\(\int \frac {-a+2 x}{(-1+b-a x+x^2) \sqrt [4]{b-a x+x^2}} \, dx\) [393]

Optimal result

Integrand size = 33, antiderivative size = 33 \[ \int \frac {-a+2 x}{\left (-1+b-a x+x^2\right ) \sqrt [4]{b-a x+x^2}} \, dx=2 \arctan \left (\sqrt [4]{b-a x+x^2}\right )-2 \text {arctanh}\left (\sqrt [4]{b-a x+x^2}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

Defer[Int][(-a + 2*x)/((-1 + b - a*x + x^2)*(b - a*x + x^2)^(1/4)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-a+2 x}{\left (-1+b-a x+x^2\right ) \sqrt [4]{b-a x+x^2}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {-a+2 x}{\left (-1+b-a x+x^2\right ) \sqrt [4]{b-a x+x^2}} \, dx=2 \left (\arctan \left (\sqrt [4]{b-a x+x^2}\right )-\text {arctanh}\left (\sqrt [4]{b-a x+x^2}\right )\right ) \]

[In]

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

[Out]

2*(ArcTan[(b - a*x + x^2)^(1/4)] - ArcTanh[(b - a*x + x^2)^(1/4)])

Maple [A] (verified)

Time = 2.74 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39

method	result	size
pseudoelliptic	\(\ln \left (\left (-a x +x^{2}+b \right )^{\frac {1}{4}}-1\right )-\ln \left (\left (-a x +x^{2}+b \right )^{\frac {1}{4}}+1\right )+2 \arctan \left (\left (-a x +x^{2}+b \right )^{\frac {1}{4}}\right )\)	\(46\)

[In]

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

[Out]

ln((-a*x+x^2+b)^(1/4)-1)-ln((-a*x+x^2+b)^(1/4)+1)+2*arctan((-a*x+x^2+b)^(1/4))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.36 \[ \int \frac {-a+2 x}{\left (-1+b-a x+x^2\right ) \sqrt [4]{b-a x+x^2}} \, dx=2 \, \arctan \left ({\left (-a x + x^{2} + b\right )}^{\frac {1}{4}}\right ) - \log \left ({\left (-a x + x^{2} + b\right )}^{\frac {1}{4}} + 1\right ) + \log \left ({\left (-a x + x^{2} + b\right )}^{\frac {1}{4}} - 1\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 4.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {-a+2 x}{\left (-1+b-a x+x^2\right ) \sqrt [4]{b-a x+x^2}} \, dx=\log {\left (\sqrt [4]{- a x + b + x^{2}} - 1 \right )} - \log {\left (\sqrt [4]{- a x + b + x^{2}} + 1 \right )} + 2 \operatorname {atan}{\left (\sqrt [4]{- a x + b + x^{2}} \right )} \]

[In]

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

[Out]

log((-a*x + b + x**2)**(1/4) - 1) - log((-a*x + b + x**2)**(1/4) + 1) + 2*atan((-a*x + b + x**2)**(1/4))

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39 \[ \int \frac {-a+2 x}{\left (-1+b-a x+x^2\right ) \sqrt [4]{b-a x+x^2}} \, dx=2 \, \arctan \left ({\left (-a x + x^{2} + b\right )}^{\frac {1}{4}}\right ) - \log \left ({\left (-a x + x^{2} + b\right )}^{\frac {1}{4}} + 1\right ) + \log \left ({\left | {\left (-a x + x^{2} + b\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]

[In]

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

[Out]

2*arctan((-a*x + x^2 + b)^(1/4)) - log((-a*x + x^2 + b)^(1/4) + 1) + log(abs((-a*x + x^2 + b)^(1/4) - 1))

Mupad [B] (verification not implemented)

Time = 5.35 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {-a+2 x}{\left (-1+b-a x+x^2\right ) \sqrt [4]{b-a x+x^2}} \, dx=2\,\mathrm {atan}\left ({\left (x^2-a\,x+b\right )}^{1/4}\right )-2\,\mathrm {atanh}\left ({\left (x^2-a\,x+b\right )}^{1/4}\right ) \]

[In]

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

[Out]

2*atan((b - a*x + x^2)^(1/4)) - 2*atanh((b - a*x + x^2)^(1/4))