3.4.68 \(\int \frac {-1+3 x^4}{(1-a x+x^4) \sqrt {x+x^5}} \, dx\)

Optimal. Leaf size=30 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x^5+x}}{x^4+1}\right )}{\sqrt {a}} \]

________________________________________________________________________________________

Rubi [F]  time = 1.22, 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 {-1+3 x^4}{\left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1 + 3*x^4)/((1 - a*x + x^4)*Sqrt[x + x^5]),x]

[Out]

(3*x^2*Sqrt[(1 + x)^2/x]*Sqrt[-((1 + x^4)/x^2)]*EllipticF[ArcSin[Sqrt[-((Sqrt[2] - 2*x + Sqrt[2]*x^2)/x)]/2],
-2*(1 - Sqrt[2])])/(Sqrt[2 + Sqrt[2]]*(1 + x)*Sqrt[x + x^5]) - (3*Sqrt[-((1 - x)^2/x)]*x^2*Sqrt[-((1 + x^4)/x^
2)]*EllipticF[ArcSin[Sqrt[(Sqrt[2] + 2*x + Sqrt[2]*x^2)/x]/2], -2*(1 - Sqrt[2])])/(Sqrt[2 + Sqrt[2]]*(1 - x)*S
qrt[x + x^5]) + (8*Sqrt[x]*Sqrt[1 + x^4]*Defer[Subst][Defer[Int][1/((-1 + a*x^2 - x^8)*Sqrt[1 + x^8]), x], x,
Sqrt[x]])/Sqrt[x + x^5] - (6*a*Sqrt[x]*Sqrt[1 + x^4]*Defer[Subst][Defer[Int][x^2/((-1 + a*x^2 - x^8)*Sqrt[1 +
x^8]), x], x, Sqrt[x]])/Sqrt[x + x^5]

Rubi steps

\begin {align*} \int \frac {-1+3 x^4}{\left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx &=\frac {\left (\sqrt {x} \sqrt {1+x^4}\right ) \int \frac {-1+3 x^4}{\sqrt {x} \sqrt {1+x^4} \left (1-a x+x^4\right )} \, dx}{\sqrt {x+x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \frac {-1+3 x^8}{\sqrt {1+x^8} \left (1-a x^2+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {3}{\sqrt {1+x^8}}-\frac {4-3 a x^2}{\sqrt {1+x^8} \left (1-a x^2+x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}\\ &=-\frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \frac {4-3 a x^2}{\sqrt {1+x^8} \left (1-a x^2+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}+\frac {\left (6 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}\\ &=-\frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \left (-\frac {4}{\left (-1+a x^2-x^8\right ) \sqrt {1+x^8}}+\frac {3 a x^2}{\left (-1+a x^2-x^8\right ) \sqrt {1+x^8}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}+\frac {\left (3 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}+\frac {\left (3 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{\sqrt {1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}\\ &=\frac {3 x^2 \sqrt {\frac {(1+x)^2}{x}} \sqrt {-\frac {1+x^4}{x^2}} F\left (\sin ^{-1}\left (\frac {1}{2} \sqrt {-\frac {\sqrt {2}-2 x+\sqrt {2} x^2}{x}}\right )|-2 \left (1-\sqrt {2}\right )\right )}{\sqrt {2+\sqrt {2}} (1+x) \sqrt {x+x^5}}-\frac {3 \sqrt {-\frac {(1-x)^2}{x}} x^2 \sqrt {-\frac {1+x^4}{x^2}} F\left (\sin ^{-1}\left (\frac {1}{2} \sqrt {\frac {\sqrt {2}+2 x+\sqrt {2} x^2}{x}}\right )|-2 \left (1-\sqrt {2}\right )\right )}{\sqrt {2+\sqrt {2}} (1-x) \sqrt {x+x^5}}+\frac {\left (8 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+a x^2-x^8\right ) \sqrt {1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}-\frac {\left (6 a \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+a x^2-x^8\right ) \sqrt {1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 0.47, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+3 x^4}{\left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-1 + 3*x^4)/((1 - a*x + x^4)*Sqrt[x + x^5]),x]

[Out]

Integrate[(-1 + 3*x^4)/((1 - a*x + x^4)*Sqrt[x + x^5]), x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.51, size = 30, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x+x^5}}{1+x^4}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + 3*x^4)/((1 - a*x + x^4)*Sqrt[x + x^5]),x]

[Out]

(-2*ArcTanh[(Sqrt[a]*Sqrt[x + x^5])/(1 + x^4)])/Sqrt[a]

________________________________________________________________________________________

fricas [A]  time = 0.54, size = 127, normalized size = 4.23 \begin {gather*} \left [\frac {\log \left (\frac {x^{8} + 6 \, a x^{5} + a^{2} x^{2} + 2 \, x^{4} - 4 \, \sqrt {x^{5} + x} {\left (x^{4} + a x + 1\right )} \sqrt {a} + 6 \, a x + 1}{x^{8} - 2 \, a x^{5} + a^{2} x^{2} + 2 \, x^{4} - 2 \, a x + 1}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {x^{5} + x} {\left (x^{4} + a x + 1\right )} \sqrt {-a}}{2 \, {\left (a x^{5} + a x\right )}}\right )}{a}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*log((x^8 + 6*a*x^5 + a^2*x^2 + 2*x^4 - 4*sqrt(x^5 + x)*(x^4 + a*x + 1)*sqrt(a) + 6*a*x + 1)/(x^8 - 2*a*x^
5 + a^2*x^2 + 2*x^4 - 2*a*x + 1))/sqrt(a), sqrt(-a)*arctan(1/2*sqrt(x^5 + x)*(x^4 + a*x + 1)*sqrt(-a)/(a*x^5 +
a*x))/a]

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, x^{4} - 1}{\sqrt {x^{5} + x} {\left (x^{4} - a x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((3*x^4 - 1)/(sqrt(x^5 + x)*(x^4 - a*x + 1)), x)

________________________________________________________________________________________

maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {3 x^{4}-1}{\left (x^{4}-a x +1\right ) \sqrt {x^{5}+x}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, x^{4} - 1}{\sqrt {x^{5} + x} {\left (x^{4} - a x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((3*x^4 - 1)/(sqrt(x^5 + x)*(x^4 - a*x + 1)), x)

________________________________________________________________________________________

mupad [B]  time = 0.42, size = 37, normalized size = 1.23 \begin {gather*} \frac {\ln \left (\frac {a\,x-2\,\sqrt {a}\,\sqrt {x^5+x}+x^4+1}{x^4-a\,x+1}\right )}{\sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((a*x - 2*a^(1/2)*(x + x^5)^(1/2) + x^4 + 1)/(x^4 - a*x + 1))/a^(1/2)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________