3.2106 \(\int \frac {1}{a+\frac {b}{x^5}} \, dx\)

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-1/5*b^(1/5)*ln(b^(1/5)+a^(1/5)*x)/a^(6/5)+1/20*b^(1/5)*ln(b^(2/5)+a^(2/5)*x^2-1/2*a^(1/5)*b^(1/5)*x*(-5^(1
/2)+1))*(-5^(1/2)+1)/a^(6/5)+1/20*b^(1/5)*ln(b^(2/5)+a^(2/5)*x^2-1/2*a^(1/5)*b^(1/5)*x*(5^(1/2)+1))*(5^(1/2)+1
)/a^(6/5)-1/10*b^(1/5)*arctan(1/5*a^(1/5)*x*(50+10*5^(1/2))^(1/2)/b^(1/5)-1/5*(25+10*5^(1/2))^(1/2))*(10-2*5^(
1/2))^(1/2)/a^(6/5)-1/10*b^(1/5)*arctan(1/5*(25-10*5^(1/2))^(1/2)+2*a^(1/5)*x*2^(1/2)/(5+5^(1/2))^(1/2)/b^(1/5
))*(10+2*5^(1/2))^(1/2)/a^(6/5)

________________________________________________________________________________________

Rubi [A]  time = 0.66, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {193, 321, 201, 634, 618, 204, 628, 31} \[ \frac {\left (1-\sqrt {5}\right ) \sqrt [5]{b} \log \left (a^{2/5} x^2-\frac {1}{2} \left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5}\right )}{20 a^{6/5}}+\frac {\left (1+\sqrt {5}\right ) \sqrt [5]{b} \log \left (a^{2/5} x^2-\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+b^{2/5}\right )}{20 a^{6/5}}-\frac {\sqrt [5]{b} \log \left (\sqrt [5]{a} x+\sqrt [5]{b}\right )}{5 a^{6/5}}-\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \sqrt [5]{b} \tan ^{-1}\left (\frac {2 \sqrt {\frac {2}{5+\sqrt {5}}} \sqrt [5]{a} x}{\sqrt [5]{b}}+\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}\right )}{5 a^{6/5}}+\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \sqrt [5]{b} \tan ^{-1}\left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}-\frac {\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \sqrt [5]{a} x}{\sqrt [5]{b}}\right )}{5 a^{6/5}}+\frac {x}{a} \]

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)*Lo
g[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))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 193

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && LtQ[n, 0]
&& IntegerQ[p]

