3.2761 \(\int \frac{(c x)^{-1-\frac{n}{4}}}{a+b x^n} \, dx\)

Optimal. Leaf size=315 \[ -\frac{\sqrt [4]{b} x^{n/4} (c x)^{-n/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{5/4} c n}+\frac{\sqrt [4]{b} x^{n/4} (c x)^{-n/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{5/4} c n}-\frac{\sqrt{2} \sqrt [4]{b} x^{n/4} (c x)^{-n/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{5/4} c n}+\frac{\sqrt{2} \sqrt [4]{b} x^{n/4} (c x)^{-n/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}+1\right )}{a^{5/4} c n}-\frac{4 (c x)^{-n/4}}{a c n} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.506051, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476 \[ -\frac{\sqrt [4]{b} x^{n/4} (c x)^{-n/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{5/4} c n}+\frac{\sqrt [4]{b} x^{n/4} (c x)^{-n/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{5/4} c n}-\frac{\sqrt{2} \sqrt [4]{b} x^{n/4} (c x)^{-n/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{5/4} c n}+\frac{\sqrt{2} \sqrt [4]{b} x^{n/4} (c x)^{-n/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}+1\right )}{a^{5/4} c n}-\frac{4 (c x)^{-n/4}}{a c n} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 77.1497, size = 260, normalized size = 0.83 \[ - \frac{4 \left (c x\right )^{- \frac{n}{4}}}{a c n} - \frac{\sqrt{2} \sqrt [4]{b} x^{\frac{n}{4}} \left (c x\right )^{- \frac{n}{4}} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{- \frac{n}{4}} + \sqrt{a} x^{- \frac{n}{2}} + \sqrt{b} \right )}}{2 a^{\frac{5}{4}} c n} + \frac{\sqrt{2} \sqrt [4]{b} x^{\frac{n}{4}} \left (c x\right )^{- \frac{n}{4}} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{- \frac{n}{4}} + \sqrt{a} x^{- \frac{n}{2}} + \sqrt{b} \right )}}{2 a^{\frac{5}{4}} c n} + \frac{\sqrt{2} \sqrt [4]{b} x^{\frac{n}{4}} \left (c x\right )^{- \frac{n}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{a} x^{- \frac{n}{4}}}{\sqrt [4]{b}} - 1 \right )}}{a^{\frac{5}{4}} c n} + \frac{\sqrt{2} \sqrt [4]{b} x^{\frac{n}{4}} \left (c x\right )^{- \frac{n}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{a} x^{- \frac{n}{4}}}{\sqrt [4]{b}} + 1 \right )}}{a^{\frac{5}{4}} c n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.0454645, size = 71, normalized size = 0.23 \[ \frac{(c x)^{-n/4} \left (b x^{n/4} \text{RootSum}\left [\text{$\#$1}^4 a+b\&,\frac{4 \log \left (x^{-n/4}-\text{$\#$1}\right )+n \log (x)}{\text{$\#$1}^3}\&\right ]-16 a\right )}{4 a^2 c n} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.089, size = 0, normalized size = 0. \[ \int{\frac{1}{a+b{x}^{n}} \left ( cx \right ) ^{-1-{\frac{n}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.257466, size = 381, normalized size = 1.21 \[ -\frac{4 \, a n \left (-\frac{b c^{-n - 4}}{a^{5} n^{4}}\right )^{\frac{1}{4}} \arctan \left (\frac{a n \left (-\frac{b c^{-n - 4}}{a^{5} n^{4}}\right )^{\frac{1}{4}}}{x e^{\left (-\frac{1}{4} \,{\left (n + 4\right )} \log \left (c\right ) - \frac{1}{4} \,{\left (n + 4\right )} \log \left (x\right )\right )} + x \sqrt{\frac{a^{2} n^{2} \sqrt{-\frac{b c^{-n - 4}}{a^{5} n^{4}}} + x^{2} e^{\left (-\frac{1}{2} \,{\left (n + 4\right )} \log \left (c\right ) - \frac{1}{2} \,{\left (n + 4\right )} \log \left (x\right )\right )}}{x^{2}}}}\right ) - a n \left (-\frac{b c^{-n - 4}}{a^{5} n^{4}}\right )^{\frac{1}{4}} \log \left (\frac{a n \left (-\frac{b c^{-n - 4}}{a^{5} n^{4}}\right )^{\frac{1}{4}} + x e^{\left (-\frac{1}{4} \,{\left (n + 4\right )} \log \left (c\right ) - \frac{1}{4} \,{\left (n + 4\right )} \log \left (x\right )\right )}}{x}\right ) + a n \left (-\frac{b c^{-n - 4}}{a^{5} n^{4}}\right )^{\frac{1}{4}} \log \left (-\frac{a n \left (-\frac{b c^{-n - 4}}{a^{5} n^{4}}\right )^{\frac{1}{4}} - x e^{\left (-\frac{1}{4} \,{\left (n + 4\right )} \log \left (c\right ) - \frac{1}{4} \,{\left (n + 4\right )} \log \left (x\right )\right )}}{x}\right ) + 4 \, x e^{\left (-\frac{1}{4} \,{\left (n + 4\right )} \log \left (c\right ) - \frac{1}{4} \,{\left (n + 4\right )} \log \left (x\right )\right )}}{a n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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