3.2104 \(\int \frac{1}{a+\frac{b}{x^5}} \, dx\)

Optimal. Leaf size=310 \[ \frac{\left (1-\sqrt{5}\right ) \sqrt [5]{b} \log \left (a^{2/5} x^2-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5}\right )}{20 a^{6/5}}+\frac{\left (1+\sqrt{5}\right ) \sqrt [5]{b} \log \left (a^{2/5} x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5}\right )}{20 a^{6/5}}-\frac{\sqrt [5]{b} \log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )}{5 a^{6/5}}-\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \sqrt [5]{b} \tan ^{-1}\left (\frac{2 \sqrt{\frac{2}{5+\sqrt{5}}} \sqrt [5]{a} x}{\sqrt [5]{b}}+\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}\right )}{5 a^{6/5}}+\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \sqrt [5]{b} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}-\frac{\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} \sqrt [5]{a} x}{\sqrt [5]{b}}\right )}{5 a^{6/5}}+\frac{x}{a} \]

[Out]

x/a - (Sqrt[(5 + Sqrt[5])/2]*b^(1/5)*ArcTan[Sqrt[(5 - 2*Sqrt[5])/5] + (2*Sqrt[2/
(5 + Sqrt[5])]*a^(1/5)*x)/b^(1/5)])/(5*a^(6/5)) + (Sqrt[(5 - Sqrt[5])/2]*b^(1/5)
*ArcTan[Sqrt[(5 + 2*Sqrt[5])/5] - (Sqrt[(2*(5 + Sqrt[5]))/5]*a^(1/5)*x)/b^(1/5)]
)/(5*a^(6/5)) - (b^(1/5)*Log[b^(1/5) + a^(1/5)*x])/(5*a^(6/5)) + ((1 - Sqrt[5])*
b^(1/5)*Log[b^(2/5) - ((1 - Sqrt[5])*a^(1/5)*b^(1/5)*x)/2 + a^(2/5)*x^2])/(20*a^
(6/5)) + ((1 + Sqrt[5])*b^(1/5)*Log[b^(2/5) - ((1 + Sqrt[5])*a^(1/5)*b^(1/5)*x)/
2 + a^(2/5)*x^2])/(20*a^(6/5))

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Rubi [A]  time = 1.42243, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.889 \[ \frac{\left (1-\sqrt{5}\right ) \sqrt [5]{b} \log \left (a^{2/5} x^2-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5}\right )}{20 a^{6/5}}+\frac{\left (1+\sqrt{5}\right ) \sqrt [5]{b} \log \left (a^{2/5} x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5}\right )}{20 a^{6/5}}-\frac{\sqrt [5]{b} \log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )}{5 a^{6/5}}-\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \sqrt [5]{b} \tan ^{-1}\left (\frac{2 \sqrt{\frac{2}{5+\sqrt{5}}} \sqrt [5]{a} x}{\sqrt [5]{b}}+\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}\right )}{5 a^{6/5}}+\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \sqrt [5]{b} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}-\frac{\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} \sqrt [5]{a} x}{\sqrt [5]{b}}\right )}{5 a^{6/5}}+\frac{x}{a} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^5)^(-1),x]

[Out]

x/a - (Sqrt[(5 + Sqrt[5])/2]*b^(1/5)*ArcTan[Sqrt[(5 - 2*Sqrt[5])/5] + (2*Sqrt[2/
(5 + Sqrt[5])]*a^(1/5)*x)/b^(1/5)])/(5*a^(6/5)) + (Sqrt[(5 - Sqrt[5])/2]*b^(1/5)
*ArcTan[Sqrt[(5 + 2*Sqrt[5])/5] - (Sqrt[(2*(5 + Sqrt[5]))/5]*a^(1/5)*x)/b^(1/5)]
)/(5*a^(6/5)) - (b^(1/5)*Log[b^(1/5) + a^(1/5)*x])/(5*a^(6/5)) + ((1 - Sqrt[5])*
b^(1/5)*Log[b^(2/5) - ((1 - Sqrt[5])*a^(1/5)*b^(1/5)*x)/2 + a^(2/5)*x^2])/(20*a^
(6/5)) + ((1 + Sqrt[5])*b^(1/5)*Log[b^(2/5) - ((1 + Sqrt[5])*a^(1/5)*b^(1/5)*x)/
2 + a^(2/5)*x^2])/(20*a^(6/5))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**5),x)

[Out]

