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

Optimal. Leaf size=45 \[ \frac {2 \sqrt {x^5+x}}{x}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x^5+x}}{x^4+1}\right ) \]

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Rubi [F]  time = 1.24, 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 (1+x^4\right ) \left (-1+3 x^4\right )}{x \left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*(1 + x^4))/Sqrt[x + x^5] - (8*x^4*Sqrt[1 + x^4]*Hypergeometric2F1[1/2, 7/8, 15/8, -x^4])/(7*Sqrt[x + x^5])
+ (2*a*Sqrt[x]*Sqrt[1 + x^4]*Defer[Subst][Defer[Int][Sqrt[1 + x^8]/(-1 + a*x^2 - x^8), x], x, Sqrt[x]])/Sqrt[x
+ x^5] + (8*Sqrt[x]*Sqrt[1 + x^4]*Defer[Subst][Defer[Int][(x^6*Sqrt[1 + x^8])/(1 - a*x^2 + x^8), x], x, Sqrt[
x]])/Sqrt[x + x^5]

Rubi steps

\begin {align*} \int \frac {\left (1+x^4\right ) \left (-1+3 x^4\right )}{x \left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx &=\frac {\left (\sqrt {x} \sqrt {1+x^4}\right ) \int \frac {\sqrt {1+x^4} \left (-1+3 x^4\right )}{x^{3/2} \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 {\sqrt {1+x^8} \left (-1+3 x^8\right )}{x^2 \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 {\sqrt {1+x^8}}{x^2}+\frac {\left (a-4 x^6\right ) \sqrt {1+x^8}}{-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 \frac {\sqrt {1+x^8}}{x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}+\frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \frac {\left (a-4 x^6\right ) \sqrt {1+x^8}}{-1+a x^2-x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}\\ &=\frac {2 \left (1+x^4\right )}{\sqrt {x+x^5}}+\frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {a \sqrt {1+x^8}}{-1+a x^2-x^8}+\frac {4 x^6 \sqrt {1+x^8}}{1-a x^2+x^8}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}-\frac {\left (8 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\sqrt {1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}\\ &=\frac {2 \left (1+x^4\right )}{\sqrt {x+x^5}}-\frac {8 x^4 \sqrt {1+x^4} \, _2F_1\left (\frac {1}{2},\frac {7}{8};\frac {15}{8};-x^4\right )}{7 \sqrt {x+x^5}}+\frac {\left (8 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt {1+x^8}}{1-a x^2+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}+\frac {\left (2 a \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^8}}{-1+a x^2-x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}\\ \end {align*}

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Mathematica [F]  time = 0.62, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^4\right ) \left (-1+3 x^4\right )}{x \left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.34, size = 45, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {x+x^5}}{x}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x+x^5}}{1+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.75, size = 146, normalized size = 3.24 \begin {gather*} \left [\frac {\sqrt {a} x \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 ) + 4 \, \sqrt {x^{5} + x}}{2 \, x}, \frac {\sqrt {-a} x \arctan \left (\frac {2 \, \sqrt {x^{5} + x} \sqrt {-a}}{x^{4} + a x + 1}\right ) + 2 \, \sqrt {x^{5} + x}}{x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.71, size = 50, normalized size = 1.11 \begin {gather*} \frac {2\,\sqrt {x^5+x}}{x}+\sqrt {a}\,\ln \left (\frac {a\,x-2\,\sqrt {a}\,\sqrt {x^5+x}+x^4+1}{x^4-a\,x+1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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