\(\int \frac {2+3 x^5}{\sqrt {-1+x^5} (-1-a x^2+x^5)} \, dx\) [244]

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*Sqrt[1 - x^5]*Hypergeometric2F1[1/5, 1/2, 6/5, x^5])/Sqrt[-1 + x^5] - 5*Defer[Int][1/((1 + a*x^2 - x^5)*S
qrt[-1 + x^5]), x] - 3*a*Defer[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 \\ & = 3 \int \frac {1}{\sqrt {-1+x^5}} \, dx+\int \frac {5+3 a x^2}{\sqrt {-1+x^5} \left (-1-a x^2+x^5\right )} \, dx \\ & = \frac {\left (3 \sqrt {1-x^5}\right ) \int \frac {1}{\sqrt {1-x^5}} \, dx}{\sqrt {-1+x^5}}+\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 \\ & = \frac {3 x \sqrt {1-x^5} \operatorname {Hypergeometric2F1}\left (\frac {1}{5},\frac {1}{2},\frac {6}{5},x^5\right )}{\sqrt {-1+x^5}}-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 = 3.90 (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 = 0.90 (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 = 0.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 5.75 \[ \int \frac {2+3 x^5}{\sqrt {-1+x^5} \left (-1-a x^2+x^5\right )} \, dx=\left [\frac {\log \left (\frac {x^{10} + 6 \, a x^{7} + a^{2} x^{4} - 2 \, x^{5} - 6 \, a x^{2} - 4 \, {\left (x^{6} + a x^{3} - x\right )} \sqrt {x^{5} - 1} \sqrt {a} + 1}{x^{10} - 2 \, a x^{7} + a^{2} x^{4} - 2 \, x^{5} + 2 \, a x^{2} + 1}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {{\left (x^{5} + a x^{2} - 1\right )} \sqrt {x^{5} - 1} \sqrt {-a}}{2 \, {\left (a x^{6} - a x\right )}}\right )}{a}\right ] \]

[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]

\[ \int \frac {2+3 x^5}{\sqrt {-1+x^5} \left (-1-a x^2+x^5\right )} \, dx=\int \frac {3 x^{5} + 2}{\sqrt {\left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )} \left (- a x^{2} + x^{5} - 1\right )}\, dx \]

[In]

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

[Out]

Integral((3*x**5 + 2)/(sqrt((x - 1)*(x**4 + x**3 + x**2 + x + 1))*(-a*x**2 + 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 = 6.01 (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)*(a*x^2 - x^5 + 1)),x)

[Out]

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