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

Optimal. Leaf size=33 \[ 2 \tan ^{-1}\left (\sqrt [4]{-a x+b+x^2}\right )-2 \tanh ^{-1}\left (\sqrt [4]{-a x+b+x^2}\right ) \]

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Rubi [F]  time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-a+2 x}{\left (-1+b-a x+x^2\right ) \sqrt [4]{b-a x+x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \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\\ \end {align*}

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Mathematica [A]  time = 0.27, size = 33, normalized size = 1.00 \begin {gather*} 2 \left (\tan ^{-1}\left (\sqrt [4]{-a x+b+x^2}\right )-\tanh ^{-1}\left (\sqrt [4]{-a x+b+x^2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[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)])

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IntegrateAlgebraic [A]  time = 0.03, size = 33, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (\sqrt [4]{b-a x+x^2}\right )-2 \tanh ^{-1}\left (\sqrt [4]{b-a x+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

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fricas [A]  time = 0.49, size = 45, normalized size = 1.36 \begin {gather*} 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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.42, size = 46, normalized size = 1.39 \begin {gather*} 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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {-a +2 x}{\left (-a x +x^{2}+b -1\right ) \left (-a x +x^{2}+b \right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.55, size = 29, normalized size = 0.88 \begin {gather*} 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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 12.22, size = 44, normalized size = 1.33 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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