3.8.100 \(\int \frac {(6+x^4) \sqrt {-2 x+x^4+x^5}}{(-2+x^4) (-2-x^3+x^4)} \, dx\)

Optimal. Leaf size=61 \[ 2 \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^5+x^4-2 x}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^5+x^4-2 x}}{x^4+x^3-2}\right ) \]

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Rubi [F]  time = 7.95, 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 {\left (6+x^4\right ) \sqrt {-2 x+x^4+x^5}}{\left (-2+x^4\right ) \left (-2-x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((6 + x^4)*Sqrt[-2*x + x^4 + x^5])/((-2 + x^4)*(-2 - x^3 + x^4)),x]

[Out]

((-I)*Sqrt[-2*x + x^4 + x^5]*Defer[Subst][Defer[Int][Sqrt[-2 + x^6 + x^8]/(I - x), x], x, Sqrt[x]])/(Sqrt[x]*S
qrt[-2 + x^3 + x^4]) + (I*Sqrt[-2*x + x^4 + x^5]*Defer[Subst][Defer[Int][Sqrt[-2 + x^6 + x^8]/(I*2^(1/8) - x),
x], x, Sqrt[x]])/(2^(3/8)*Sqrt[x]*Sqrt[-2 + x^3 + x^4]) + (Sqrt[-2*x + x^4 + x^5]*Defer[Subst][Defer[Int][Sqr
t[-2 + x^6 + x^8]/(2^(1/8) - x), x], x, Sqrt[x]])/(2^(3/8)*Sqrt[x]*Sqrt[-2 + x^3 + x^4]) - ((1 + I)*Sqrt[-2*x
+ x^4 + x^5]*Defer[Subst][Defer[Int][Sqrt[-2 + x^6 + x^8]/((-1)^(1/4)*2^(1/8) - x), x], x, Sqrt[x]])/(2^(7/8)*
Sqrt[x]*Sqrt[-2 + x^3 + x^4]) - ((1 - I)*Sqrt[-2*x + x^4 + x^5]*Defer[Subst][Defer[Int][Sqrt[-2 + x^6 + x^8]/(
-((-1)^(3/4)*2^(1/8)) - x), x], x, Sqrt[x]])/(2^(7/8)*Sqrt[x]*Sqrt[-2 + x^3 + x^4]) - (I*Sqrt[-2*x + x^4 + x^5
]*Defer[Subst][Defer[Int][Sqrt[-2 + x^6 + x^8]/(I + x), x], x, Sqrt[x]])/(Sqrt[x]*Sqrt[-2 + x^3 + x^4]) + (I*S
qrt[-2*x + x^4 + x^5]*Defer[Subst][Defer[Int][Sqrt[-2 + x^6 + x^8]/(I*2^(1/8) + x), x], x, Sqrt[x]])/(2^(3/8)*
Sqrt[x]*Sqrt[-2 + x^3 + x^4]) + (Sqrt[-2*x + x^4 + x^5]*Defer[Subst][Defer[Int][Sqrt[-2 + x^6 + x^8]/(2^(1/8)
+ x), x], x, Sqrt[x]])/(2^(3/8)*Sqrt[x]*Sqrt[-2 + x^3 + x^4]) - ((1 + I)*Sqrt[-2*x + x^4 + x^5]*Defer[Subst][D
efer[Int][Sqrt[-2 + x^6 + x^8]/((-1)^(1/4)*2^(1/8) + x), x], x, Sqrt[x]])/(2^(7/8)*Sqrt[x]*Sqrt[-2 + x^3 + x^4
]) - ((1 - I)*Sqrt[-2*x + x^4 + x^5]*Defer[Subst][Defer[Int][Sqrt[-2 + x^6 + x^8]/(-((-1)^(3/4)*2^(1/8)) + x),
x], x, Sqrt[x]])/(2^(7/8)*Sqrt[x]*Sqrt[-2 + x^3 + x^4]) - (4*Sqrt[-2*x + x^4 + x^5]*Defer[Subst][Defer[Int][S
qrt[-2 + x^6 + x^8]/(-2 + 2*x^2 - 2*x^4 + x^6), x], x, Sqrt[x]])/(Sqrt[x]*Sqrt[-2 + x^3 + x^4]) + (2*Sqrt[-2*x
+ x^4 + x^5]*Defer[Subst][Defer[Int][(x^2*Sqrt[-2 + x^6 + x^8])/(-2 + 2*x^2 - 2*x^4 + x^6), x], x, Sqrt[x]])/
(Sqrt[x]*Sqrt[-2 + x^3 + x^4]) + (2*Sqrt[-2*x + x^4 + x^5]*Defer[Subst][Defer[Int][(x^4*Sqrt[-2 + x^6 + x^8])/
(-2 + 2*x^2 - 2*x^4 + x^6), x], x, Sqrt[x]])/(Sqrt[x]*Sqrt[-2 + x^3 + x^4])

