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

Optimal. Leaf size=26 \[ -\frac {2 \tanh ^{-1}\left (\frac {x}{\sqrt {a} \sqrt {x^5-x}}\right )}{\sqrt {a}} \]

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Rubi [F]  time = 1.34, 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 + 3*I)*x^2*Sqrt[-(((-1)^(3/4)*(1 + (-1)^(1/4)*x)^2)/x)]*Sqrt[(I*(1 - x^4))/x^2]*EllipticF[ArcSin[Sqrt[((-1
)^(3/4)*(Sqrt[2] - 2*(-1)^(1/4)*x + I*Sqrt[2]*x^2))/x]/2], -2*(1 - Sqrt[2])])/(Sqrt[2*(2 + Sqrt[2])]*a*(1 + (-
1)^(1/4)*x)*Sqrt[-x + x^5]) - ((3 + 3*I)*x^2*Sqrt[((-1)^(3/4)*(1 - (-1)^(1/4)*x)^2)/x]*Sqrt[(I*(1 - x^4))/x^2]
*EllipticF[ArcSin[Sqrt[-(((-1)^(3/4)*(Sqrt[2] + 2*(-1)^(1/4)*x + I*Sqrt[2]*x^2))/x)]/2], -2*(1 - Sqrt[2])])/(S
qrt[2*(2 + Sqrt[2])]*a*(1 - (-1)^(1/4)*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-\sqrt [4]{-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+\sqrt [4]{-1} x^2}{\sqrt {-1+x^8}} \, dx,x,\sqrt {x}\right )}{a \sqrt {-x+x^5}}\\ &=\frac {(3+3 i) x^2 \sqrt {-\frac {(-1)^{3/4} \left (1+\sqrt [4]{-1} x\right )^2}{x}} \sqrt {\frac {i \left (1-x^4\right )}{x^2}} F\left (\sin ^{-1}\left (\frac {1}{2} \sqrt {\frac {(-1)^{3/4} \left (\sqrt {2}-2 \sqrt [4]{-1} x+i \sqrt {2} x^2\right )}{x}}\right )|-2 \left (1-\sqrt {2}\right )\right )}{\sqrt {2 \left (2+\sqrt {2}\right )} a \left (1+\sqrt [4]{-1} x\right ) \sqrt {-x+x^5}}-\frac {(3+3 i) x^2 \sqrt {\frac {(-1)^{3/4} \left (1-\sqrt [4]{-1} x\right )^2}{x}} \sqrt {\frac {i \left (1-x^4\right )}{x^2}} F\left (\sin ^{-1}\left (\frac {1}{2} \sqrt {-\frac {(-1)^{3/4} \left (\sqrt {2}+2 \sqrt [4]{-1} x+i \sqrt {2} x^2\right )}{x}}\right )|-2 \left (1-\sqrt {2}\right )\right )}{\sqrt {2 \left (2+\sqrt {2}\right )} a \left (1-\sqrt [4]{-1} x\right ) \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*}

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Mathematica [F]  time = 0.75, 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]

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IntegrateAlgebraic [A]  time = 1.15, size = 26, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {x}{\sqrt {a} \sqrt {-x+x^5}}\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[x/(Sqrt[a]*Sqrt[-x + x^5])])/Sqrt[a]

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fricas [B]  time = 0.52, size = 146, normalized size = 5.62 \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]

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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)

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maple [F]  time = 0.04, 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)

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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)

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mupad [B]  time = 0.44, size = 42, normalized size = 1.62 \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^5 - x)^(1/2)*(a + x - a*x^4)),x)

[Out]

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

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

[Out]

Timed out

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