Timed out

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Mathematica [A]  time = 0.370442, size = 267, normalized size = 0.86 \[ \frac{-\left (\sqrt{5}-1\right ) \sqrt [5]{b} \log \left (a^{2/5} x^2+\frac{1}{2} \left (\sqrt{5}-1\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5}\right )+\left (1+\sqrt{5}\right ) \sqrt [5]{b} \log \left (a^{2/5} x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5}\right )-4 \sqrt [5]{b} \log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )-2 \sqrt{2 \left (5+\sqrt{5}\right )} \sqrt [5]{b} \tan ^{-1}\left (\frac{4 \sqrt [5]{a} x+\left (\sqrt{5}-1\right ) \sqrt [5]{b}}{\sqrt{2 \left (5+\sqrt{5}\right )} \sqrt [5]{b}}\right )-2 \sqrt{10-2 \sqrt{5}} \sqrt [5]{b} \tan ^{-1}\left (\frac{4 \sqrt [5]{a} x-\left (1+\sqrt{5}\right ) \sqrt [5]{b}}{\sqrt{10-2 \sqrt{5}} \sqrt [5]{b}}\right )+20 \sqrt [5]{a} x}{20 a^{6/5}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^5)^(-1),x]

[Out]

(20*a^(1/5)*x - 2*Sqrt[2*(5 + Sqrt[5])]*b^(1/5)*ArcTan[((-1 + Sqrt[5])*b^(1/5) +
4*a^(1/5)*x)/(Sqrt[2*(5 + Sqrt[5])]*b^(1/5))] - 2*Sqrt[10 - 2*Sqrt[5]]*b^(1/5)*
ArcTan[(-((1 + Sqrt[5])*b^(1/5)) + 4*a^(1/5)*x)/(Sqrt[10 - 2*Sqrt[5]]*b^(1/5))]
- 4*b^(1/5)*Log[b^(1/5) + a^(1/5)*x] - (-1 + Sqrt[5])*b^(1/5)*Log[b^(2/5) + ((-1
+ Sqrt[5])*a^(1/5)*b^(1/5)*x)/2 + a^(2/5)*x^2] + (1 + Sqrt[5])*b^(1/5)*Log[b^(2
/5) - ((1 + Sqrt[5])*a^(1/5)*b^(1/5)*x)/2 + a^(2/5)*x^2])/(20*a^(6/5))

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Maple [B]  time = 0.089, size = 911, normalized size = 2.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^5),x)

[Out]

x/a-b/a^2/(b/a)^(4/5)/(5^(1/2)-5)/(5+5^(1/2))*ln(-(b/a)^(1/5)*x*5^(1/2)+2*(b/a)^
(2/5)-(b/a)^(1/5)*x+2*x^2)*5^(1/2)-b/a^2/(b/a)^(4/5)/(5^(1/2)-5)/(5+5^(1/2))*ln(
-(b/a)^(1/5)*x*5^(1/2)+2*(b/a)^(2/5)-(b/a)^(1/5)*x+2*x^2)+20*b/a^2/(b/a)^(3/5)/(
5^(1/2)-5)/(5+5^(1/2))/(10*(b/a)^(2/5)-2*(b/a)^(2/5)*5^(1/2))^(1/2)*arctan(-1/(1
0*(b/a)^(2/5)-2*(b/a)^(2/5)*5^(1/2))^(1/2)*(b/a)^(1/5)*5^(1/2)-1/(10*(b/a)^(2/5)
-2*(b/a)^(2/5)*5^(1/2))^(1/2)*(b/a)^(1/5)+4/(10*(b/a)^(2/5)-2*(b/a)^(2/5)*5^(1/2
))^(1/2)*x)-4*b/a^2/(b/a)^(3/5)/(5^(1/2)-5)/(5+5^(1/2))/(10*(b/a)^(2/5)-2*(b/a)^
(2/5)*5^(1/2))^(1/2)*arctan(-1/(10*(b/a)^(2/5)-2*(b/a)^(2/5)*5^(1/2))^(1/2)*(b/a
)^(1/5)*5^(1/2)-1/(10*(b/a)^(2/5)-2*(b/a)^(2/5)*5^(1/2))^(1/2)*(b/a)^(1/5)+4/(10
*(b/a)^(2/5)-2*(b/a)^(2/5)*5^(1/2))^(1/2)*x)*5^(1/2)+4*b/a^2/(b/a)^(4/5)/(5^(1/2
)-5)/(5+5^(1/2))*ln(x+(b/a)^(1/5))+b/a^2/(b/a)^(4/5)/(5^(1/2)-5)/(5+5^(1/2))*ln(
(b/a)^(1/5)*x*5^(1/2)+2*(b/a)^(2/5)-(b/a)^(1/5)*x+2*x^2)*5^(1/2)-b/a^2/(b/a)^(4/
5)/(5^(1/2)-5)/(5+5^(1/2))*ln((b/a)^(1/5)*x*5^(1/2)+2*(b/a)^(2/5)-(b/a)^(1/5)*x+
2*x^2)+20*b/a^2/(b/a)^(3/5)/(5^(1/2)-5)/(5+5^(1/2))/(10*(b/a)^(2/5)+2*(b/a)^(2/5
)*5^(1/2))^(1/2)*arctan(1/(10*(b/a)^(2/5)+2*(b/a)^(2/5)*5^(1/2))^(1/2)*(b/a)^(1/
5)*5^(1/2)-1/(10*(b/a)^(2/5)+2*(b/a)^(2/5)*5^(1/2))^(1/2)*(b/a)^(1/5)+4/(10*(b/a
)^(2/5)+2*(b/a)^(2/5)*5^(1/2))^(1/2)*x)+4*b/a^2/(b/a)^(3/5)/(5^(1/2)-5)/(5+5^(1/
2))/(10*(b/a)^(2/5)+2*(b/a)^(2/5)*5^(1/2))^(1/2)*arctan(1/(10*(b/a)^(2/5)+2*(b/a
)^(2/5)*5^(1/2))^(1/2)*(b/a)^(1/5)*5^(1/2)-1/(10*(b/a)^(2/5)+2*(b/a)^(2/5)*5^(1/
2))^(1/2)*(b/a)^(1/5)+4/(10*(b/a)^(2/5)+2*(b/a)^(2/5)*5^(1/2))^(1/2)*x)*5^(1/2)

