\(\int \frac {1+x}{(-1+2 x) \sqrt {x+x^4}} \, dx\) [379]

Optimal result

Integrand size = 20, antiderivative size = 31 \[ \int \frac {1+x}{(-1+2 x) \sqrt {x+x^4}} \, dx=-\frac {2}{3} \text {arctanh}\left (\frac {\frac {1}{3}+\frac {2 x}{3}+\frac {x^2}{3}}{\sqrt {x+x^4}}\right ) \]

[Out]

-2/3*arctanh((1/3+2/3*x+1/3*x^2)/(x^4+x)^(1/2))

Rubi [F]

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

[In]

Int[(1 + x)/((-1 + 2*x)*Sqrt[x + x^4]),x]

[Out]

(x*(1 + x)*Sqrt[(1 - x + x^2)/(1 + (1 + Sqrt[3])*x)^2]*EllipticF[ArcCos[(1 + (1 - Sqrt[3])*x)/(1 + (1 + Sqrt[3
])*x)], (2 + Sqrt[3])/4])/(2*3^(1/4)*Sqrt[(x*(1 + x))/(1 + (1 + Sqrt[3])*x)^2]*Sqrt[x + x^4]) - (3*Sqrt[x]*Sqr
t[1 + x^3]*Defer[Subst][Defer[Int][1/((1 - Sqrt[2]*x)*Sqrt[1 + x^6]), x], x, Sqrt[x]])/(2*Sqrt[x + x^4]) - (3*
Sqrt[x]*Sqrt[1 + x^3]*Defer[Subst][Defer[Int][1/((1 + Sqrt[2]*x)*Sqrt[1 + x^6]), x], x, Sqrt[x]])/(2*Sqrt[x +
x^4])

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {1+x^3}\right ) \int \frac {1+x}{\sqrt {x} (-1+2 x) \sqrt {1+x^3}} \, dx}{\sqrt {x+x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1+x^2}{\left (-1+2 x^2\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {1+x^6}}+\frac {3}{2 \left (-1+2 x^2\right ) \sqrt {1+x^6}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}} \\ & = \frac {\left (\sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}}+\frac {\left (3 \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+2 x^2\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}} \\ & = \frac {x (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1+\left (1-\sqrt {3}\right ) x}{1+\left (1+\sqrt {3}\right ) x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 \sqrt [4]{3} \sqrt {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {x+x^4}}+\frac {\left (3 \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \left (-\frac {1}{2 \left (1-\sqrt {2} x\right ) \sqrt {1+x^6}}-\frac {1}{2 \left (1+\sqrt {2} x\right ) \sqrt {1+x^6}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^4}} \\ & = \frac {x (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1+\left (1-\sqrt {3}\right ) x}{1+\left (1+\sqrt {3}\right ) x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 \sqrt [4]{3} \sqrt {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {x+x^4}}-\frac {\left (3 \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\sqrt {2} x\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x+x^4}}-\frac {\left (3 \sqrt {x} \sqrt {1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\sqrt {2} x\right ) \sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x+x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.93 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {1+x}{(-1+2 x) \sqrt {x+x^4}} \, dx=-\frac {2}{3} \text {arctanh}\left (\frac {\frac {1}{3}+\frac {2 x}{3}+\frac {x^2}{3}}{\sqrt {x+x^4}}\right ) \]

[In]

Integrate[(1 + x)/((-1 + 2*x)*Sqrt[x + x^4]),x]

[Out]

(-2*ArcTanh[(1/3 + (2*x)/3 + x^2/3)/Sqrt[x + x^4]])/3

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(45\) vs. \(2(21)=42\).

Time = 2.76 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48

method	result	size
trager	\(-\frac {\ln \left (\frac {10 x^{3}+6 x \sqrt {x^{4}+x}-6 x^{2}+6 \sqrt {x^{4}+x}+12 x +1}{\left (-1+2 x \right )^{3}}\right )}{3}\)	\(46\)
default	\(-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}+\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+2 \operatorname {EllipticPi}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {3}{2}+\frac {3 i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(491\)
elliptic	\(-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}+\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+2 \operatorname {EllipticPi}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {3}{2}+\frac {3 i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(491\)

[In]

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

[Out]

-1/3*ln((10*x^3+6*x*(x^4+x)^(1/2)-6*x^2+6*(x^4+x)^(1/2)+12*x+1)/(-1+2*x)^3)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).

Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.55 \[ \int \frac {1+x}{(-1+2 x) \sqrt {x+x^4}} \, dx=\frac {1}{3} \, \log \left (\frac {10 \, x^{3} - 6 \, x^{2} - 6 \, \sqrt {x^{4} + x} {\left (x + 1\right )} + 12 \, x + 1}{8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1}\right ) \]

[In]

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

[Out]

1/3*log((10*x^3 - 6*x^2 - 6*sqrt(x^4 + x)*(x + 1) + 12*x + 1)/(8*x^3 - 12*x^2 + 6*x - 1))

Sympy [F]

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

[In]

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

[Out]

Integral((x + 1)/(sqrt(x*(x + 1)*(x**2 - x + 1))*(2*x - 1)), x)

Maxima [F]

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

[In]

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

[Out]

integrate((x + 1)/(sqrt(x^4 + x)*(2*x - 1)), x)

Giac [F]

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

[In]

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

[Out]

integrate((x + 1)/(sqrt(x^4 + x)*(2*x - 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1+x}{(-1+2 x) \sqrt {x+x^4}} \, dx=\int \frac {x+1}{\left (2\,x-1\right )\,\sqrt {x^4+x}} \,d x \]

[In]

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

[Out]

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