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

Optimal result

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

[Out]

-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)

Rubi [F]

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

[In]

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

[Out]

-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]])/(Sqrt
[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3]) + (4*Sqrt[x]*(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*x + a^2*x^3]) - (16*b^4*Sqrt[x]*Sqrt[b^2 + a^2*
x^2]*Defer[Subst][Defer[Int][1/(Sqrt[b^2 + a^2*x^4]*(b^4 - a*b^3*x^2 + a^2*b^2*x^4 - a^3*b*x^6 + a^4*x^8)), x]
, x, Sqrt[x]])/(5*Sqrt[b^2*x + a^2*x^3]) + (12*a*b^3*Sqrt[x]*Sqrt[b^2 + a^2*x^2]*Defer[Subst][Defer[Int][x^2/(
Sqrt[b^2 + a^2*x^4]*(b^4 - a*b^3*x^2 + a^2*b^2*x^4 - a^3*b*x^6 + a^4*x^8)), x], x, Sqrt[x]])/(5*Sqrt[b^2*x + a
^2*x^3]) - (8*a^2*b^2*Sqrt[x]*Sqrt[b^2 + a^2*x^2]*Defer[Subst][Defer[Int][x^4/(Sqrt[b^2 + a^2*x^4]*(b^4 - a*b^
3*x^2 + a^2*b^2*x^4 - a^3*b*x^6 + a^4*x^8)), x], x, Sqrt[x]])/(5*Sqrt[b^2*x + a^2*x^3]) + (4*a^3*b*Sqrt[x]*Sqr
t[b^2 + a^2*x^2]*Defer[Subst][Defer[Int][x^6/(Sqrt[b^2 + a^2*x^4]*(b^4 - a*b^3*x^2 + a^2*b^2*x^4 - a^3*b*x^6 +
a^4*x^8)), x], x, Sqrt[x]])/(5*Sqrt[b^2*x + a^2*x^3])

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \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 {b^2 x+a^2 x^3}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {-b^5+a^5 x^{10}}{\sqrt {b^2+a^2 x^4} \left (b^5+a^5 x^{10}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt {b^2+a^2 x^4}}-\frac {2 b^5}{\sqrt {b^2+a^2 x^4} \left (b^5+a^5 x^{10}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 b^5 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4} \left (b^5+a^5 x^{10}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}} \\ & = \frac {\sqrt {x} (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 )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 b^5 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{5 b^4 \left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}}+\frac {4 b^3-3 a b^2 x^2+2 a^2 b x^4-a^3 x^6}{5 b^4 \sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}} \\ & = \frac {\sqrt {x} (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 )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {4 b^3-3 a b^2 x^2+2 a^2 b x^4-a^3 x^6}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}} \\ & = \frac {\sqrt {x} (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 )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {b-a x^2}{\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {4 b^3}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )}-\frac {3 a b^2 x^2}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )}+\frac {2 a^2 b x^4}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )}-\frac {a^3 x^6}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}} \\ & = \frac {4 \sqrt {x} (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 x+a^2 x^3}}-\frac {\left (2 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{b+2 a b^2 x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{5 \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 a^3 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {x^6}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (8 a^2 b^2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}+\frac {\left (12 a b^3 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (16 b^4 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}} \\ & = -\frac {\sqrt {2} \sqrt {x} \sqrt {b^2+a^2 x^2} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{5 \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {4 \sqrt {x} (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 x+a^2 x^3}}+\frac {\left (4 a^3 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {x^6}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (8 a^2 b^2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}+\frac {\left (12 a b^3 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (16 b^4 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4} \left (b^4-a b^3 x^2+a^2 b^2 x^4-a^3 b x^6+a^4 x^8\right )} \, dx,x,\sqrt {x}\right )}{5 \sqrt {b^2 x+a^2 x^3}} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

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

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	\(\text {Expression too large to display}\)	\(2185\)

[In]

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

[Out]

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))

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

[In]

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")

[Out]

[-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))*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*sqr
t(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)) + 1/20*sqrt(2)*sqrt(-1/(a*b))*log((a^4*x^4 - 12*a^3*b*x^3 + 6*a^2*
b^2*x^2 - 12*a*b^3*x + b^4 + 4*sqrt(2)*(a^3*b*x^2 - 2*a^2*b^2*x + a*b^3)*sqrt(a^2*x^3 + b^2*x)*sqrt(-1/(a*b)))
/(a^4*x^4 + 4*a^3*b*x^3 + 6*a^2*b^2*x^2 + 4*a*b^3*x + b^4)), -1/10*sqrt(2)*sqrt(1/(a*b))*arctan(2*sqrt(2)*sqrt
(a^2*x^3 + b^2*x)*a*b*sqrt(1/(a*b))/(a^2*x^2 - 2*a*b*x + b^2)) - 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*s
qrt(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))*l
og(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
qrt(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))]

Sympy [F]

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

[In]

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

[Out]

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)

Maxima [F]

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

[In]

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")

[Out]

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

Giac [F]

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

[In]

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")

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}