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Maxima [A]  time = 1.59824, size = 441, normalized size = 1.42 \[ -\frac{\frac{\sqrt{5} b^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )} \log \left (\frac{4 \, a^{\frac{2}{5}} x - a^{\frac{1}{5}} b^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )} - a^{\frac{1}{5}} b^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10}}{4 \, a^{\frac{2}{5}} x - a^{\frac{1}{5}} b^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )} + a^{\frac{1}{5}} b^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10}}\right )}{a^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10}} + \frac{\sqrt{5} b^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )} \log \left (\frac{4 \, a^{\frac{2}{5}} x + a^{\frac{1}{5}} b^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )} - a^{\frac{1}{5}} b^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10}}{4 \, a^{\frac{2}{5}} x + a^{\frac{1}{5}} b^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )} + a^{\frac{1}{5}} b^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10}}\right )}{a^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10}} - \frac{b^{\frac{1}{5}}{\left (\sqrt{5} + 3\right )} \log \left (2 \, a^{\frac{2}{5}} x^{2} - a^{\frac{1}{5}} b^{\frac{1}{5}} x{\left (\sqrt{5} + 1\right )} + 2 \, b^{\frac{2}{5}}\right )}{a^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )}} - \frac{b^{\frac{1}{5}}{\left (\sqrt{5} - 3\right )} \log \left (2 \, a^{\frac{2}{5}} x^{2} + a^{\frac{1}{5}} b^{\frac{1}{5}} x{\left (\sqrt{5} - 1\right )} + 2 \, b^{\frac{2}{5}}\right )}{a^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )}} + \frac{2 \, b^{\frac{1}{5}} \log \left (a^{\frac{1}{5}} x + b^{\frac{1}{5}}\right )}{a^{\frac{1}{5}}}}{10 \, a} + \frac{x}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a + b/x^5),x, algorithm="maxima")

[Out]

