\(\int \frac {2-3 x^5}{\sqrt {1+x^5} (1-a x^2+x^5)} \, dx\) [243]

Optimal result

Integrand size = 30, antiderivative size = 24 \[ \int \frac {2-3 x^5}{\sqrt {1+x^5} \left (1-a x^2+x^5\right )} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {1+x^5}}\right )}{\sqrt {a}} \]

[Out]

2*arctanh(a^(1/2)*x/(x^5+1)^(1/2))/a^(1/2)

Rubi [F]

\[ \int \frac {2-3 x^5}{\sqrt {1+x^5} \left (1-a x^2+x^5\right )} \, dx=\int \frac {2-3 x^5}{\sqrt {1+x^5} \left (1-a x^2+x^5\right )} \, dx \]

[In]

Int[(2 - 3*x^5)/(Sqrt[1 + x^5]*(1 - a*x^2 + x^5)),x]

[Out]

-3*x*Hypergeometric2F1[1/5, 1/2, 6/5, -x^5] - 5*Defer[Int][1/((-1 + a*x^2 - x^5)*Sqrt[1 + x^5]), x] + 3*a*Defe
r[Int][x^2/((-1 + a*x^2 - x^5)*Sqrt[1 + x^5]), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3}{\sqrt {1+x^5}}+\frac {5-3 a x^2}{\sqrt {1+x^5} \left (1-a x^2+x^5\right )}\right ) \, dx \\ & = -\left (3 \int \frac {1}{\sqrt {1+x^5}} \, dx\right )+\int \frac {5-3 a x^2}{\sqrt {1+x^5} \left (1-a x^2+x^5\right )} \, dx \\ & = -3 x \operatorname {Hypergeometric2F1}\left (\frac {1}{5},\frac {1}{2},\frac {6}{5},-x^5\right )+\int \left (-\frac {5}{\left (-1+a x^2-x^5\right ) \sqrt {1+x^5}}+\frac {3 a x^2}{\left (-1+a x^2-x^5\right ) \sqrt {1+x^5}}\right ) \, dx \\ & = -3 x \operatorname {Hypergeometric2F1}\left (\frac {1}{5},\frac {1}{2},\frac {6}{5},-x^5\right )-5 \int \frac {1}{\left (-1+a x^2-x^5\right ) \sqrt {1+x^5}} \, dx+(3 a) \int \frac {x^2}{\left (-1+a x^2-x^5\right ) \sqrt {1+x^5}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 3.96 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {2-3 x^5}{\sqrt {1+x^5} \left (1-a x^2+x^5\right )} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {1+x^5}}\right )}{\sqrt {a}} \]

[In]

Integrate[(2 - 3*x^5)/(Sqrt[1 + x^5]*(1 - a*x^2 + x^5)),x]

[Out]

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

Maple [A] (verified)

Time = 0.86 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88

method	result	size
pseudoelliptic	\(\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{5}+1}}{x \sqrt {a}}\right )}{\sqrt {a}}\)	\(21\)

[In]

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

[Out]

2/a^(1/2)*arctanh((x^5+1)^(1/2)/x/a^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (18) = 36\).

Time = 0.29 (sec) , antiderivative size = 136, normalized size of antiderivative = 5.67 \[ \int \frac {2-3 x^5}{\sqrt {1+x^5} \left (1-a x^2+x^5\right )} \, dx=\left [\frac {\log \left (\frac {x^{10} + 6 \, a x^{7} + a^{2} x^{4} + 2 \, x^{5} + 6 \, a x^{2} + 4 \, {\left (x^{6} + a x^{3} + x\right )} \sqrt {x^{5} + 1} \sqrt {a} + 1}{x^{10} - 2 \, a x^{7} + a^{2} x^{4} + 2 \, x^{5} - 2 \, a x^{2} + 1}\right )}{2 \, \sqrt {a}}, -\frac {\sqrt {-a} \arctan \left (\frac {{\left (x^{5} + a x^{2} + 1\right )} \sqrt {x^{5} + 1} \sqrt {-a}}{2 \, {\left (a x^{6} + a x\right )}}\right )}{a}\right ] \]

[In]

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

[Out]

[1/2*log((x^10 + 6*a*x^7 + a^2*x^4 + 2*x^5 + 6*a*x^2 + 4*(x^6 + a*x^3 + x)*sqrt(x^5 + 1)*sqrt(a) + 1)/(x^10 -
2*a*x^7 + a^2*x^4 + 2*x^5 - 2*a*x^2 + 1))/sqrt(a), -sqrt(-a)*arctan(1/2*(x^5 + a*x^2 + 1)*sqrt(x^5 + 1)*sqrt(-
a)/(a*x^6 + a*x))/a]

Sympy [F]

\[ \int \frac {2-3 x^5}{\sqrt {1+x^5} \left (1-a x^2+x^5\right )} \, dx=- \int \frac {3 x^{5}}{- a x^{2} \sqrt {x^{5} + 1} + x^{5} \sqrt {x^{5} + 1} + \sqrt {x^{5} + 1}}\, dx - \int \left (- \frac {2}{- a x^{2} \sqrt {x^{5} + 1} + x^{5} \sqrt {x^{5} + 1} + \sqrt {x^{5} + 1}}\right )\, dx \]

[In]

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

[Out]

-Integral(3*x**5/(-a*x**2*sqrt(x**5 + 1) + x**5*sqrt(x**5 + 1) + sqrt(x**5 + 1)), x) - Integral(-2/(-a*x**2*sq
rt(x**5 + 1) + x**5*sqrt(x**5 + 1) + sqrt(x**5 + 1)), x)

Maxima [F]

\[ \int \frac {2-3 x^5}{\sqrt {1+x^5} \left (1-a x^2+x^5\right )} \, dx=\int { -\frac {3 \, x^{5} - 2}{{\left (x^{5} - a x^{2} + 1\right )} \sqrt {x^{5} + 1}} \,d x } \]

[In]

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

[Out]

-integrate((3*x^5 - 2)/((x^5 - a*x^2 + 1)*sqrt(x^5 + 1)), x)

Giac [F]

\[ \int \frac {2-3 x^5}{\sqrt {1+x^5} \left (1-a x^2+x^5\right )} \, dx=\int { -\frac {3 \, x^{5} - 2}{{\left (x^{5} - a x^{2} + 1\right )} \sqrt {x^{5} + 1}} \,d x } \]

[In]

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

[Out]

integrate(-(3*x^5 - 2)/((x^5 - a*x^2 + 1)*sqrt(x^5 + 1)), x)

Mupad [B] (verification not implemented)

Time = 5.96 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83 \[ \int \frac {2-3 x^5}{\sqrt {1+x^5} \left (1-a x^2+x^5\right )} \, dx=\frac {\ln \left (\frac {a\,x^2+x^5+2\,\sqrt {a}\,x\,\sqrt {x^5+1}+1}{4\,x^5-4\,a\,x^2+4}\right )}{\sqrt {a}} \]

[In]

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

[Out]

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