Integrand size = 44, antiderivative size = 239 \[ \int \frac {b^5+a^5 x^5}{\sqrt {b^2 x+a^2 x^3} \left (-b^5+a^5 x^5\right )} \, dx=-\frac {2 \sqrt {2 \left (-1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{5 \sqrt {a} \sqrt {b}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{5 \sqrt {a} \sqrt {b}}-\frac {2 \sqrt {2 \left (1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{5 \sqrt {a} \sqrt {b}} \]
-2/5*(-2+2*5^(1/2))^(1/2)*arctan(1/2*(2+2*5^(1/2))^(1/2)*a^(1/2)*b^(1/2)*( a^2*x^3+b^2*x)^(1/2)/(a^2*x^2+b^2))/a^(1/2)/b^(1/2)-1/5*2^(1/2)*arctanh(2^ (1/2)*a^(1/2)*b^(1/2)*(a^2*x^3+b^2*x)^(1/2)/(a^2*x^2+b^2))/a^(1/2)/b^(1/2) -2/5*(2+2*5^(1/2))^(1/2)*arctanh(1/2*(-2+2*5^(1/2))^(1/2)*a^(1/2)*b^(1/2)* (a^2*x^3+b^2*x)^(1/2)/(a^2*x^2+b^2))/a^(1/2)/b^(1/2)
Time = 1.51 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.90 \[ \int \frac {b^5+a^5 x^5}{\sqrt {b^2 x+a^2 x^3} \left (-b^5+a^5 x^5\right )} \, dx=-\frac {\sqrt {2} \sqrt {x} \sqrt {b^2+a^2 x^2} \left (2 \sqrt {-1+\sqrt {5}} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )+2 \sqrt {1+\sqrt {5}} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )\right )}{5 \sqrt {a} \sqrt {b} \sqrt {x \left (b^2+a^2 x^2\right )}} \]
-1/5*(Sqrt[2]*Sqrt[x]*Sqrt[b^2 + a^2*x^2]*(2*Sqrt[-1 + Sqrt[5]]*ArcTan[(Sq rt[(1 + Sqrt[5])/2]*Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[b^2 + a^2*x^2]] + ArcTan h[(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[b^2 + a^2*x^2]] + 2*Sqrt[1 + Sqrt [5]]*ArcTanh[(Sqrt[(-1 + Sqrt[5])/2]*Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[b^2 + a ^2*x^2]]))/(Sqrt[a]*Sqrt[b]*Sqrt[x*(b^2 + a^2*x^2)])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 5.81 (sec) , antiderivative size = 1405, normalized size of antiderivative = 5.88, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2467, 25, 2035, 7276, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {a^5 x^5+b^5}{\sqrt {a^2 x^3+b^2 x} \left (a^5 x^5-b^5\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {a^2 x^2+b^2} \int -\frac {b^5+a^5 x^5}{\sqrt {x} \sqrt {b^2+a^2 x^2} \left (b^5-a^5 x^5\right )}dx}{\sqrt {a^2 x^3+b^2 x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {a^2 x^2+b^2} \int \frac {b^5+a^5 x^5}{\sqrt {x} \sqrt {b^2+a^2 x^2} \left (b^5-a^5 x^5\right )}dx}{\sqrt {a^2 x^3+b^2 x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2+b^2} \int \frac {b^5+a^5 x^5}{\sqrt {b^2+a^2 x^2} \left (b^5-a^5 x^5\right )}d\sqrt {x}}{\sqrt {a^2 x^3+b^2 x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2+b^2} \int \left (\frac {2 b^5}{\sqrt {b^2+a^2 x^2} \left (b^5-a^5 x^5\right )}-\frac {1}{\sqrt {b^2+a^2 x^2}}\right )d\sqrt {x}}{\sqrt {a^2 x^3+b^2 x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {b^2+a^2 x^2} \left (\frac {i \left (1-\sqrt [5]{-1}\right ) \arctan \left (\frac {\sqrt [10]{-1} \sqrt {1-(-1)^{3/5}} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{5 \left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \sqrt {1-(-1)^{3/5}} \sqrt {a} \sqrt {b}}-\frac {(-1)^{9/10} \arctan \left (\frac {\sqrt [10]{-1} \sqrt {1-(-1)^{3/5}} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{5 \sqrt {1-(-1)^{3/5}} \sqrt {a} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{5 \sqrt {2} \sqrt {a} \sqrt {b}}-\frac {(-1)^{4/5} \left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left (1-(-1)^{3/5}\right ) \text {arctanh}\left (\frac {\sqrt [5]{-1} \sqrt {1-\sqrt [5]{-1}} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{5 \sqrt {1-\sqrt [5]{-1}} \sqrt {a} \sqrt {b}}-\frac {(-1)^{4/5} \text {arctanh}\left (\frac {\sqrt [5]{-1} \sqrt {1-\sqrt [5]{-1}} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{5 \sqrt {1-\sqrt [5]{-1}} \sqrt {a} \sqrt {b}}+\frac {\left (1-(-1)^{4/5}\right ) (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{5 \left (1+(-1)^{3/5}\right ) \sqrt {a} \sqrt {b} \sqrt {b^2+a^2 x^2}}+\frac {\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{5 \sqrt {a} \sqrt {b} \sqrt {b^2+a^2 x^2}}+\frac {(b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{5 \left (1+(-1)^{2/5}\right ) \sqrt {a} \sqrt {b} \sqrt {b^2+a^2 x^2}}+\frac {(b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{5 \left (1-\sqrt [5]{-1}\right ) \sqrt {a} \sqrt {b} \sqrt {b^2+a^2 x^2}}-\frac {2 (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{5 \sqrt {a} \sqrt {b} \sqrt {b^2+a^2 x^2}}-\frac {\sqrt [5]{-1} \left (1+\sqrt [5]{-1}\right ) \left (1+(-1)^{2/5}\right ) (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticPi}\left (\frac {1}{4} (-1)^{4/5} \left (1-\sqrt [5]{-1}\right )^2,2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{10 \sqrt {a} \sqrt {b} \sqrt {b^2+a^2 x^2}}-\frac {\left (1-(-1)^{2/5}\right ) (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticPi}\left (-\frac {1}{4} (-1)^{3/5} \left (1+(-1)^{2/5}\right )^2,2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{10 \left (1+(-1)^{2/5}\right ) \sqrt {a} \sqrt {b} \sqrt {b^2+a^2 x^2}}+\frac {\sqrt [5]{-1} \left (1-\sqrt [5]{-1}\right )^2 \left (1+\sqrt [5]{-1}\right ) (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticPi}\left (\frac {1}{4} (-1)^{2/5} \left (1-(-1)^{3/5}\right )^2,2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{10 \sqrt {a} \sqrt {b} \sqrt {b^2+a^2 x^2}}+\frac {(-1)^{3/5} \left (1+\sqrt [5]{-1}\right ) (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticPi}\left (-\frac {1}{4} \sqrt [5]{-1} \left (1+(-1)^{4/5}\right )^2,2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{10 \left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \sqrt {a} \sqrt {b} \sqrt {b^2+a^2 x^2}}\right )}{\sqrt {a^2 x^3+b^2 x}}\) |
(-2*Sqrt[x]*Sqrt[b^2 + a^2*x^2]*(-1/5*((-1)^(9/10)*ArcTan[((-1)^(1/10)*Sqr t[1 - (-1)^(3/5)]*Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[b^2 + a^2*x^2]])/(Sqrt[1 - (-1)^(3/5)]*Sqrt[a]*Sqrt[b]) + ((I/5)*(1 - (-1)^(1/5))*ArcTan[((-1)^(1/10 )*Sqrt[1 - (-1)^(3/5)]*Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[b^2 + a^2*x^2]])/((1 - (-1)^(1/5) + (-1)^(2/5))*Sqrt[1 - (-1)^(3/5)]*Sqrt[a]*Sqrt[b]) + ArcTanh [(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[b^2 + a^2*x^2]]/(5*Sqrt[2]*Sqrt[a] *Sqrt[b]) - ((-1)^(4/5)*ArcTanh[((-1)^(1/5)*Sqrt[1 - (-1)^(1/5)]*Sqrt[a]*S qrt[b]*Sqrt[x])/Sqrt[b^2 + a^2*x^2]])/(5*Sqrt[1 - (-1)^(1/5)]*Sqrt[a]*Sqrt [b]) - ((-1)^(4/5)*(1 - (-1)^(1/5) + (-1)^(2/5))*(1 - (-1)^(3/5))*ArcTanh[ ((-1)^(1/5)*Sqrt[1 - (-1)^(1/5)]*Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[b^2 + a^2*x ^2]])/(5*Sqrt[1 - (-1)^(1/5)]*Sqrt[a]*Sqrt[b]) - (2*(b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2]) /(5*Sqrt[a]*Sqrt[b]*Sqrt[b^2 + a^2*x^2]) + ((b + a*x)*Sqrt[(b^2 + a^2*x^2) /(b + a*x)^2]*EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(5*(1 - (-1)^(1/5))*Sqrt[a]*Sqrt[b]*Sqrt[b^2 + a^2*x^2]) + ((b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2] )/(5*(1 + (-1)^(2/5))*Sqrt[a]*Sqrt[b]*Sqrt[b^2 + a^2*x^2]) + ((1 - (-1)^(1 /5) + (-1)^(2/5))*(b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticF[2* ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(5*Sqrt[a]*Sqrt[b]*Sqrt[b^2 + a^2 *x^2]) + ((1 - (-1)^(4/5))*(b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]*...