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]]
, k, u}, Simp[u = Int[(r - s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (r*
Int[1/(r + s*x), x])/(a*n) + Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[
(n - 3)/2, 0] && PosQ[a/b]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {1}{a+\frac {b}{x^5}} \, dx &=\int \frac {x^5}{b+a x^5} \, dx\\ &=\frac {x}{a}-\frac {b \int \frac {1}{b+a x^5} \, dx}{a}\\ &=\frac {x}{a}-\frac {\sqrt [5]{b} \int \frac {1}{\sqrt [5]{b}+\sqrt [5]{a} x} \, dx}{5 a}-\frac {\left (2 \sqrt [5]{b}\right ) \int \frac {\sqrt [5]{b}-\frac {1}{4} \left (1-\sqrt {5}\right ) \sqrt [5]{a} x}{b^{2/5}-\frac {1}{2} \left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2} \, dx}{5 a}-\frac {\left (2 \sqrt [5]{b}\right ) \int \frac {\sqrt [5]{b}-\frac {1}{4} \left (1+\sqrt {5}\right ) \sqrt [5]{a} x}{b^{2/5}-\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2} \, dx}{5 a}\\ &=\frac {x}{a}-\frac {\sqrt [5]{b} \log \left (\sqrt [5]{b}+\sqrt [5]{a} x\right )}{5 a^{6/5}}+\frac {\left (\left (1-\sqrt {5}\right ) \sqrt [5]{b}\right ) \int \frac {-\frac {1}{2} \left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}+2 a^{2/5} x}{b^{2/5}-\frac {1}{2} \left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2} \, dx}{20 a^{6/5}}+\frac {\left (\left (1+\sqrt {5}\right ) \sqrt [5]{b}\right ) \int \frac {-\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}+2 a^{2/5} x}{b^{2/5}-\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2} \, dx}{20 a^{6/5}}-\frac {\left (\left (5-\sqrt {5}\right ) b^{2/5}\right ) \int \frac {1}{b^{2/5}-\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2} \, dx}{20 a}-\frac {\left (\left (5+\sqrt {5}\right ) b^{2/5}\right ) \int \frac {1}{b^{2/5}-\frac {1}{2} \left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b} x+a^{2/5} x^2} \, dx}{20 a}\\ &=\frac {x}{a}-\frac {\sqrt [5]{b} \log \left (\sqrt [5]{b}+\sqrt [5]{a} x\right )}{5 a^{6/5}}+\frac {\left (1+\sqrt {5}\right ) \sqrt [5]{b} \log \left (2 b^{2/5}-\sqrt [5]{a} \sqrt [5]{b} x-\sqrt {5} \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5} x^2\right )}{20 a^{6/5}}+\frac {\left (1-\sqrt {5}\right ) \sqrt [5]{b} \log \left (2 b^{2/5}-\sqrt [5]{a} \sqrt [5]{b} x+\sqrt {5} \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5} x^2\right )}{20 a^{6/5}}+\frac {\left (\left (5-\sqrt {5}\right ) b^{2/5}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2} \left (5-\sqrt {5}\right ) a^{2/5} b^{2/5}-x^2} \, dx,x,-\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}+2 a^{2/5} x\right )}{10 a}+\frac {\left (\left (5+\sqrt {5}\right ) b^{2/5}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2} \left (5+\sqrt {5}\right ) a^{2/5} b^{2/5}-x^2} \, dx,x,-\frac {1}{2} \left (1-\sqrt {5}\right ) \sqrt [5]{a} \sqrt [5]{b}+2 a^{2/5} x\right )}{10 a}\\ &=\frac {x}{a}+\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \sqrt [5]{b} \tan ^{-1}\left (\frac {\sqrt {5+\sqrt {5}} \left (\left (1+\sqrt {5}\right ) \sqrt [5]{b}-4 \sqrt [5]{a} x\right )}{2 \sqrt {10} \sqrt [5]{b}}\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 {2 \sqrt {\frac {2}{5+\sqrt {5}}} \sqrt [5]{a} x}{\sqrt [5]{b}}\right )}{5 a^{6/5}}-\frac {\sqrt [5]{b} \log \left (\sqrt [5]{b}+\sqrt [5]{a} x\right )}{5 a^{6/5}}+\frac {\left (1+\sqrt {5}\right ) \sqrt [5]{b} \log \left (2 b^{2/5}-\sqrt [5]{a} \sqrt [5]{b} x-\sqrt {5} \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5} x^2\right )}{20 a^{6/5}}+\frac {\left (1-\sqrt {5}\right ) \sqrt [5]{b} \log \left (2 b^{2/5}-\sqrt [5]{a} \sqrt [5]{b} x+\sqrt {5} \sqrt [5]{a} \sqrt [5]{b} x+2 a^{2/5} x^2\right )}{20 a^{6/5}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.29, 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 + Sqr
t[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))