Rubi steps

\begin {align*} \int \frac {\left (6+x^4\right ) \sqrt {-2 x+x^4+x^5}}{\left (-2+x^4\right ) \left (-2-x^3+x^4\right )} \, dx &=\frac {\sqrt {-2 x+x^4+x^5} \int \frac {\sqrt {x} \left (6+x^4\right ) \sqrt {-2+x^3+x^4}}{\left (-2+x^4\right ) \left (-2-x^3+x^4\right )} \, dx}{\sqrt {x} \sqrt {-2+x^3+x^4}}\\ &=\frac {\sqrt {-2 x+x^4+x^5} \int \left (\frac {\sqrt {x} \sqrt {-2+x^3+x^4}}{1+x}+\frac {\sqrt {x} \left (-1+3 x-x^2\right ) \sqrt {-2+x^3+x^4}}{-2+2 x-2 x^2+x^3}-\frac {4 x^{3/2} \sqrt {-2+x^3+x^4}}{-2+x^4}\right ) \, dx}{\sqrt {x} \sqrt {-2+x^3+x^4}}\\ &=\frac {\sqrt {-2 x+x^4+x^5} \int \frac {\sqrt {x} \sqrt {-2+x^3+x^4}}{1+x} \, dx}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\sqrt {-2 x+x^4+x^5} \int \frac {\sqrt {x} \left (-1+3 x-x^2\right ) \sqrt {-2+x^3+x^4}}{-2+2 x-2 x^2+x^3} \, dx}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (4 \sqrt {-2 x+x^4+x^5}\right ) \int \frac {x^{3/2} \sqrt {-2+x^3+x^4}}{-2+x^4} \, dx}{\sqrt {x} \sqrt {-2+x^3+x^4}}\\ &=\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-2+x^6+x^8}}{1+x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-1+3 x^2-x^4\right ) \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (4 \sqrt {-2 x+x^4+x^5}\right ) \int \left (-\frac {x^{3/2} \sqrt {-2+x^3+x^4}}{2 \sqrt {2} \left (\sqrt {2}-x^2\right )}-\frac {x^{3/2} \sqrt {-2+x^3+x^4}}{2 \sqrt {2} \left (\sqrt {2}+x^2\right )}\right ) \, dx}{\sqrt {x} \sqrt {-2+x^3+x^4}}\\ &=\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (\sqrt {-2+x^6+x^8}-\frac {\sqrt {-2+x^6+x^8}}{1+x^2}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (-\sqrt {-2+x^6+x^8}-\frac {\left (2-x^2-x^4\right ) \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (\sqrt {2} \sqrt {-2 x+x^4+x^5}\right ) \int \frac {x^{3/2} \sqrt {-2+x^3+x^4}}{\sqrt {2}-x^2} \, dx}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (\sqrt {2} \sqrt {-2 x+x^4+x^5}\right ) \int \frac {x^{3/2} \sqrt {-2+x^3+x^4}}{\sqrt {2}+x^2} \, dx}{\sqrt {x} \sqrt {-2+x^3+x^4}}\\ &=-\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{1+x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\left (2-x^2-x^4\right ) \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (\sqrt {2} \sqrt {-2 x+x^4+x^5}\right ) \int \left (\frac {i x^{3/2} \sqrt {-2+x^3+x^4}}{2 \sqrt [4]{2} \left (i \sqrt [4]{2}-x\right )}+\frac {i x^{3/2} \sqrt {-2+x^3+x^4}}{2 \sqrt [4]{2} \left (i \sqrt [4]{2}+x\right )}\right ) \, dx}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (\sqrt {2} \sqrt {-2 x+x^4+x^5}\right ) \int \left (\frac {x^{3/2} \sqrt {-2+x^3+x^4}}{2 \sqrt [4]{2} \left (\sqrt [4]{2}-x\right )}+\frac {x^{3/2} \sqrt {-2+x^3+x^4}}{2 \sqrt [4]{2} \left (\sqrt [4]{2}+x\right )}\right ) \, dx}{\sqrt {x} \sqrt {-2+x^3+x^4}}\\ &=-\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {i \sqrt {-2+x^6+x^8}}{2 (i-x)}+\frac {i \sqrt {-2+x^6+x^8}}{2 (i+x)}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {2 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6}-\frac {x^2 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6}-\frac {x^4 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \int \frac {x^{3/2} \sqrt {-2+x^3+x^4}}{i \sqrt [4]{2}-x} \, dx}{2^{3/4} \sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \int \frac {x^{3/2} \sqrt {-2+x^3+x^4}}{i \sqrt [4]{2}+x} \, dx}{2^{3/4} \sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\sqrt {-2 x+x^4+x^5} \int \frac {x^{3/2} \sqrt {-2+x^3+x^4}}{\sqrt [4]{2}-x} \, dx}{2^{3/4} \sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\sqrt {-2 x+x^4+x^5} \int \frac {x^{3/2} \sqrt {-2+x^3+x^4}}{\sqrt [4]{2}+x} \, dx}{2^{3/4} \sqrt {x} \sqrt {-2+x^3+x^4}}\\ &=-\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i-x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i+x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (4 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (i \sqrt [4]{2} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-2+x^6+x^8}}{i \sqrt [4]{2}-x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (i \sqrt [4]{2} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-2+x^6+x^8}}{i \sqrt [4]{2}+x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (\sqrt [4]{2} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-2+x^6+x^8}}{\sqrt [4]{2}-x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (\sqrt [4]{2} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-2+x^6+x^8}}{\sqrt [4]{2}+x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}\\ &=-\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i-x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i+x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (4 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (i \sqrt [4]{2} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (-i \sqrt [4]{2} \sqrt {-2+x^6+x^8}-x^2 \sqrt {-2+x^6+x^8}-\frac {\sqrt {2} \sqrt {-2+x^6+x^8}}{i \sqrt [4]{2}-x^2}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (i \sqrt [4]{2} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (-i \sqrt [4]{2} \sqrt {-2+x^6+x^8}+x^2 \sqrt {-2+x^6+x^8}-\frac {\sqrt {2} \sqrt {-2+x^6+x^8}}{i \sqrt [4]{2}+x^2}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (\sqrt [4]{2} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (-\sqrt [4]{2} \sqrt {-2+x^6+x^8}-x^2 \sqrt {-2+x^6+x^8}+\frac {\sqrt {2} \sqrt {-2+x^6+x^8}}{\sqrt [4]{2}-x^2}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (\sqrt [4]{2} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (-\sqrt [4]{2} \sqrt {-2+x^6+x^8}+x^2 \sqrt {-2+x^6+x^8}+\frac {\sqrt {2} \sqrt {-2+x^6+x^8}}{\sqrt [4]{2}+x^2}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}\\ &=-\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i-x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i+x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (4 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (i 2^{3/4} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i \sqrt [4]{2}-x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (i 2^{3/4} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i \sqrt [4]{2}+x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2^{3/4} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{\sqrt [4]{2}-x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2^{3/4} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{\sqrt [4]{2}+x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}\\ &=-\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i-x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i+x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (4 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (i 2^{3/4} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (-\frac {(-1)^{3/4} \sqrt {-2+x^6+x^8}}{2 \sqrt [8]{2} \left (\sqrt [4]{-1} \sqrt [8]{2}-x\right )}-\frac {(-1)^{3/4} \sqrt {-2+x^6+x^8}}{2 \sqrt [8]{2} \left (\sqrt [4]{-1} \sqrt [8]{2}+x\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (i 2^{3/4} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (-\frac {\sqrt [4]{-1} \sqrt {-2+x^6+x^8}}{2 \sqrt [8]{2} \left (-(-1)^{3/4} \sqrt [8]{2}-x\right )}-\frac {\sqrt [4]{-1} \sqrt {-2+x^6+x^8}}{2 \sqrt [8]{2} \left (-(-1)^{3/4} \sqrt [8]{2}+x\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2^{3/4} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {i \sqrt {-2+x^6+x^8}}{2 \sqrt [8]{2} \left (i \sqrt [8]{2}-x\right )}+\frac {i \sqrt {-2+x^6+x^8}}{2 \sqrt [8]{2} \left (i \sqrt [8]{2}+x\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2^{3/4} \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt {-2+x^6+x^8}}{2 \sqrt [8]{2} \left (\sqrt [8]{2}-x\right )}+\frac {\sqrt {-2+x^6+x^8}}{2 \sqrt [8]{2} \left (\sqrt [8]{2}+x\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}\\ &=-\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i-x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i+x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (2 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left (4 \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{-2+2 x^2-2 x^4+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left ((1-i) \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{-(-1)^{3/4} \sqrt [8]{2}-x} \, dx,x,\sqrt {x}\right )}{2^{7/8} \sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left ((1-i) \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{-(-1)^{3/4} \sqrt [8]{2}+x} \, dx,x,\sqrt {x}\right )}{2^{7/8} \sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left ((1+i) \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{\sqrt [4]{-1} \sqrt [8]{2}-x} \, dx,x,\sqrt {x}\right )}{2^{7/8} \sqrt {x} \sqrt {-2+x^3+x^4}}-\frac {\left ((1+i) \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{\sqrt [4]{-1} \sqrt [8]{2}+x} \, dx,x,\sqrt {x}\right )}{2^{7/8} \sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i \sqrt [8]{2}-x} \, dx,x,\sqrt {x}\right )}{2^{3/8} \sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\left (i \sqrt {-2 x+x^4+x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{i \sqrt [8]{2}+x} \, dx,x,\sqrt {x}\right )}{2^{3/8} \sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\sqrt {-2 x+x^4+x^5} \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{\sqrt [8]{2}-x} \, dx,x,\sqrt {x}\right )}{2^{3/8} \sqrt {x} \sqrt {-2+x^3+x^4}}+\frac {\sqrt {-2 x+x^4+x^5} \operatorname {Subst}\left (\int \frac {\sqrt {-2+x^6+x^8}}{\sqrt [8]{2}+x} \, dx,x,\sqrt {x}\right )}{2^{3/8} \sqrt {x} \sqrt {-2+x^3+x^4}}\\ \end {align*}