3.27.68.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 1.44 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.72
| method | result | size |
| default | \(\frac {8 \arctan \left (\frac {2 \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x \sqrt {-2 \sqrt {5}\, \sqrt {a^{2} b^{2}}+2 a b}}\right )}{5 \sqrt {-2 \sqrt {5}\, \sqrt {a^{2} b^{2}}+2 a b}}+\frac {8 \arctan \left (\frac {2 \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x \sqrt {2 \sqrt {5}\, \sqrt {a^{2} b^{2}}+2 a b}}\right )}{5 \sqrt {2 \sqrt {5}\, \sqrt {a^{2} b^{2}}+2 a b}}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \sqrt {2}}{2 x \sqrt {a b}}\right )}{5 \sqrt {a b}}\) | \(171\) |
| pseudoelliptic | \(\frac {8 \arctan \left (\frac {2 \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x \sqrt {-2 \sqrt {5}\, \sqrt {a^{2} b^{2}}+2 a b}}\right )}{5 \sqrt {-2 \sqrt {5}\, \sqrt {a^{2} b^{2}}+2 a b}}+\frac {8 \arctan \left (\frac {2 \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x \sqrt {2 \sqrt {5}\, \sqrt {a^{2} b^{2}}+2 a b}}\right )}{5 \sqrt {2 \sqrt {5}\, \sqrt {a^{2} b^{2}}+2 a b}}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \sqrt {2}}{2 x \sqrt {a b}}\right )}{5 \sqrt {a b}}\) | \(171\) |
| elliptic | \(\text {Expression too large to display}\) | \(2129\) |
8/5/(-2*5^(1/2)*(a^2*b^2)^(1/2)+2*a*b)^(1/2)*arctan(2*(x*(a^2*x^2+b^2))^(1 /2)/x/(-2*5^(1/2)*(a^2*b^2)^(1/2)+2*a*b)^(1/2))+8/5/(2*5^(1/2)*(a^2*b^2)^( 1/2)+2*a*b)^(1/2)*arctan(2*(x*(a^2*x^2+b^2))^(1/2)/x/(2*5^(1/2)*(a^2*b^2)^ (1/2)+2*a*b)^(1/2))-1/5*2^(1/2)/(a*b)^(1/2)*arctanh(1/2*(x*(a^2*x^2+b^2))^ (1/2)/x*2^(1/2)/(a*b)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 997 vs. \(2 (177) = 354\).