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

giac [A]  time = 0.20, size = 268, normalized size = 0.86 \[ \frac {\left (-\frac {b}{a}\right )^{\frac {1}{5}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{5}} \right |}\right )}{5 \, a} + \frac {x}{a} - \frac {\left (-a^{4} b\right )^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} + 10} \arctan \left (-\frac {{\left (\sqrt {5} - 1\right )} \left (-\frac {b}{a}\right )^{\frac {1}{5}} - 4 \, x}{\sqrt {2 \, \sqrt {5} + 10} \left (-\frac {b}{a}\right )^{\frac {1}{5}}}\right )}{10 \, a^{2}} - \frac {\left (-a^{4} b\right )^{\frac {1}{5}} \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (\frac {{\left (\sqrt {5} + 1\right )} \left (-\frac {b}{a}\right )^{\frac {1}{5}} + 4 \, x}{\sqrt {-2 \, \sqrt {5} + 10} \left (-\frac {b}{a}\right )^{\frac {1}{5}}}\right )}{10 \, a^{2}} - \frac {\left (-a^{4} b\right )^{\frac {1}{5}} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {5} \left (-\frac {b}{a}\right )^{\frac {1}{5}} + \left (-\frac {b}{a}\right )^{\frac {1}{5}}\right )} + \left (-\frac {b}{a}\right )^{\frac {2}{5}}\right )}{5 \, a^{2} {\left (\sqrt {5} - 1\right )}} + \frac {\left (-a^{4} b\right )^{\frac {1}{5}} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {5} \left (-\frac {b}{a}\right )^{\frac {1}{5}} - \left (-\frac {b}{a}\right )^{\frac {1}{5}}\right )} + \left (-\frac {b}{a}\right )^{\frac {2}{5}}\right )}{5 \, a^{2} {\left (\sqrt {5} + 1\right )}} \]

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)*log(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) + 1
0)*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)*lo
g(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)*l
og(x^2 - 1/2*x*(sqrt(5)*(-b/a)^(1/5) - (-b/a)^(1/5)) + (-b/a)^(2/5))/(a^2*(sqrt(5) + 1))

________________________________________________________________________________________

maple [B]  time = 0.09, size = 911, normalized size = 2.94 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^5),x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 2.03, size = 250, normalized size = 0.81 \[ -\frac {\frac {2 \, \sqrt {5} b^{\frac {1}{5}} {\left (\sqrt {5} + 1\right )} \arctan \left (\frac {4 \, a^{\frac {2}{5}} x + a^{\frac {1}{5}} b^{\frac {1}{5}} {\left (\sqrt {5} - 1\right )}}{a^{\frac {1}{5}} b^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} + 10}}\right )}{a^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} + 10}} + \frac {2 \, \sqrt {5} b^{\frac {1}{5}} {\left (\sqrt {5} - 1\right )} \arctan \left (\frac {4 \, a^{\frac {2}{5}} x - a^{\frac {1}{5}} b^{\frac {1}{5}} {\left (\sqrt {5} + 1\right )}}{a^{\frac {1}{5}} b^{\frac {1}{5}} \sqrt {-2 \, \sqrt {5} + 10}}\right )}{a^{\frac {1}{5}} \sqrt {-2 \, \sqrt {5} + 10}} - \frac {b^{\frac {1}{5}} {\left (\sqrt {5} + 3\right )} \log \left (2 \, a^{\frac {2}{5}} x^{2} - a^{\frac {1}{5}} b^{\frac {1}{5}} x {\left (\sqrt {5} + 1\right )} + 2 \, b^{\frac {2}{5}}\right )}{a^{\frac {1}{5}} {\left (\sqrt {5} + 1\right )}} - \frac {b^{\frac {1}{5}} {\left (\sqrt {5} - 3\right )} \log \left (2 \, a^{\frac {2}{5}} x^{2} + a^{\frac {1}{5}} b^{\frac {1}{5}} x {\left (\sqrt {5} - 1\right )} + 2 \, b^{\frac {2}{5}}\right )}{a^{\frac {1}{5}} {\left (\sqrt {5} - 1\right )}} + \frac {2 \, b^{\frac {1}{5}} \log \left (a^{\frac {1}{5}} x + b^{\frac {1}{5}}\right )}{a^{\frac {1}{5}}}}{10 \, a} + \frac {x}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.63, size = 290, normalized size = 0.94 \[ \frac {x}{a}+\frac {{\left (-b\right )}^{1/5}\,\ln \left ({\left (-b\right )}^{16/5}+a^{1/5}\,b^3\,x\right )}{5\,a^{6/5}}-\frac {{\left (-b\right )}^{1/5}\,\ln \left (25\,a^{4/5}\,{\left (-b\right )}^{16/5}\,\left (\frac {\sqrt {-2\,\sqrt {5}-10}}{20}-\frac {\sqrt {5}}{20}+\frac {1}{20}\right )-5\,a\,b^3\,x\right )\,\left (\frac {\sqrt {-2\,\sqrt {5}-10}}{20}-\frac {\sqrt {5}}{20}+\frac {1}{20}\right )}{a^{6/5}}+\frac {{\left (-b\right )}^{1/5}\,\ln \left (25\,a^{4/5}\,{\left (-b\right )}^{16/5}\,\left (\frac {\sqrt {5}}{20}+\frac {\sqrt {-2\,\sqrt {5}-10}}{20}-\frac {1}{20}\right )+5\,a\,b^3\,x\right )\,\left (\frac {\sqrt {5}}{20}+\frac {\sqrt {-2\,\sqrt {5}-10}}{20}-\frac {1}{20}\right )}{a^{6/5}}-\frac {{\left (-b\right )}^{1/5}\,\ln \left (25\,a^{4/5}\,{\left (-b\right )}^{16/5}\,\left (\frac {\sqrt {5}}{20}-\frac {\sqrt {2\,\sqrt {5}-10}}{20}+\frac {1}{20}\right )-5\,a\,b^3\,x\right )\,\left (\frac {\sqrt {5}}{20}-\frac {\sqrt {2\,\sqrt {5}-10}}{20}+\frac {1}{20}\right )}{a^{6/5}}-\frac {{\left (-b\right )}^{1/5}\,\ln \left (25\,a^{4/5}\,{\left (-b\right )}^{16/5}\,\left (\frac {\sqrt {5}}{20}+\frac {\sqrt {2\,\sqrt {5}-10}}{20}+\frac {1}{20}\right )-5\,a\,b^3\,x\right )\,\left (\frac {\sqrt {5}}{20}+\frac {\sqrt {2\,\sqrt {5}-10}}{20}+\frac {1}{20}\right )}{a^{6/5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b/x^5),x)

