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

Optimal. Leaf size=35 \[ \tan ^{-1}\left (\sqrt [4]{2 a x+2 b+x^2}\right )-\tanh ^{-1}\left (\sqrt [4]{2 a x+2 b+x^2}\right ) \]

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Rubi [F]  time = 0.02, 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+x}{\left (-1+2 b+2 a x+x^2\right ) \sqrt [4]{2 b+2 a x+x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.27, size = 35, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\sqrt [4]{2 a x+2 b+x^2}\right )-\tanh ^{-1}\left (\sqrt [4]{2 a x+2 b+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.03, size = 35, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\sqrt [4]{2 b+2 a x+x^2}\right )-\tanh ^{-1}\left (\sqrt [4]{2 b+2 a x+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.47, size = 51, normalized size = 1.46 \begin {gather*} \arctan \left ({\left (2 \, a x + x^{2} + 2 \, b\right )}^{\frac {1}{4}}\right ) - \frac {1}{2} \, \log \left ({\left (2 \, a x + x^{2} + 2 \, b\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{2} \, \log \left ({\left (2 \, a x + x^{2} + 2 \, b\right )}^{\frac {1}{4}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.46, size = 60, normalized size = 1.71 \begin {gather*} \arctan \left (2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a x + \frac {1}{2} \, x^{2} + b\right )}^{\frac {1}{4}}\right ) - \frac {1}{2} \, \log \left (\left (\frac {1}{2}\right )^{\frac {1}{4}} + {\left (a x + \frac {1}{2} \, x^{2} + b\right )}^{\frac {1}{4}}\right ) + \frac {1}{2} \, \log \left ({\left | -\left (\frac {1}{2}\right )^{\frac {1}{4}} + {\left (a x + \frac {1}{2} \, x^{2} + b\right )}^{\frac {1}{4}} \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(2*(1/2)^(3/4)*(a*x + 1/2*x^2 + b)^(1/4)) - 1/2*log((1/2)^(1/4) + (a*x + 1/2*x^2 + b)^(1/4)) + 1/2*log(a
bs(-(1/2)^(1/4) + (a*x + 1/2*x^2 + b)^(1/4)))

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maple [F]  time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {a +x}{\left (2 a x +x^{2}+2 b -1\right ) \left (2 a x +x^{2}+2 b \right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a+x)/(2*a*x+x^2+2*b-1)/(2*a*x+x^2+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 + x}{{\left (2 \, a x + x^{2} + 2 \, b\right )}^{\frac {1}{4}} {\left (2 \, a x + x^{2} + 2 \, b - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.61, size = 31, normalized size = 0.89 \begin {gather*} \mathrm {atan}\left ({\left (x^2+2\,a\,x+2\,b\right )}^{1/4}\right )-\mathrm {atanh}\left ({\left (x^2+2\,a\,x+2\,b\right )}^{1/4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 11.81, size = 56, normalized size = 1.60 \begin {gather*} \frac {\log {\left (\sqrt [4]{2 a x + 2 b + x^{2}} - 1 \right )}}{2} - \frac {\log {\left (\sqrt [4]{2 a x + 2 b + x^{2}} + 1 \right )}}{2} + \operatorname {atan}{\left (\sqrt [4]{2 a x + 2 b + x^{2}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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