\(\int \frac {-1+3 x^4}{(a-x+a x^4) \sqrt {x+x^5}} \, dx\) [369]

Optimal result

Integrand size = 29, antiderivative size = 30 \[ \int \frac {-1+3 x^4}{\left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {x+x^5}}{\sqrt {a} \left (1+x^4\right )}\right )}{\sqrt {a}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(3*x^2*Sqrt[(1 + x)^2/x]*Sqrt[-((1 + x^4)/x^2)]*EllipticF[ArcSin[Sqrt[-((Sqrt[2] - 2*x + Sqrt[2]*x^2)/x)]/2],
-2*(1 - Sqrt[2])])/(Sqrt[2 + Sqrt[2]]*a*(1 + x)*Sqrt[x + x^5]) - (3*Sqrt[-((1 - x)^2/x)]*x^2*Sqrt[-((1 + x^4)/
x^2)]*EllipticF[ArcSin[Sqrt[(Sqrt[2] + 2*x + Sqrt[2]*x^2)/x]/2], -2*(1 - Sqrt[2])])/(Sqrt[2 + Sqrt[2]]*a*(1 -
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*} \text {integral}& = \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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {Subst}\left (\int \frac {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 ) \text {Subst}\left (\int \frac {1+x^2}{\sqrt {1+x^8}} \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^5}} \\ & = \frac {3 x^2 \sqrt {\frac {(1+x)^2}{x}} \sqrt {-\frac {1+x^4}{x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {-\frac {\sqrt {2}-2 x+\sqrt {2} x^2}{x}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{\sqrt {2+\sqrt {2}} a (1+x) \sqrt {x+x^5}}-\frac {3 \sqrt {-\frac {(1-x)^2}{x}} x^2 \sqrt {-\frac {1+x^4}{x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {\frac {\sqrt {2}+2 x+\sqrt {2} x^2}{x}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{\sqrt {2+\sqrt {2}} a (1-x) \sqrt {x+x^5}}-\frac {\left (8 \sqrt {x} \sqrt {1+x^4}\right ) \text {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 ) \text {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*}

Mathematica [A] (verified)

Time = 10.82 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {-1+3 x^4}{\left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {x+x^5}}{\sqrt {a} \left (1+x^4\right )}\right )}{\sqrt {a}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.87 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.77

method	result	size
pseudoelliptic	\(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{4}+1\right )}\, \sqrt {a}}{x}\right )}{\sqrt {a}}\)	\(23\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 4.57 \[ \int \frac {-1+3 x^4}{\left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=\left [\frac {\log \left (\frac {a^{2} x^{8} + 2 \, a^{2} x^{4} + 6 \, a x^{5} - 4 \, {\left (a x^{4} + a + x\right )} \sqrt {x^{5} + x} \sqrt {a} + a^{2} + 6 \, a x + x^{2}}{a^{2} x^{8} + 2 \, a^{2} x^{4} - 2 \, a x^{5} + a^{2} - 2 \, a x + x^{2}}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {{\left (a x^{4} + a + x\right )} \sqrt {x^{5} + x} \sqrt {-a}}{2 \, {\left (a x^{5} + a x\right )}}\right )}{a}\right ] \]

[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]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [B] (verification not implemented)

Time = 5.54 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27 \[ \int \frac {-1+3 x^4}{\left (a-x+a x^4\right ) \sqrt {x+x^5}} \, dx=\frac {\ln \left (\frac {a+x-2\,\sqrt {a}\,\sqrt {x^5+x}+a\,x^4}{a\,x^4-x+a}\right )}{\sqrt {a}} \]

[In]

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

[Out]

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