3.2757 \(\int \frac{(c x)^{-1-\frac{3 n}{4}}}{a+b x^n} \, dx\)

Optimal. Leaf size=317 \[ \frac{b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt{a}+\sqrt{b} x^{n/2}\right )}{\sqrt{2} a^{7/4} c n}-\frac{b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt{a}+\sqrt{b} x^{n/2}\right )}{\sqrt{2} a^{7/4} c n}+\frac{\sqrt{2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{a^{7/4} c n}-\frac{\sqrt{2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}+1\right )}{a^{7/4} c n}-\frac{4 (c x)^{-3 n/4}}{3 a c n} \]

[Out]

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

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Rubi [A]  time = 0.548636, antiderivative size = 317, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ \frac{b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt{a}+\sqrt{b} x^{n/2}\right )}{\sqrt{2} a^{7/4} c n}-\frac{b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt{a}+\sqrt{b} x^{n/2}\right )}{\sqrt{2} a^{7/4} c n}+\frac{\sqrt{2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{a^{7/4} c n}-\frac{\sqrt{2} b^{3/4} x^{3 n/4} (c x)^{-3 n/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}+1\right )}{a^{7/4} c n}-\frac{4 (c x)^{-3 n/4}}{3 a c n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 76.9261, size = 277, normalized size = 0.87 \[ - \frac{4 \left (c x\right )^{- \frac{3 n}{4}}}{3 a c n} + \frac{\sqrt{2} b^{\frac{3}{4}} x^{\frac{3 n}{4}} \left (c x\right )^{- \frac{3 n}{4}} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{\frac{n}{4}} + \sqrt{a} + \sqrt{b} x^{\frac{n}{2}} \right )}}{2 a^{\frac{7}{4}} c n} - \frac{\sqrt{2} b^{\frac{3}{4}} x^{\frac{3 n}{4}} \left (c x\right )^{- \frac{3 n}{4}} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{\frac{n}{4}} + \sqrt{a} + \sqrt{b} x^{\frac{n}{2}} \right )}}{2 a^{\frac{7}{4}} c n} + \frac{\sqrt{2} b^{\frac{3}{4}} x^{\frac{3 n}{4}} \left (c x\right )^{- \frac{3 n}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x^{\frac{n}{4}}}{\sqrt [4]{a}} \right )}}{a^{\frac{7}{4}} c n} - \frac{\sqrt{2} b^{\frac{3}{4}} x^{\frac{3 n}{4}} \left (c x\right )^{- \frac{3 n}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x^{\frac{n}{4}}}{\sqrt [4]{a}} \right )}}{a^{\frac{7}{4}} c n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.049287, size = 72, normalized size = 0.23 \[ \frac{(c x)^{-3 n/4} \left (3 b x^{3 n/4} \text{RootSum}\left [\text{$\#$1}^4 a+b\&,\frac{4 \log \left (x^{-n/4}-\text{$\#$1}\right )+n \log (x)}{\text{$\#$1}}\&\right ]-16 a\right )}{12 a^2 c n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.091, size = 0, normalized size = 0. \[ \int{\frac{1}{a+b{x}^{n}} \left ( cx \right ) ^{-1-{\frac{3\,n}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.59834, size = 525, normalized size = 1.66 \[ \frac{12 \, a n \left (-\frac{b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{5} n^{3} \left (-\frac{b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac{3}{4}}}{b^{2} c^{-2 \, n - \frac{8}{3}} x^{\frac{1}{3}} e^{\left (-\frac{1}{12} \,{\left (3 \, n + 4\right )} \log \left (c\right ) - \frac{1}{12} \,{\left (3 \, n + 4\right )} \log \left (x\right )\right )} + x^{\frac{1}{3}} \sqrt{-\frac{a^{3} b^{3} c^{-3 \, n - 4} n^{2} \sqrt{-\frac{b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}} - b^{4} c^{-4 \, n - \frac{16}{3}} x^{\frac{2}{3}} e^{\left (-\frac{1}{6} \,{\left (3 \, n + 4\right )} \log \left (c\right ) - \frac{1}{6} \,{\left (3 \, n + 4\right )} \log \left (x\right )\right )}}{x^{\frac{2}{3}}}}}\right ) + 3 \, a n \left (-\frac{b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac{1}{4}} \log \left (\frac{a^{5} n^{3} \left (-\frac{b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac{3}{4}} + b^{2} c^{-2 \, n - \frac{8}{3}} x^{\frac{1}{3}} e^{\left (-\frac{1}{12} \,{\left (3 \, n + 4\right )} \log \left (c\right ) - \frac{1}{12} \,{\left (3 \, n + 4\right )} \log \left (x\right )\right )}}{x^{\frac{1}{3}}}\right ) - 3 \, a n \left (-\frac{b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac{1}{4}} \log \left (-\frac{a^{5} n^{3} \left (-\frac{b^{3} c^{-3 \, n - 4}}{a^{7} n^{4}}\right )^{\frac{3}{4}} - b^{2} c^{-2 \, n - \frac{8}{3}} x^{\frac{1}{3}} e^{\left (-\frac{1}{12} \,{\left (3 \, n + 4\right )} \log \left (c\right ) - \frac{1}{12} \,{\left (3 \, n + 4\right )} \log \left (x\right )\right )}}{x^{\frac{1}{3}}}\right ) - 4 \, x e^{\left (-\frac{1}{4} \,{\left (3 \, n + 4\right )} \log \left (c\right ) - \frac{1}{4} \,{\left (3 \, n + 4\right )} \log \left (x\right )\right )}}{3 \, a n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(12*a*n*(-b^3*c^(-3*n - 4)/(a^7*n^4))^(1/4)*arctan(a^5*n^3*(-b^3*c^(-3*n - 4
)/(a^7*n^4))^(3/4)/(b^2*c^(-2*n - 8/3)*x^(1/3)*e^(-1/12*(3*n + 4)*log(c) - 1/12*
(3*n + 4)*log(x)) + x^(1/3)*sqrt(-(a^3*b^3*c^(-3*n - 4)*n^2*sqrt(-b^3*c^(-3*n -
4)/(a^7*n^4)) - b^4*c^(-4*n - 16/3)*x^(2/3)*e^(-1/6*(3*n + 4)*log(c) - 1/6*(3*n
+ 4)*log(x)))/x^(2/3)))) + 3*a*n*(-b^3*c^(-3*n - 4)/(a^7*n^4))^(1/4)*log((a^5*n^
3*(-b^3*c^(-3*n - 4)/(a^7*n^4))^(3/4) + b^2*c^(-2*n - 8/3)*x^(1/3)*e^(-1/12*(3*n
+ 4)*log(c) - 1/12*(3*n + 4)*log(x)))/x^(1/3)) - 3*a*n*(-b^3*c^(-3*n - 4)/(a^7*
n^4))^(1/4)*log(-(a^5*n^3*(-b^3*c^(-3*n - 4)/(a^7*n^4))^(3/4) - b^2*c^(-2*n - 8/
3)*x^(1/3)*e^(-1/12*(3*n + 4)*log(c) - 1/12*(3*n + 4)*log(x)))/x^(1/3)) - 4*x*e^
(-1/4*(3*n + 4)*log(c) - 1/4*(3*n + 4)*log(x)))/(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((c*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. \[ \int \frac{\left (c x\right )^{-\frac{3}{4} \, n - 1}}{b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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