3.1274 \(\int \frac{1}{a+b x^5} \, dx\)

Optimal. Leaf size=301 \[ -\frac{\left (1-\sqrt{5}\right ) \log \left (2 a^{2/5}-\left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}-\frac{\left (1+\sqrt{5}\right ) \log \left (2 a^{2/5}-\left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+2 b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}+\frac{\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}-\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\left (1-\sqrt{5}\right ) \sqrt [5]{a}-4 \sqrt [5]{b} x}{\sqrt{2 \left (5+\sqrt{5}\right )} \sqrt [5]{a}}\right )}{5 a^{4/5} \sqrt [5]{b}}-\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{\left (1+\sqrt{5}\right ) \sqrt [5]{a}-4 \sqrt [5]{b} x}{\sqrt{2 \left (5-\sqrt{5}\right )} \sqrt [5]{a}}\right )}{5 a^{4/5} \sqrt [5]{b}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 1.16663, antiderivative size = 305, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ -\frac{\left (1-\sqrt{5}\right ) \log \left (a^{2/5}-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}-\frac{\left (1+\sqrt{5}\right ) \log \left (a^{2/5}-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}+\frac{\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}+\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{2 \sqrt{\frac{2}{5+\sqrt{5}}} \sqrt [5]{b} x}{\sqrt [5]{a}}+\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}\right )}{5 a^{4/5} \sqrt [5]{b}}-\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}-\frac{\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} \sqrt [5]{b} x}{\sqrt [5]{a}}\right )}{5 a^{4/5} \sqrt [5]{b}} \]

Warning: Unable to verify antiderivative.

[In]  Int[(a + b*x^5)^(-1),x]

[Out]

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

_______________________________________________________________________________________

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/(b*x**5+a),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.308953, size = 311, normalized size = 1.03 \[ -\frac{\left (1-\sqrt{5}\right ) \log \left (a^{2/5}-\frac{1}{2} \left (1-\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}-\frac{\left (1+\sqrt{5}\right ) \log \left (a^{2/5}-\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5} x^2\right )}{20 a^{4/5} \sqrt [5]{b}}+\frac{\log \left (\sqrt [5]{a}+\sqrt [5]{b} x\right )}{5 a^{4/5} \sqrt [5]{b}}+\frac{2 \sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}} \tan ^{-1}\left (\frac{\sqrt [5]{b} x-\frac{1}{4} \left (1-\sqrt{5}\right ) \sqrt [5]{a}}{\sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}} \sqrt [5]{a}}\right )}{5 a^{4/5} \sqrt [5]{b}}+\frac{2 \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}} \tan ^{-1}\left (\frac{\sqrt [5]{b} x-\frac{1}{4} \left (1+\sqrt{5}\right ) \sqrt [5]{a}}{\sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}} \sqrt [5]{a}}\right )}{5 a^{4/5} \sqrt [5]{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^5)^(-1),x]

[Out]

(2*Sqrt[5/8 + Sqrt[5]/8]*ArcTan[(-((1 - Sqrt[5])*a^(1/5))/4 + b^(1/5)*x)/(Sqrt[5
/8 + Sqrt[5]/8]*a^(1/5))])/(5*a^(4/5)*b^(1/5)) + (2*Sqrt[5/8 - Sqrt[5]/8]*ArcTan
[(-((1 + Sqrt[5])*a^(1/5))/4 + b^(1/5)*x)/(Sqrt[5/8 - Sqrt[5]/8]*a^(1/5))])/(5*a
^(4/5)*b^(1/5)) + Log[a^(1/5) + b^(1/5)*x]/(5*a^(4/5)*b^(1/5)) - ((1 - Sqrt[5])*
Log[a^(2/5) - ((1 - Sqrt[5])*a^(1/5)*b^(1/5)*x)/2 + b^(2/5)*x^2])/(20*a^(4/5)*b^
(1/5)) - ((1 + Sqrt[5])*Log[a^(2/5) - ((1 + Sqrt[5])*a^(1/5)*b^(1/5)*x)/2 + b^(2
/5)*x^2])/(20*a^(4/5)*b^(1/5))