[Out]

x/a + ((-b)^(1/5)*log((-b)^(16/5) + a^(1/5)*b^3*x))/(5*a^(6/5)) - ((-b)^(1/5)*log(25*a^(4/5)*(-b)^(16/5)*((- 2
*5^(1/2) - 10)^(1/2)/20 - 5^(1/2)/20 + 1/20) - 5*a*b^3*x)*((- 2*5^(1/2) - 10)^(1/2)/20 - 5^(1/2)/20 + 1/20))/a
^(6/5) + ((-b)^(1/5)*log(25*a^(4/5)*(-b)^(16/5)*(5^(1/2)/20 + (- 2*5^(1/2) - 10)^(1/2)/20 - 1/20) + 5*a*b^3*x)
*(5^(1/2)/20 + (- 2*5^(1/2) - 10)^(1/2)/20 - 1/20))/a^(6/5) - ((-b)^(1/5)*log(25*a^(4/5)*(-b)^(16/5)*(5^(1/2)/
20 - (2*5^(1/2) - 10)^(1/2)/20 + 1/20) - 5*a*b^3*x)*(5^(1/2)/20 - (2*5^(1/2) - 10)^(1/2)/20 + 1/20))/a^(6/5) -
((-b)^(1/5)*log(25*a^(4/5)*(-b)^(16/5)*(5^(1/2)/20 + (2*5^(1/2) - 10)^(1/2)/20 + 1/20) - 5*a*b^3*x)*(5^(1/2)/
20 + (2*5^(1/2) - 10)^(1/2)/20 + 1/20))/a^(6/5)

________________________________________________________________________________________

sympy [A]  time = 0.20, 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

________________________________________________________________________________________