3.27.67 \(\int \frac {-b^5+a^5 x^5}{\sqrt {b^2 x+a^2 x^3} (b^5+a^5 x^5)} \, dx\) [2667]

3.27.67.1 Optimal result

Integrand size = 44, antiderivative size = 239 \[ \int \frac {-b^5+a^5 x^5}{\sqrt {b^2 x+a^2 x^3} \left (b^5+a^5 x^5\right )} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{5 \sqrt {a} \sqrt {b}}-\frac {2 \sqrt {2 \left (1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{5 \sqrt {a} \sqrt {b}}-\frac {2 \sqrt {2 \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{5 \sqrt {a} \sqrt {b}} \]

output

-1/5*2^(1/2)*arctan(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)*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)- 
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)
 

3.27.67.2 Mathematica [A] (verified)

Time = 1.62 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.90 \[ \int \frac {-b^5+a^5 x^5}{\sqrt {b^2 x+a^2 x^3} \left (b^5+a^5 x^5\right )} \, dx=-\frac {\sqrt {2} \sqrt {x} \sqrt {b^2+a^2 x^2} \left (\arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )+2 \sqrt {1+\sqrt {5}} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )+2 \sqrt {-1+\sqrt {5}} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )\right )}{5 \sqrt {a} \sqrt {b} \sqrt {x \left (b^2+a^2 x^2\right )}} \]

input

Integrate[(-b^5 + a^5*x^5)/(Sqrt[b^2*x + a^2*x^3]*(b^5 + a^5*x^5)),x]
 

output

-1/5*(Sqrt[2]*Sqrt[x]*Sqrt[b^2 + a^2*x^2]*(ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[b] 
*Sqrt[x])/Sqrt[b^2 + a^2*x^2]] + 2*Sqrt[1 + Sqrt[5]]*ArcTan[(Sqrt[(-1 + Sq 
rt[5])/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)])
 

3.27.67.3 Rubi [F]

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

\(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2+b^2} \int \left (\frac {2 b^5}{\sqrt {b^2+a^2 x^2} \left (b^5+a^5 x^5\right )}-\frac {1}{\sqrt {b^2+a^2 x^2}}\right )d\sqrt {x}}{\sqrt {a^2 x^3+b^2 x}}\)

\(\Big \downarrow \) 7299

\(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2+b^2} \int \left (\frac {2 b^5}{\sqrt {b^2+a^2 x^2} \left (b^5+a^5 x^5\right )}-\frac {1}{\sqrt {b^2+a^2 x^2}}\right )d\sqrt {x}}{\sqrt {a^2 x^3+b^2 x}}\)

input

Int[(-b^5 + a^5*x^5)/(Sqrt[b^2*x + a^2*x^3]*(b^5 + a^5*x^5)),x]
 

output

$Aborted
 

3.27.67.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2035

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]
 

rule 2467

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]
 

rule 7276

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]
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

3.27.67.4 Maple [A] (verified)

Time = 1.82 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.72

method	result	size
default	\(\frac {8 \arctan \left (\frac {2 \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x \sqrt {-2 \sqrt {5}\, \sqrt {a^{2} b^{2}}-2 a b}}\right )}{5 \sqrt {-2 \sqrt {5}\, \sqrt {a^{2} b^{2}}-2 a b}}+\frac {8 \arctan \left (\frac {2 \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x \sqrt {2 \sqrt {5}\, \sqrt {a^{2} b^{2}}-2 a b}}\right )}{5 \sqrt {2 \sqrt {5}\, \sqrt {a^{2} b^{2}}-2 a b}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \sqrt {2}}{2 x \sqrt {a b}}\right )}{5 \sqrt {a b}}\)	\(171\)
pseudoelliptic	\(\frac {8 \arctan \left (\frac {2 \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x \sqrt {-2 \sqrt {5}\, \sqrt {a^{2} b^{2}}-2 a b}}\right )}{5 \sqrt {-2 \sqrt {5}\, \sqrt {a^{2} b^{2}}-2 a b}}+\frac {8 \arctan \left (\frac {2 \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x \sqrt {2 \sqrt {5}\, \sqrt {a^{2} b^{2}}-2 a b}}\right )}{5 \sqrt {2 \sqrt {5}\, \sqrt {a^{2} b^{2}}-2 a b}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \sqrt {2}}{2 x \sqrt {a b}}\right )}{5 \sqrt {a b}}\)	\(171\)
elliptic	\(\text {Expression too large to display}\)	\(2185\)

