3.2764 \(\int \frac{(c x)^{-1-\frac{5 n}{4}}}{a+b x^n} \, dx\)

Optimal. Leaf size=341 \[ \frac{b^{5/4} x^{5 n/4} (c x)^{-5 n/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{9/4} c n}-\frac{b^{5/4} x^{5 n/4} (c x)^{-5 n/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{9/4} c n}+\frac{\sqrt{2} b^{5/4} x^{5 n/4} (c x)^{-5 n/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{9/4} c n}-\frac{\sqrt{2} b^{5/4} x^{5 n/4} (c x)^{-5 n/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}+1\right )}{a^{9/4} c n}+\frac{4 b x^n (c x)^{-5 n/4}}{a^2 c n}-\frac{4 (c x)^{-5 n/4}}{5 a c n} \]

[Out]

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

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Rubi [A]  time = 0.551685, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.524 \[ \frac{b^{5/4} x^{5 n/4} (c x)^{-5 n/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{9/4} c n}-\frac{b^{5/4} x^{5 n/4} (c x)^{-5 n/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{-n/4}+\sqrt{a} x^{-n/2}+\sqrt{b}\right )}{\sqrt{2} a^{9/4} c n}+\frac{\sqrt{2} b^{5/4} x^{5 n/4} (c x)^{-5 n/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}\right )}{a^{9/4} c n}-\frac{\sqrt{2} b^{5/4} x^{5 n/4} (c x)^{-5 n/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} x^{-n/4}}{\sqrt [4]{b}}+1\right )}{a^{9/4} c n}+\frac{4 b x^n (c x)^{-5 n/4}}{a^2 c n}-\frac{4 (c x)^{-5 n/4}}{5 a c n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 87.6908, size = 299, normalized size = 0.88 \[ - \frac{4 \left (c x\right )^{- \frac{5 n}{4}}}{5 a c n} + \frac{4 b x^{n} \left (c x\right )^{- \frac{5 n}{4}}}{a^{2} c n} + \frac{\sqrt{2} b^{\frac{5}{4}} x^{\frac{5 n}{4}} \left (c x\right )^{- \frac{5 n}{4}} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{- \frac{n}{4}} + \sqrt{a} x^{- \frac{n}{2}} + \sqrt{b} \right )}}{2 a^{\frac{9}{4}} c n} - \frac{\sqrt{2} b^{\frac{5}{4}} x^{\frac{5 n}{4}} \left (c x\right )^{- \frac{5 n}{4}} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{- \frac{n}{4}} + \sqrt{a} x^{- \frac{n}{2}} + \sqrt{b} \right )}}{2 a^{\frac{9}{4}} c n} - \frac{\sqrt{2} b^{\frac{5}{4}} x^{\frac{5 n}{4}} \left (c x\right )^{- \frac{5 n}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{a} x^{- \frac{n}{4}}}{\sqrt [4]{b}} - 1 \right )}}{a^{\frac{9}{4}} c n} - \frac{\sqrt{2} b^{\frac{5}{4}} x^{\frac{5 n}{4}} \left (c x\right )^{- \frac{5 n}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{a} x^{- \frac{n}{4}}}{\sqrt [4]{b}} + 1 \right )}}{a^{\frac{9}{4}} c n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0585601, size = 82, normalized size = 0.24 \[ \frac{(c x)^{-5 n/4} \left (-5 b^2 x^{5 n/4} \text{RootSum}\left [\text{$\#$1}^4 a+b\&,\frac{4 \log \left (x^{-n/4}-\text{$\#$1}\right )+n \log (x)}{\text{$\#$1}^3}\&\right ]-16 a \left (a-5 b x^n\right )\right )}{20 a^3 c n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.086, size = 0, normalized size = 0. \[ \int{\frac{1}{a+b{x}^{n}} \left ( cx \right ) ^{-1-{\frac{5\,n}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.375324, size = 549, normalized size = 1.61 \[ \frac{20 \, a^{2} n \left (-\frac{b^{5} c^{-5 \, n - 4}}{a^{9} n^{4}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{2} n x^{\frac{4}{5}} \left (-\frac{b^{5} c^{-5 \, n - 4}}{a^{9} n^{4}}\right )^{\frac{1}{4}}}{b c^{-n - \frac{4}{5}} x e^{\left (-\frac{1}{20} \,{\left (5 \, n + 4\right )} \log \left (c\right ) - \frac{1}{20} \,{\left (5 \, n + 4\right )} \log \left (x\right )\right )} + x \sqrt{\frac{a^{4} n^{2} x^{\frac{3}{5}} \sqrt{-\frac{b^{5} c^{-5 \, n - 4}}{a^{9} n^{4}}} + b^{2} c^{-2 \, n - \frac{8}{5}} x e^{\left (-\frac{1}{10} \,{\left (5 \, n + 4\right )} \log \left (c\right ) - \frac{1}{10} \,{\left (5 \, n + 4\right )} \log \left (x\right )\right )}}{x}}}\right ) - 5 \, a^{2} n \left (-\frac{b^{5} c^{-5 \, n - 4}}{a^{9} n^{4}}\right )^{\frac{1}{4}} \log \left (\frac{a^{2} n x^{\frac{4}{5}} \left (-\frac{b^{5} c^{-5 \, n - 4}}{a^{9} n^{4}}\right )^{\frac{1}{4}} + b c^{-n - \frac{4}{5}} x e^{\left (-\frac{1}{20} \,{\left (5 \, n + 4\right )} \log \left (c\right ) - \frac{1}{20} \,{\left (5 \, n + 4\right )} \log \left (x\right )\right )}}{x}\right ) + 5 \, a^{2} n \left (-\frac{b^{5} c^{-5 \, n - 4}}{a^{9} n^{4}}\right )^{\frac{1}{4}} \log \left (-\frac{a^{2} n x^{\frac{4}{5}} \left (-\frac{b^{5} c^{-5 \, n - 4}}{a^{9} n^{4}}\right )^{\frac{1}{4}} - b c^{-n - \frac{4}{5}} x e^{\left (-\frac{1}{20} \,{\left (5 \, n + 4\right )} \log \left (c\right ) - \frac{1}{20} \,{\left (5 \, n + 4\right )} \log \left (x\right )\right )}}{x}\right ) + 20 \, b c^{-n - \frac{4}{5}} x^{\frac{1}{5}} e^{\left (-\frac{1}{20} \,{\left (5 \, n + 4\right )} \log \left (c\right ) - \frac{1}{20} \,{\left (5 \, n + 4\right )} \log \left (x\right )\right )} - 4 \, a x e^{\left (-\frac{1}{4} \,{\left (5 \, n + 4\right )} \log \left (c\right ) - \frac{1}{4} \,{\left (5 \, n + 4\right )} \log \left (x\right )\right )}}{5 \, a^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(20*a^2*n*(-b^5*c^(-5*n - 4)/(a^9*n^4))^(1/4)*arctan(a^2*n*x^(4/5)*(-b^5*c^(
-5*n - 4)/(a^9*n^4))^(1/4)/(b*c^(-n - 4/5)*x*e^(-1/20*(5*n + 4)*log(c) - 1/20*(5
*n + 4)*log(x)) + x*sqrt((a^4*n^2*x^(3/5)*sqrt(-b^5*c^(-5*n - 4)/(a^9*n^4)) + b^
2*c^(-2*n - 8/5)*x*e^(-1/10*(5*n + 4)*log(c) - 1/10*(5*n + 4)*log(x)))/x))) - 5*
a^2*n*(-b^5*c^(-5*n - 4)/(a^9*n^4))^(1/4)*log((a^2*n*x^(4/5)*(-b^5*c^(-5*n - 4)/
(a^9*n^4))^(1/4) + b*c^(-n - 4/5)*x*e^(-1/20*(5*n + 4)*log(c) - 1/20*(5*n + 4)*l
og(x)))/x) + 5*a^2*n*(-b^5*c^(-5*n - 4)/(a^9*n^4))^(1/4)*log(-(a^2*n*x^(4/5)*(-b
^5*c^(-5*n - 4)/(a^9*n^4))^(1/4) - b*c^(-n - 4/5)*x*e^(-1/20*(5*n + 4)*log(c) -
1/20*(5*n + 4)*log(x)))/x) + 20*b*c^(-n - 4/5)*x^(1/5)*e^(-1/20*(5*n + 4)*log(c)
- 1/20*(5*n + 4)*log(x)) - 4*a*x*e^(-1/4*(5*n + 4)*log(c) - 1/4*(5*n + 4)*log(x
)))/(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((c*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. \[ \int \frac{\left (c x\right )^{-\frac{5}{4} \, n - 1}}{b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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