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

Optimal. Leaf size=52 \[ \sqrt [4]{2} \tan ^{-1}\left (\sqrt [4]{2} \sqrt [4]{a x+b+x^2}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\sqrt [4]{2} \sqrt [4]{a x+b+x^2}\right ) \]

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

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.28, size = 45, normalized size = 0.87 \begin {gather*} \sqrt [4]{2} \left (\tan ^{-1}\left (\sqrt [4]{2} \sqrt [4]{x (a+x)+b}\right )-\tanh ^{-1}\left (\sqrt [4]{2} \sqrt [4]{x (a+x)+b}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.13, size = 52, normalized size = 1.00 \begin {gather*} \sqrt [4]{2} \tan ^{-1}\left (\sqrt [4]{2} \sqrt [4]{b+a x+x^2}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\sqrt [4]{2} \sqrt [4]{b+a x+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.47, size = 94, normalized size = 1.81 \begin {gather*} -\frac {1}{2} \cdot 8^{\frac {3}{4}} \arctan \left (\frac {1}{8} \cdot 8^{\frac {3}{4}} \sqrt {2 \, \sqrt {2} + 4 \, \sqrt {a x + x^{2} + b}} - \frac {1}{4} \cdot 8^{\frac {3}{4}} {\left (a x + x^{2} + b\right )}^{\frac {1}{4}}\right ) - \frac {1}{8} \cdot 8^{\frac {3}{4}} \log \left (8^{\frac {1}{4}} + 2 \, {\left (a x + x^{2} + b\right )}^{\frac {1}{4}}\right ) + \frac {1}{8} \cdot 8^{\frac {3}{4}} \log \left (-8^{\frac {1}{4}} + 2 \, {\left (a x + x^{2} + b\right )}^{\frac {1}{4}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*8^(3/4)*arctan(1/8*8^(3/4)*sqrt(2*sqrt(2) + 4*sqrt(a*x + x^2 + b)) - 1/4*8^(3/4)*(a*x + x^2 + b)^(1/4)) -
1/8*8^(3/4)*log(8^(1/4) + 2*(a*x + x^2 + b)^(1/4)) + 1/8*8^(3/4)*log(-8^(1/4) + 2*(a*x + x^2 + b)^(1/4))

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giac [A]  time = 0.30, size = 65, normalized size = 1.25 \begin {gather*} \frac {1}{4} \cdot 8^{\frac {3}{4}} \arctan \left (2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a x + x^{2} + b\right )}^{\frac {1}{4}}\right ) - \frac {1}{8} \cdot 8^{\frac {3}{4}} \log \left (\left (\frac {1}{2}\right )^{\frac {1}{4}} + {\left (a x + x^{2} + b\right )}^{\frac {1}{4}}\right ) + \frac {1}{8} \cdot 8^{\frac {3}{4}} \log \left ({\left | -\left (\frac {1}{2}\right )^{\frac {1}{4}} + {\left (a x + x^{2} + b\right )}^{\frac {1}{4}} \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*8^(3/4)*arctan(2*(1/2)^(3/4)*(a*x + x^2 + b)^(1/4)) - 1/8*8^(3/4)*log((1/2)^(1/4) + (a*x + x^2 + b)^(1/4))
+ 1/8*8^(3/4)*log(abs(-(1/2)^(1/4) + (a*x + x^2 + b)^(1/4)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + 2 \, x}{{\left (2 \, a x + 2 \, x^{2} + 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)^(1/4)/(2*a*x+2*x^2+2*b-1),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.99, size = 39, normalized size = 0.75 \begin {gather*} 2^{1/4}\,\left (\mathrm {atan}\left ({\left (2\,x^2+2\,a\,x+2\,b\right )}^{1/4}\right )-\mathrm {atanh}\left ({\left (2\,x^2+2\,a\,x+2\,b\right )}^{1/4}\right )\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)*(2*b + 2*a*x + 2*x^2 - 1)),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + 2 x}{\sqrt [4]{a x + b + x^{2}} \left (2 a x + 2 b + 2 x^{2} - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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