-1/10*(sqrt(5)*b^(1/5)*(sqrt(5) - 1)*log((4*a^(2/5)*x - a^(1/5)*b^(1/5)*(sqrt(5)
+ 1) - a^(1/5)*b^(1/5)*sqrt(2*sqrt(5) - 10))/(4*a^(2/5)*x - a^(1/5)*b^(1/5)*(sq
rt(5) + 1) + a^(1/5)*b^(1/5)*sqrt(2*sqrt(5) - 10)))/(a^(1/5)*sqrt(2*sqrt(5) - 10
)) + sqrt(5)*b^(1/5)*(sqrt(5) + 1)*log((4*a^(2/5)*x + a^(1/5)*b^(1/5)*(sqrt(5) -
1) - a^(1/5)*b^(1/5)*sqrt(-2*sqrt(5) - 10))/(4*a^(2/5)*x + a^(1/5)*b^(1/5)*(sqr
t(5) - 1) + a^(1/5)*b^(1/5)*sqrt(-2*sqrt(5) - 10)))/(a^(1/5)*sqrt(-2*sqrt(5) - 1
0)) - b^(1/5)*(sqrt(5) + 3)*log(2*a^(2/5)*x^2 - a^(1/5)*b^(1/5)*x*(sqrt(5) + 1)
+ 2*b^(2/5))/(a^(1/5)*(sqrt(5) + 1)) - b^(1/5)*(sqrt(5) - 3)*log(2*a^(2/5)*x^2 +
a^(1/5)*b^(1/5)*x*(sqrt(5) - 1) + 2*b^(2/5))/(a^(1/5)*(sqrt(5) - 1)) + 2*b^(1/5
)*log(a^(1/5)*x + b^(1/5))/a^(1/5))/a + x/a

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a + b/x^5),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 1.3344, size = 22, normalized size = 0.07 \[ \operatorname{RootSum}{\left (3125 t^{5} a^{6} + b, \left ( t \mapsto t \log{\left (- 5 t a + x \right )} \right )\right )} + \frac{x}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**5),x)

[Out]

RootSum(3125*_t**5*a**6 + b, Lambda(_t, _t*log(-5*_t*a + x))) + x/a

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GIAC/XCAS [A]  time = 0.226639, size = 362, normalized size = 1.17 \[ \frac{\left (-\frac{b}{a}\right )^{\frac{1}{5}}{\rm ln}\left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{5}} \right |}\right )}{5 \, a} + \frac{x}{a} - \frac{\left (-a^{4} b\right )^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} + 10} \arctan \left (-\frac{{\left (\sqrt{5} - 1\right )} \left (-\frac{b}{a}\right )^{\frac{1}{5}} - 4 \, x}{\sqrt{2 \, \sqrt{5} + 10} \left (-\frac{b}{a}\right )^{\frac{1}{5}}}\right )}{10 \, a^{2}} - \frac{\left (-a^{4} b\right )^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} + 10} \arctan \left (\frac{{\left (\sqrt{5} + 1\right )} \left (-\frac{b}{a}\right )^{\frac{1}{5}} + 4 \, x}{\sqrt{-2 \, \sqrt{5} + 10} \left (-\frac{b}{a}\right )^{\frac{1}{5}}}\right )}{10 \, a^{2}} - \frac{\left (-a^{4} b\right )^{\frac{1}{5}}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{5} \left (-\frac{b}{a}\right )^{\frac{1}{5}} + \left (-\frac{b}{a}\right )^{\frac{1}{5}}\right )} + \left (-\frac{b}{a}\right )^{\frac{2}{5}}\right )}{5 \, a^{2}{\left (\sqrt{5} - 1\right )}} + \frac{\left (-a^{4} b\right )^{\frac{1}{5}}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{5} \left (-\frac{b}{a}\right )^{\frac{1}{5}} - \left (-\frac{b}{a}\right )^{\frac{1}{5}}\right )} + \left (-\frac{b}{a}\right )^{\frac{2}{5}}\right )}{5 \, a^{2}{\left (\sqrt{5} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a + b/x^5),x, algorithm="giac")

[Out]

1/5*(-b/a)^(1/5)*ln(abs(x - (-b/a)^(1/5)))/a + x/a - 1/10*(-a^4*b)^(1/5)*sqrt(2*
sqrt(5) + 10)*arctan(-((sqrt(5) - 1)*(-b/a)^(1/5) - 4*x)/(sqrt(2*sqrt(5) + 10)*(
-b/a)^(1/5)))/a^2 - 1/10*(-a^4*b)^(1/5)*sqrt(-2*sqrt(5) + 10)*arctan(((sqrt(5) +
1)*(-b/a)^(1/5) + 4*x)/(sqrt(-2*sqrt(5) + 10)*(-b/a)^(1/5)))/a^2 - 1/5*(-a^4*b)
^(1/5)*ln(x^2 + 1/2*x*(sqrt(5)*(-b/a)^(1/5) + (-b/a)^(1/5)) + (-b/a)^(2/5))/(a^2
*(sqrt(5) - 1)) + 1/5*(-a^4*b)^(1/5)*ln(x^2 - 1/2*x*(sqrt(5)*(-b/a)^(1/5) - (-b/
a)^(1/5)) + (-b/a)^(2/5))/(a^2*(sqrt(5) + 1))