \(\int \frac{x^{-1+\frac{n}{4}}}{b x^n+c x^{2 n}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.386315, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36 \[ \frac{c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt{b}+\sqrt{c} x^{n/2}\right )}{\sqrt{2} b^{7/4} n}-\frac{c^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt{b}+\sqrt{c} x^{n/2}\right )}{\sqrt{2} b^{7/4} n}+\frac{\sqrt{2} c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{b}}\right )}{b^{7/4} n}-\frac{\sqrt{2} c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{b}}+1\right )}{b^{7/4} n}-\frac{4 x^{-3 n/4}}{3 b n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + n/4)/(b*x^n + c*x^(2*n)),x]

[Out]

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

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Rubi in Sympy [A]  time = 64.2371, size = 206, normalized size = 0.87 \[ - \frac{4 x^{- \frac{3 n}{4}}}{3 b n} + \frac{\sqrt{2} c^{\frac{3}{4}} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} x^{\frac{n}{4}} + \sqrt{b} + \sqrt{c} x^{\frac{n}{2}} \right )}}{2 b^{\frac{7}{4}} n} - \frac{\sqrt{2} c^{\frac{3}{4}} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} x^{\frac{n}{4}} + \sqrt{b} + \sqrt{c} x^{\frac{n}{2}} \right )}}{2 b^{\frac{7}{4}} n} + \frac{\sqrt{2} c^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x^{\frac{n}{4}}}{\sqrt [4]{b}} \right )}}{b^{\frac{7}{4}} n} - \frac{\sqrt{2} c^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x^{\frac{n}{4}}}{\sqrt [4]{b}} \right )}}{b^{\frac{7}{4}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+1/4*n)/(b*x**n+c*x**(2*n)),x)

[Out]

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

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Mathematica [C]  time = 0.0436591, size = 60, normalized size = 0.25 \[ \frac{3 c \text{RootSum}\left [\text{$\#$1}^4 b+c\&,\frac{4 \log \left (x^{-n/4}-\text{$\#$1}\right )+n \log (x)}{\text{$\#$1}}\&\right ]-16 b x^{-3 n/4}}{12 b^2 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + n/4)/(b*x^n + c*x^(2*n)),x]

[Out]

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

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Maple [C]  time = 0.322, size = 54, normalized size = 0.2 \[ -{\frac{4}{3\,bn} \left ({x}^{{\frac{n}{4}}} \right ) ^{-3}}+\sum _{{\it \_R}={\it RootOf} \left ({b}^{7}{n}^{4}{{\it \_Z}}^{4}+{c}^{3} \right ) }{\it \_R}\,\ln \left ({x}^{{\frac{n}{4}}}-{\frac{{b}^{2}n{\it \_R}}{c}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+1/4*n)/(b*x^n+c*x^(2*n)),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.306976, size = 335, normalized size = 1.42 \[ \frac{12 \, b n x^{3} x^{\frac{3}{4} \, n - 3} \left (-\frac{c^{3}}{b^{7} n^{4}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{2} n \left (-\frac{c^{3}}{b^{7} n^{4}}\right )^{\frac{1}{4}}}{c x x^{\frac{1}{4} \, n - 1} + x \sqrt{\frac{b^{4} n^{2} \sqrt{-\frac{c^{3}}{b^{7} n^{4}}} + c^{2} x^{2} x^{\frac{1}{2} \, n - 2}}{x^{2}}}}\right ) - 3 \, b n x^{3} x^{\frac{3}{4} \, n - 3} \left (-\frac{c^{3}}{b^{7} n^{4}}\right )^{\frac{1}{4}} \log \left (\frac{b^{2} n \left (-\frac{c^{3}}{b^{7} n^{4}}\right )^{\frac{1}{4}} + c x x^{\frac{1}{4} \, n - 1}}{x}\right ) + 3 \, b n x^{3} x^{\frac{3}{4} \, n - 3} \left (-\frac{c^{3}}{b^{7} n^{4}}\right )^{\frac{1}{4}} \log \left (-\frac{b^{2} n \left (-\frac{c^{3}}{b^{7} n^{4}}\right )^{\frac{1}{4}} - c x x^{\frac{1}{4} \, n - 1}}{x}\right ) - 4}{3 \, b n x^{3} x^{\frac{3}{4} \, n - 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(1/4*n - 1)/(c*x^(2*n) + b*x^n),x, algorithm="fricas")

[Out]

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

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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)/(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. \[ \int \frac{x^{\frac{1}{4} \, n - 1}}{c x^{2 \, n} + b x^{n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(1/4*n - 1)/(c*x^(2*n) + b*x^n),x, algorithm="giac")

[Out]

integrate(x^(1/4*n - 1)/(c*x^(2*n) + b*x^n), x)