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Mathematica [F]  time = 1.94, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (6+x^4\right ) \sqrt {-2 x+x^4+x^5}}{\left (-2+x^4\right ) \left (-2-x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((6 + x^4)*Sqrt[-2*x + x^4 + x^5])/((-2 + x^4)*(-2 - x^3 + x^4)),x]

[Out]

Integrate[((6 + x^4)*Sqrt[-2*x + x^4 + x^5])/((-2 + x^4)*(-2 - x^3 + x^4)), x]

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IntegrateAlgebraic [A]  time = 0.17, size = 61, normalized size = 1.00 \begin {gather*} 2 \tanh ^{-1}\left (\frac {x^2}{\sqrt {-2 x+x^4+x^5}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {-2 x+x^4+x^5}}{-2+x^3+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((6 + x^4)*Sqrt[-2*x + x^4 + x^5])/((-2 + x^4)*(-2 - x^3 + x^4)),x]

[Out]

2*ArcTanh[x^2/Sqrt[-2*x + x^4 + x^5]] - 2*Sqrt[2]*ArcTanh[(Sqrt[2]*x*Sqrt[-2*x + x^4 + x^5])/(-2 + x^3 + x^4)]

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fricas [B]  time = 0.50, size = 121, normalized size = 1.98 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (\frac {x^{8} + 14 \, x^{7} + 17 \, x^{6} - 4 \, x^{4} - 28 \, x^{3} - 4 \, \sqrt {2} {\left (x^{5} + 3 \, x^{4} - 2 \, x\right )} \sqrt {x^{5} + x^{4} - 2 \, x} + 4}{x^{8} - 2 \, x^{7} + x^{6} - 4 \, x^{4} + 4 \, x^{3} + 4}\right ) + \log \left (\frac {x^{4} + 2 \, x^{3} + 2 \, \sqrt {x^{5} + x^{4} - 2 \, x} x - 2}{x^{4} - 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+6)*(x^5+x^4-2*x)^(1/2)/(x^4-2)/(x^4-x^3-2),x, algorithm="fricas")

[Out]

1/2*sqrt(2)*log((x^8 + 14*x^7 + 17*x^6 - 4*x^4 - 28*x^3 - 4*sqrt(2)*(x^5 + 3*x^4 - 2*x)*sqrt(x^5 + x^4 - 2*x)
+ 4)/(x^8 - 2*x^7 + x^6 - 4*x^4 + 4*x^3 + 4)) + log((x^4 + 2*x^3 + 2*sqrt(x^5 + x^4 - 2*x)*x - 2)/(x^4 - 2))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{5} + x^{4} - 2 \, x} {\left (x^{4} + 6\right )}}{{\left (x^{4} - x^{3} - 2\right )} {\left (x^{4} - 2\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+6)*(x^5+x^4-2*x)^(1/2)/(x^4-2)/(x^4-x^3-2),x, algorithm="giac")

[Out]

integrate(sqrt(x^5 + x^4 - 2*x)*(x^4 + 6)/((x^4 - x^3 - 2)*(x^4 - 2)), x)

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maple [C]  time = 0.64, size = 111, normalized size = 1.82

method	result	size
trager	\(\ln \left (\frac {x^{4}+2 x^{3}+2 x \sqrt {x^{5}+x^{4}-2 x}-2}{x^{4}-2}\right )+\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{4}-3 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{3}+4 x \sqrt {x^{5}+x^{4}-2 x}+2 \RootOf \left (\textit {\_Z}^{2}-2\right )}{\left (1+x \right ) \left (x^{3}-2 x^{2}+2 x -2\right )}\right )\)	\(111\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+6)*(x^5+x^4-2*x)^(1/2)/(x^4-2)/(x^4-x^3-2),x,method=_RETURNVERBOSE)

[Out]

ln((x^4+2*x^3+2*x*(x^5+x^4-2*x)^(1/2)-2)/(x^4-2))+RootOf(_Z^2-2)*ln((-RootOf(_Z^2-2)*x^4-3*RootOf(_Z^2-2)*x^3+
4*x*(x^5+x^4-2*x)^(1/2)+2*RootOf(_Z^2-2))/(1+x)/(x^3-2*x^2+2*x-2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{5} + x^{4} - 2 \, x} {\left (x^{4} + 6\right )}}{{\left (x^{4} - x^{3} - 2\right )} {\left (x^{4} - 2\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+6)*(x^5+x^4-2*x)^(1/2)/(x^4-2)/(x^4-x^3-2),x, algorithm="maxima")

[Out]

integrate(sqrt(x^5 + x^4 - 2*x)*(x^4 + 6)/((x^4 - x^3 - 2)*(x^4 - 2)), x)

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mupad [B]  time = 3.44, size = 81, normalized size = 1.33 \begin {gather*} \ln \left (\frac {2\,x\,\sqrt {x\,\left (x^4+x^3-2\right )}+2\,x^3+x^4-2}{x^4-2}\right )+\sqrt {2}\,\ln \left (\frac {3\,x^3+x^4-2\,\sqrt {2}\,x\,\sqrt {x\,\left (x^4+x^3-2\right )}-2}{-x^4+x^3+2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^4 + 6)*(x^4 - 2*x + x^5)^(1/2))/((x^4 - 2)*(x^3 - x^4 + 2)),x)

[Out]

log((2*x*(x*(x^3 + x^4 - 2))^(1/2) + 2*x^3 + x^4 - 2)/(x^4 - 2)) + 2^(1/2)*log((3*x^3 + x^4 - 2*2^(1/2)*x*(x*(
x^3 + x^4 - 2))^(1/2) - 2)/(x^3 - x^4 + 2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+6)*(x**5+x**4-2*x)**(1/2)/(x**4-2)/(x**4-x**3-2),x)

[Out]

Timed out

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