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

Optimal. Leaf size=88 \[ \frac{a \tan ^{-1}\left (\frac{x}{\sqrt{2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt{2}}+\frac{a \tanh ^{-1}\left (\frac{x}{\sqrt{2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt{2}}+b \tan ^{-1}\left (\sqrt [4]{-x^2-1}\right )-b \tanh ^{-1}\left (\sqrt [4]{-x^2-1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.130639, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{a \tan ^{-1}\left (\frac{x}{\sqrt{2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt{2}}+\frac{a \tanh ^{-1}\left (\frac{x}{\sqrt{2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt{2}}+b \tan ^{-1}\left (\sqrt [4]{-x^2-1}\right )-b \tanh ^{-1}\left (\sqrt [4]{-x^2-1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 29.508, size = 199, normalized size = 2.26 \[ - \frac{\sqrt{2} a x \left (1 - i\right ) \Pi \left (i; \operatorname{asin}{\left (\frac{\sqrt{2} \left (1 + i\right ) \sqrt [4]{- x^{2} - 1}}{2} \right )}\middle | -1\right )}{2 \sqrt{- i \sqrt{- x^{2} - 1} + 1} \sqrt{i \sqrt{- x^{2} - 1} + 1}} - \frac{\sqrt{2} a \sqrt{- x^{2}} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt [4]{- x^{2} - 1}}{\sqrt{- x^{2}}} \right )}}{4 x} - \frac{a \sqrt{- \frac{x^{2}}{\left (\sqrt{- x^{2} - 1} + 1\right )^{2}}} \left (\sqrt{- x^{2} - 1} + 1\right ) F\left (2 \operatorname{atan}{\left (\sqrt [4]{- x^{2} - 1} \right )}\middle | \frac{1}{2}\right )}{4 x} + b \operatorname{atan}{\left (\sqrt [4]{- x^{2} - 1} \right )} - b \operatorname{atanh}{\left (\sqrt [4]{- x^{2} - 1} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)/(-x**2-1)**(1/4)/(x**2+2),x)

[Out]

-sqrt(2)*a*x*(1 - I)*elliptic_pi(I, asin(sqrt(2)*(1 + I)*(-x**2 - 1)**(1/4)/2),
-1)/(2*sqrt(-I*sqrt(-x**2 - 1) + 1)*sqrt(I*sqrt(-x**2 - 1) + 1)) - sqrt(2)*a*sqr
t(-x**2)*atanh(sqrt(2)*(-x**2 - 1)**(1/4)/sqrt(-x**2))/(4*x) - a*sqrt(-x**2/(sqr
t(-x**2 - 1) + 1)**2)*(sqrt(-x**2 - 1) + 1)*elliptic_f(2*atan((-x**2 - 1)**(1/4)
), 1/2)/(4*x) + b*atan((-x**2 - 1)**(1/4)) - b*atanh((-x**2 - 1)**(1/4))

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Mathematica [C]  time = 0.554591, size = 221, normalized size = 2.51 \[ \frac{2 x \left (-\frac{3 a F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-x^2,-\frac{x^2}{2}\right )}{x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-x^2,-\frac{x^2}{2}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-x^2,-\frac{x^2}{2}\right )\right )-6 F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-x^2,-\frac{x^2}{2}\right )}-\frac{2 b x F_1\left (1;\frac{1}{4},1;2;-x^2,-\frac{x^2}{2}\right )}{x^2 \left (2 F_1\left (2;\frac{1}{4},2;3;-x^2,-\frac{x^2}{2}\right )+F_1\left (2;\frac{5}{4},1;3;-x^2,-\frac{x^2}{2}\right )\right )-8 F_1\left (1;\frac{1}{4},1;2;-x^2,-\frac{x^2}{2}\right )}\right )}{\sqrt [4]{-x^2-1} \left (x^2+2\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*x*((-3*a*AppellF1[1/2, 1/4, 1, 3/2, -x^2, -x^2/2])/(-6*AppellF1[1/2, 1/4, 1,
3/2, -x^2, -x^2/2] + x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, -x^2, -x^2/2] + AppellF1[
3/2, 5/4, 1, 5/2, -x^2, -x^2/2])) - (2*b*x*AppellF1[1, 1/4, 1, 2, -x^2, -x^2/2])
/(-8*AppellF1[1, 1/4, 1, 2, -x^2, -x^2/2] + x^2*(2*AppellF1[2, 1/4, 2, 3, -x^2,
-x^2/2] + AppellF1[2, 5/4, 1, 3, -x^2, -x^2/2]))))/((-1 - x^2)^(1/4)*(2 + x^2))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{bx+a}{{x}^{2}+2}{\frac{1}{\sqrt [4]{-{x}^{2}-1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{b x + a}{{\left (x^{2} + 2\right )}{\left (-x^{2} - 1\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{b x + a}{{\left (x^{2} + 2\right )}{\left (-x^{2} - 1\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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