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

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

________________________________________________________________________________________

Rubi [F]  time = 1.12, 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 (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [F]  time = 0.44, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+3 x^4}{\left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 1.04, size = 30, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {x+x^5}}{\sqrt {a} \left (1+x^4\right )}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.53, size = 137, normalized size = 4.57 \begin {gather*} \left [\frac {\log \left (\frac {a^{2} x^{8} + 2 \, a^{2} x^{4} + 6 \, a x^{5} - 4 \, {\left (a x^{4} + a + x\right )} \sqrt {x^{5} + x} \sqrt {a} + a^{2} + 6 \, a x + x^{2}}{a^{2} x^{8} + 2 \, a^{2} x^{4} - 2 \, a x^{5} + a^{2} - 2 \, a x + x^{2}}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {{\left (a x^{4} + a + x\right )} \sqrt {x^{5} + x} \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)/(a*x^4+a-x)/(x^5+x)^(1/2),x, algorithm="fricas")

[Out]

[1/2*log((a^2*x^8 + 2*a^2*x^4 + 6*a*x^5 - 4*(a*x^4 + a + x)*sqrt(x^5 + x)*sqrt(a) + a^2 + 6*a*x + x^2)/(a^2*x^
8 + 2*a^2*x^4 - 2*a*x^5 + a^2 - 2*a*x + x^2))/sqrt(a), sqrt(-a)*arctan(1/2*(a*x^4 + a + x)*sqrt(x^5 + x)*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}{{\left (a x^{4} + a - x\right )} \sqrt {x^{5} + x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((3*x^4-1)/(a*x^4+a-x)/(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}{{\left (a x^{4} + a - x\right )} \sqrt {x^{5} + x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.42, size = 38, normalized size = 1.27 \begin {gather*} \frac {\ln \left (\frac {a+x-2\,\sqrt {a}\,\sqrt {x^5+x}+a\,x^4}{a\,x^4-x+a}\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)*(a - x + a*x^4)),x)

[Out]

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

[Out]

Timed out

________________________________________________________________________________________