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

Optimal. Leaf size=239 \[ -\frac {2 \sqrt {2 \left (\sqrt {5}-1\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}{a^2 x^2+b^2}\right )}{5 \sqrt {a} \sqrt {b}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}{a^2 x^2+b^2}\right )}{5 \sqrt {a} \sqrt {b}}-\frac {2 \sqrt {2 \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}{a^2 x^2+b^2}\right )}{5 \sqrt {a} \sqrt {b}} \]

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Rubi [F]  time = 3.38, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[b^2 + a^2*x^2]])/(Sqr
t[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*ArcTa
n[(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]*Sq
rt[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*} \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 {\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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}} F\left (2 \tan ^{-1}\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 ) \operatorname {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}} F\left (2 \tan ^{-1}\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 ) \operatorname {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 ) \operatorname {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}} F\left (2 \tan ^{-1}\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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}} F\left (2 \tan ^{-1}\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \tanh ^{-1}\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}} F\left (2 \tan ^{-1}\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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*}

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Mathematica [C]  time = 1.92, size = 574, normalized size = 2.40 \begin {gather*} \frac {2 \sqrt [10]{-1} x^{3/2} \sqrt {\frac {b^2}{a^2 x^2}+1} \left (\left (1+\sqrt [5]{-1}\right )^2 \left (1-3 \sqrt [5]{-1}+(-1)^{2/5}\right ) F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+2 \left ((-1)^{2/5} \Pi \left (i;\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+\left (1-\sqrt [5]{-1}+2 (-1)^{2/5}-(-1)^{3/5}+(-1)^{4/5}\right ) \Pi \left (\sqrt [10]{-1};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )-(-1)^{4/5} \Pi \left (-(-1)^{3/10};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+(-1)^{3/5} \Pi \left (-(-1)^{3/10};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+\sqrt [5]{-1} \Pi \left (-(-1)^{3/10};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (-(-1)^{3/10};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+(-1)^{2/5} \Pi \left (-(-1)^{7/10};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )-(-1)^{4/5} \Pi \left ((-1)^{9/10};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+(-1)^{3/5} \Pi \left ((-1)^{9/10};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )+\sqrt [5]{-1} \Pi \left ((-1)^{9/10};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left ((-1)^{9/10};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )\right )\right )}{\left (\sqrt [5]{-1}-1\right )^2 \left (1+\sqrt [5]{-1}\right )^4 \left (1+(-1)^{2/5}\right ) \left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \sqrt {\frac {i b}{a}} \sqrt {x \left (a^2 x^2+b^2\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*(-1)^(1/10)*Sqrt[1 + b^2/(a^2*x^2)]*x^(3/2)*((1 + (-1)^(1/5))^2*(1 - 3*(-1)^(1/5) + (-1)^(2/5))*EllipticF[I
*ArcSinh[Sqrt[(I*b)/a]/Sqrt[x]], -1] + 2*((-1)^(2/5)*EllipticPi[I, I*ArcSinh[Sqrt[(I*b)/a]/Sqrt[x]], -1] + (1
- (-1)^(1/5) + 2*(-1)^(2/5) - (-1)^(3/5) + (-1)^(4/5))*EllipticPi[(-1)^(1/10), I*ArcSinh[Sqrt[(I*b)/a]/Sqrt[x]
], -1] - EllipticPi[-(-1)^(3/10), I*ArcSinh[Sqrt[(I*b)/a]/Sqrt[x]], -1] + (-1)^(1/5)*EllipticPi[-(-1)^(3/10),
I*ArcSinh[Sqrt[(I*b)/a]/Sqrt[x]], -1] + (-1)^(3/5)*EllipticPi[-(-1)^(3/10), I*ArcSinh[Sqrt[(I*b)/a]/Sqrt[x]],
-1] - (-1)^(4/5)*EllipticPi[-(-1)^(3/10), I*ArcSinh[Sqrt[(I*b)/a]/Sqrt[x]], -1] + (-1)^(2/5)*EllipticPi[-(-1)^
(7/10), I*ArcSinh[Sqrt[(I*b)/a]/Sqrt[x]], -1] - EllipticPi[(-1)^(9/10), I*ArcSinh[Sqrt[(I*b)/a]/Sqrt[x]], -1]
+ (-1)^(1/5)*EllipticPi[(-1)^(9/10), I*ArcSinh[Sqrt[(I*b)/a]/Sqrt[x]], -1] + (-1)^(3/5)*EllipticPi[(-1)^(9/10)
, I*ArcSinh[Sqrt[(I*b)/a]/Sqrt[x]], -1] - (-1)^(4/5)*EllipticPi[(-1)^(9/10), I*ArcSinh[Sqrt[(I*b)/a]/Sqrt[x]],
-1])))/((-1 + (-1)^(1/5))^2*(1 + (-1)^(1/5))^4*(1 + (-1)^(2/5))*(1 - (-1)^(1/5) + (-1)^(2/5))*Sqrt[(I*b)/a]*S
qrt[x*(b^2 + a^2*x^2)])