Time = 0.48 (sec) , antiderivative size = 2075, normalized size of antiderivative = 8.68 \[ \int \frac {b^5+a^5 x^5}{\sqrt {b^2 x+a^2 x^3} \left (-b^5+a^5 x^5\right )} \, dx=\text {Too large to display} \]
[1/10*sqrt(2)*sqrt((5*sqrt(1/5)*a*b*sqrt(1/(a^2*b^2)) + 1)/(a*b))*log(2*(2 *a^4*x^4 + 6*a^2*b^2*x^2 + 2*b^4 + sqrt(2)*(a^3*b*x^2 - 2*a^2*b^2*x + a*b^ 3 - 5*sqrt(1/5)*(a^4*b^2*x^2 + a^2*b^4)*sqrt(1/(a^2*b^2)))*sqrt(a^2*x^3 + b^2*x)*sqrt((5*sqrt(1/5)*a*b*sqrt(1/(a^2*b^2)) + 1)/(a*b)) + 10*sqrt(1/5)* (a^4*b^2*x^3 + a^2*b^4*x)*sqrt(1/(a^2*b^2)))/(a^4*x^4 + a^3*b*x^3 + a^2*b^ 2*x^2 + a*b^3*x + b^4)) - 1/10*sqrt(2)*sqrt((5*sqrt(1/5)*a*b*sqrt(1/(a^2*b ^2)) + 1)/(a*b))*log(2*(2*a^4*x^4 + 6*a^2*b^2*x^2 + 2*b^4 - sqrt(2)*(a^3*b *x^2 - 2*a^2*b^2*x + a*b^3 - 5*sqrt(1/5)*(a^4*b^2*x^2 + a^2*b^4)*sqrt(1/(a ^2*b^2)))*sqrt(a^2*x^3 + b^2*x)*sqrt((5*sqrt(1/5)*a*b*sqrt(1/(a^2*b^2)) + 1)/(a*b)) + 10*sqrt(1/5)*(a^4*b^2*x^3 + a^2*b^4*x)*sqrt(1/(a^2*b^2)))/(a^4 *x^4 + a^3*b*x^3 + a^2*b^2*x^2 + a*b^3*x + b^4)) + 1/10*sqrt(2)*sqrt(-(5*s qrt(1/5)*a*b*sqrt(1/(a^2*b^2)) - 1)/(a*b))*log(2*(2*a^4*x^4 + 6*a^2*b^2*x^ 2 + 2*b^4 + sqrt(2)*(a^3*b*x^2 - 2*a^2*b^2*x + a*b^3 + 5*sqrt(1/5)*(a^4*b^ 2*x^2 + a^2*b^4)*sqrt(1/(a^2*b^2)))*sqrt(a^2*x^3 + b^2*x)*sqrt(-(5*sqrt(1/ 5)*a*b*sqrt(1/(a^2*b^2)) - 1)/(a*b)) - 10*sqrt(1/5)*(a^4*b^2*x^3 + a^2*b^4 *x)*sqrt(1/(a^2*b^2)))/(a^4*x^4 + a^3*b*x^3 + a^2*b^2*x^2 + a*b^3*x + b^4) ) - 1/10*sqrt(2)*sqrt(-(5*sqrt(1/5)*a*b*sqrt(1/(a^2*b^2)) - 1)/(a*b))*log( 2*(2*a^4*x^4 + 6*a^2*b^2*x^2 + 2*b^4 - sqrt(2)*(a^3*b*x^2 - 2*a^2*b^2*x + a*b^3 + 5*sqrt(1/5)*(a^4*b^2*x^2 + a^2*b^4)*sqrt(1/(a^2*b^2)))*sqrt(a^2*x^ 3 + b^2*x)*sqrt(-(5*sqrt(1/5)*a*b*sqrt(1/(a^2*b^2)) - 1)/(a*b)) - 10*sq...
\[ \int \frac {b^5+a^5 x^5}{\sqrt {b^2 x+a^2 x^3} \left (-b^5+a^5 x^5\right )} \, dx=\int \frac {\left (a x + b\right ) \left (a^{4} x^{4} - a^{3} b x^{3} + a^{2} b^{2} x^{2} - a b^{3} x + b^{4}\right )}{\sqrt {x \left (a^{2} x^{2} + b^{2}\right )} \left (a x - b\right ) \left (a^{4} x^{4} + a^{3} b x^{3} + a^{2} b^{2} x^{2} + a b^{3} x + b^{4}\right )}\, dx \]
Integral((a*x + b)*(a**4*x**4 - a**3*b*x**3 + a**2*b**2*x**2 - a*b**3*x + b**4)/(sqrt(x*(a**2*x**2 + b**2))*(a*x - b)*(a**4*x**4 + a**3*b*x**3 + a** 2*b**2*x**2 + a*b**3*x + b**4)), x)
\[ \int \frac {b^5+a^5 x^5}{\sqrt {b^2 x+a^2 x^3} \left (-b^5+a^5 x^5\right )} \, dx=\int { \frac {a^{5} x^{5} + b^{5}}{{\left (a^{5} x^{5} - b^{5}\right )} \sqrt {a^{2} x^{3} + b^{2} x}} \,d x } \]
\[ \int \frac {b^5+a^5 x^5}{\sqrt {b^2 x+a^2 x^3} \left (-b^5+a^5 x^5\right )} \, dx=\int { \frac {a^{5} x^{5} + b^{5}}{{\left (a^{5} x^{5} - b^{5}\right )} \sqrt {a^{2} x^{3} + b^{2} x}} \,d x } \]
Timed out. \[ \int \frac {b^5+a^5 x^5}{\sqrt {b^2 x+a^2 x^3} \left (-b^5+a^5 x^5\right )} \, dx=\text {Hanged} \]