\(\int \frac {-2+3 x^5}{\sqrt {1+x^5} (a-x^2+a x^5)} \, dx\) [246]

Optimal result

Integrand size = 31, antiderivative size = 24 \[ \int \frac {-2+3 x^5}{\sqrt {1+x^5} \left (a-x^2+a x^5\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {x}{\sqrt {a} \sqrt {1+x^5}}\right )}{\sqrt {a}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 1.79 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-2+3 x^5}{\sqrt {1+x^5} \left (a-x^2+a x^5\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {x}{\sqrt {a} \sqrt {1+x^5}}\right )}{\sqrt {a}} \]

[In]

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

[Out]

(-2*ArcTanh[x/(Sqrt[a]*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	\(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{5}+1}\, \sqrt {a}}{x}\right )}{\sqrt {a}}\)	\(21\)

[In]

int((3*x^5-2)/(x^5+1)^(1/2)/(a*x^5-x^2+a),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. 45 vs. \(2 (18) = 36\).

Time = 0.29 (sec) , antiderivative size = 147, normalized size of antiderivative = 6.12 \[ \int \frac {-2+3 x^5}{\sqrt {1+x^5} \left (a-x^2+a x^5\right )} \, dx=\left [\frac {\log \left (\frac {a^{2} x^{10} + 6 \, a x^{7} + 2 \, a^{2} x^{5} + x^{4} + 6 \, a x^{2} - 4 \, {\left (a x^{6} + x^{3} + a x\right )} \sqrt {x^{5} + 1} \sqrt {a} + a^{2}}{a^{2} x^{10} - 2 \, a x^{7} + 2 \, a^{2} x^{5} + x^{4} - 2 \, a x^{2} + a^{2}}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {{\left (a x^{5} + x^{2} + a\right )} \sqrt {x^{5} + 1} \sqrt {-a}}{2 \, {\left (a x^{6} + a x\right )}}\right )}{a}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.98 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.92 \[ \int \frac {-2+3 x^5}{\sqrt {1+x^5} \left (a-x^2+a x^5\right )} \, dx=\frac {\ln \left (\frac {a+a\,x^5+x^2-2\,\sqrt {a}\,x\,\sqrt {x^5+1}}{4\,a\,x^5-4\,x^2+4\,a}\right )}{\sqrt {a}} \]

[In]

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

[Out]

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