_______________________________________________________________________________________

Maple [B]  time = 0.115, size = 895, normalized size = 3. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^5+a),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.60496, size = 429, normalized size = 1.43 \[ \frac{\sqrt{5}{\left (\sqrt{5} - 1\right )} \log \left (\frac{4 \, b^{\frac{2}{5}} x - a^{\frac{1}{5}} b^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )} - a^{\frac{1}{5}} b^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10}}{4 \, b^{\frac{2}{5}} x - a^{\frac{1}{5}} b^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )} + a^{\frac{1}{5}} b^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10}}\right )}{10 \, a^{\frac{4}{5}} b^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10}} + \frac{\sqrt{5}{\left (\sqrt{5} + 1\right )} \log \left (\frac{4 \, b^{\frac{2}{5}} x + a^{\frac{1}{5}} b^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )} - a^{\frac{1}{5}} b^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10}}{4 \, b^{\frac{2}{5}} x + a^{\frac{1}{5}} b^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )} + a^{\frac{1}{5}} b^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10}}\right )}{10 \, a^{\frac{4}{5}} b^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10}} - \frac{{\left (\sqrt{5} + 3\right )} \log \left (2 \, b^{\frac{2}{5}} x^{2} - a^{\frac{1}{5}} b^{\frac{1}{5}} x{\left (\sqrt{5} + 1\right )} + 2 \, a^{\frac{2}{5}}\right )}{10 \, a^{\frac{4}{5}} b^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )}} - \frac{{\left (\sqrt{5} - 3\right )} \log \left (2 \, b^{\frac{2}{5}} x^{2} + a^{\frac{1}{5}} b^{\frac{1}{5}} x{\left (\sqrt{5} - 1\right )} + 2 \, a^{\frac{2}{5}}\right )}{10 \, a^{\frac{4}{5}} b^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )}} + \frac{\log \left (b^{\frac{1}{5}} x + a^{\frac{1}{5}}\right )}{5 \, a^{\frac{4}{5}} b^{\frac{1}{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*sqrt(5)*(sqrt(5) - 1)*log((4*b^(2/5)*x - a^(1/5)*b^(1/5)*(sqrt(5) + 1) - a^
(1/5)*b^(1/5)*sqrt(2*sqrt(5) - 10))/(4*b^(2/5)*x - a^(1/5)*b^(1/5)*(sqrt(5) + 1)
+ a^(1/5)*b^(1/5)*sqrt(2*sqrt(5) - 10)))/(a^(4/5)*b^(1/5)*sqrt(2*sqrt(5) - 10))
+ 1/10*sqrt(5)*(sqrt(5) + 1)*log((4*b^(2/5)*x + a^(1/5)*b^(1/5)*(sqrt(5) - 1) -
a^(1/5)*b^(1/5)*sqrt(-2*sqrt(5) - 10))/(4*b^(2/5)*x + a^(1/5)*b^(1/5)*(sqrt(5)
- 1) + a^(1/5)*b^(1/5)*sqrt(-2*sqrt(5) - 10)))/(a^(4/5)*b^(1/5)*sqrt(-2*sqrt(5)
- 10)) - 1/10*(sqrt(5) + 3)*log(2*b^(2/5)*x^2 - a^(1/5)*b^(1/5)*x*(sqrt(5) + 1)
+ 2*a^(2/5))/(a^(4/5)*b^(1/5)*(sqrt(5) + 1)) - 1/10*(sqrt(5) - 3)*log(2*b^(2/5)*
x^2 + a^(1/5)*b^(1/5)*x*(sqrt(5) - 1) + 2*a^(2/5))/(a^(4/5)*b^(1/5)*(sqrt(5) - 1
)) + 1/5*log(b^(1/5)*x + a^(1/5))/(a^(4/5)*b^(1/5))

_______________________________________________________________________________________

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/(b*x^5 + a),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

_______________________________________________________________________________________

Sympy [A]  time = 0.408071, size = 20, normalized size = 0.07 \[ \operatorname{RootSum}{\left (3125 t^{5} a^{4} b - 1, \left ( t \mapsto t \log{\left (5 t a + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**5+a),x)

[Out]

RootSum(3125*_t**5*a**4*b - 1, Lambda(_t, _t*log(5*_t*a + x)))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.239199, size = 371, normalized size = 1.23 \[ -\frac{\left (-\frac{a}{b}\right )^{\frac{1}{5}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{5}} \right |}\right )}{5 \, a} + \frac{\left (-a b^{4}\right )^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} + 10} \arctan \left (-\frac{{\left (\sqrt{5} - 1\right )} \left (-\frac{a}{b}\right )^{\frac{1}{5}} - 4 \, x}{\sqrt{2 \, \sqrt{5} + 10} \left (-\frac{a}{b}\right )^{\frac{1}{5}}}\right )}{10 \, a b} + \frac{\left (-a b^{4}\right )^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} + 10} \arctan \left (\frac{{\left (\sqrt{5} + 1\right )} \left (-\frac{a}{b}\right )^{\frac{1}{5}} + 4 \, x}{\sqrt{-2 \, \sqrt{5} + 10} \left (-\frac{a}{b}\right )^{\frac{1}{5}}}\right )}{10 \, a b} + \frac{\left (-a b^{4}\right )^{\frac{1}{5}}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{5} \left (-\frac{a}{b}\right )^{\frac{1}{5}} + \left (-\frac{a}{b}\right )^{\frac{1}{5}}\right )} + \left (-\frac{a}{b}\right )^{\frac{2}{5}}\right )}{5 \, a b{\left (\sqrt{5} - 1\right )}} - \frac{\left (-a b^{4}\right )^{\frac{1}{5}}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{5} \left (-\frac{a}{b}\right )^{\frac{1}{5}} - \left (-\frac{a}{b}\right )^{\frac{1}{5}}\right )} + \left (-\frac{a}{b}\right )^{\frac{2}{5}}\right )}{5 \, a b{\left (\sqrt{5} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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