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IntegrateAlgebraic [A]  time = 1.20, size = 239, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {2 \left (-1+\sqrt {5}\right )} \tan ^{-1}\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} \tanh ^{-1}\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 )} \tanh ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2*(-1 + Sqrt[5])]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])/(b^2 + a^2*x^2
)])/(5*Sqrt[a]*Sqrt[b]) - (Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])/(b^2 + a^2*x^2)])/(
5*Sqrt[a]*Sqrt[b]) - (2*Sqrt[2*(1 + Sqrt[5])]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2
*x^3])/(b^2 + a^2*x^2)])/(5*Sqrt[a]*Sqrt[b])

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fricas [B]  time = 1.03, size = 2075, normalized size = 8.68

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[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 + s
qrt(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*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)) + 1/20*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*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*sqrt(1/5)*a*b*sqrt(1/(a^2*b^2)) - 1)/(a*b)) - 10*sqr
t(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))]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 0.28, size = 482, normalized size = 2.02

method	result	size
default	\(\frac {i b \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {2}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \EllipticF \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}+b^{2} x}}+\frac {2 i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{4} \textit {\_Z}^{4}+a^{3} b \,\textit {\_Z}^{3}+a^{2} b^{2} \textit {\_Z}^{2}+a \,b^{3} \textit {\_Z} +b^{4}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3} a^{3}-2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b -3 \underline {\hspace {1.25 ex}}\alpha a \,b^{2}-4 b^{3}\right ) \underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-i \underline {\hspace {1.25 ex}}\alpha a b +\underline {\hspace {1.25 ex}}\alpha a b -i b^{2}\right ) \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \EllipticPi \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, -\frac {\underline {\hspace {1.25 ex}}\alpha \left (i \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+i \underline {\hspace {1.25 ex}}\alpha a b +\underline {\hspace {1.25 ex}}\alpha a b +b^{2}\right ) a}{b^{3}}, \frac {\sqrt {2}}{2}\right )}{\left (4 \underline {\hspace {1.25 ex}}\alpha ^{3} a^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b +2 \underline {\hspace {1.25 ex}}\alpha a \,b^{2}+b^{3}\right ) \sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}\right )}{5 b^{2}}+\frac {2 i b^{2} \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {2}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \EllipticPi \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, -\frac {i b}{a \left (-\frac {i b}{a}-\frac {b}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{5 a^{2} \sqrt {a^{2} x^{3}+b^{2} x}\, \left (-\frac {i b}{a}-\frac {b}{a}\right )}\)	\(482\)
elliptic	\(\text {Expression too large to display}\)	\(2129\)

Verification of antiderivative is not currently implemented for this CAS.

[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]

I*b/a*(-I*(x+I*b/a)/b*a)^(1/2)*2^(1/2)*(I*(x-I*b/a)/b*a)^(1/2)*(I*x/b*a)^(1/2)/(a^2*x^3+b^2*x)^(1/2)*EllipticF
((-I*(x+I*b/a)/b*a)^(1/2),1/2*2^(1/2))+2/5*I/b^2*2^(1/2)*sum((-_alpha^3*a^3-2*_alpha^2*a^2*b-3*_alpha*a*b^2-4*
b^3)/(4*_alpha^3*a^3+3*_alpha^2*a^2*b+2*_alpha*a*b^2+b^3)*_alpha*(-I*_alpha*a*b+_alpha^2*a^2-I*b^2+_alpha*a*b)
*(-I*(x+I*b/a)/b*a)^(1/2)*(I*(x-I*b/a)/b*a)^(1/2)*(I*x/b*a)^(1/2)/(x*(a^2*x^2+b^2))^(1/2)*EllipticPi((-I*(x+I*
b/a)/b*a)^(1/2),-_alpha*(I*_alpha^2*a^2+I*_alpha*a*b+_alpha*a*b+b^2)*a/b^3,1/2*2^(1/2)),_alpha=RootOf(_Z^4*a^4
+_Z^3*a^3*b+_Z^2*a^2*b^2+_Z*a*b^3+b^4))+2/5*I*b^2/a^2*(-I*(x+I*b/a)/b*a)^(1/2)*2^(1/2)*(I*(x-I*b/a)/b*a)^(1/2)
*(I*x/b*a)^(1/2)/(a^2*x^3+b^2*x)^(1/2)/(-I*b/a-b/a)*EllipticPi((-I*(x+I*b/a)/b*a)^(1/2),-I*b/a/(-I*b/a-b/a),1/
2*2^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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}

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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