input

int((a^5*x^5-b^5)/(a^2*x^3+b^2*x)^(1/2)/(a^5*x^5+b^5),x,method=_RETURNVERB 
OSE)
 

output

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)*arctan(1/2*(x*(a^2*x^2+b^2))^( 
1/2)/x*2^(1/2)/(a*b)^(1/2))
 

3.27.67.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1003 vs. \(2 (177) = 354\).

Time = 0.46 (sec) , antiderivative size = 2091, normalized size of antiderivative = 8.75 \[ \int \frac {-b^5+a^5 x^5}{\sqrt {b^2 x+a^2 x^3} \left (b^5+a^5 x^5\right )} \, dx=\text {Too large to display} \]

input

integrate((a^5*x^5-b^5)/(a^2*x^3+b^2*x)^(1/2)/(a^5*x^5+b^5),x, algorithm=" 
fricas")
 

output

[-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*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))*lo 
g(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*s...
 

3.27.67.6 Sympy [F]

\[ \int \frac {-b^5+a^5 x^5}{\sqrt {b^2 x+a^2 x^3} \left (b^5+a^5 x^5\right )} \, dx=\int \frac {\left (a x - b\right ) \left (a^{4} x^{4} + a^{3} b x^{3} + a^{2} b^{2} x^{2} + a b^{3} x + b^{4}\right )}{\sqrt {x \left (a^{2} x^{2} + b^{2}\right )} \left (a x + b\right ) \left (a^{4} x^{4} - a^{3} b x^{3} + a^{2} b^{2} x^{2} - a b^{3} x + b^{4}\right )}\, dx \]

input

integrate((a**5*x**5-b**5)/(a**2*x**3+b**2*x)**(1/2)/(a**5*x**5+b**5),x)
 

output

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)
 

3.27.67.7 Maxima [F]

\[ \int \frac {-b^5+a^5 x^5}{\sqrt {b^2 x+a^2 x^3} \left (b^5+a^5 x^5\right )} \, dx=\int { \frac {a^{5} x^{5} - b^{5}}{{\left (a^{5} x^{5} + b^{5}\right )} \sqrt {a^{2} x^{3} + b^{2} x}} \,d x } \]

input

integrate((a^5*x^5-b^5)/(a^2*x^3+b^2*x)^(1/2)/(a^5*x^5+b^5),x, algorithm=" 
maxima")
 

output

integrate((a^5*x^5 - b^5)/((a^5*x^5 + b^5)*sqrt(a^2*x^3 + b^2*x)), x)
 

3.27.67.8 Giac [F]

\[ \int \frac {-b^5+a^5 x^5}{\sqrt {b^2 x+a^2 x^3} \left (b^5+a^5 x^5\right )} \, dx=\int { \frac {a^{5} x^{5} - b^{5}}{{\left (a^{5} x^{5} + b^{5}\right )} \sqrt {a^{2} x^{3} + b^{2} x}} \,d x } \]

input

integrate((a^5*x^5-b^5)/(a^2*x^3+b^2*x)^(1/2)/(a^5*x^5+b^5),x, algorithm=" 
giac")
 

output

integrate((a^5*x^5 - b^5)/((a^5*x^5 + b^5)*sqrt(a^2*x^3 + b^2*x)), x)
 

3.27.67.9 Mupad [F(-1)]

Timed out. \[ \int \frac {-b^5+a^5 x^5}{\sqrt {b^2 x+a^2 x^3} \left (b^5+a^5 x^5\right )} \, dx=\text {Hanged} \]

input

int(-(b^5 - a^5*x^5)/((b^5 + a^5*x^5)*(b^2*x + a^2*x^3)^(1/2)),x)
 

output

\text{Hanged}