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

Optimal. Leaf size=149 \[ \frac{1}{2} a \tan ^{-1}\left (\frac{1-\sqrt{1-x^2}}{x \sqrt [4]{1-x^2}}\right )+\frac{1}{2} a \tanh ^{-1}\left (\frac{\sqrt{1-x^2}+1}{x \sqrt [4]{1-x^2}}\right )+\frac{b \tan ^{-1}\left (\frac{1-\sqrt{1-x^2}}{\sqrt{2} \sqrt [4]{1-x^2}}\right )}{\sqrt{2}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{1-x^2}+1}{\sqrt{2} \sqrt [4]{1-x^2}}\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.136095, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{1}{2} a \tan ^{-1}\left (\frac{1-\sqrt{1-x^2}}{x \sqrt [4]{1-x^2}}\right )+\frac{1}{2} a \tanh ^{-1}\left (\frac{\sqrt{1-x^2}+1}{x \sqrt [4]{1-x^2}}\right )+\frac{b \tan ^{-1}\left (\frac{1-\sqrt{1-x^2}}{\sqrt{2} \sqrt [4]{1-x^2}}\right )}{\sqrt{2}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{1-x^2}+1}{\sqrt{2} \sqrt [4]{1-x^2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 39.3188, size = 170, normalized size = 1.14 \[ - \frac{i a \sqrt{x^{2}} \Pi \left (- i; \operatorname{asin}{\left (\sqrt [4]{- x^{2} + 1} \right )}\middle | -1\right )}{x} + \frac{i a \sqrt{x^{2}} \Pi \left (i; \operatorname{asin}{\left (\sqrt [4]{- x^{2} + 1} \right )}\middle | -1\right )}{x} - \frac{\sqrt{2} b \log{\left (- \sqrt{2} \sqrt [4]{- x^{2} + 1} + \sqrt{- x^{2} + 1} + 1 \right )}}{4} + \frac{\sqrt{2} b \log{\left (\sqrt{2} \sqrt [4]{- x^{2} + 1} + \sqrt{- x^{2} + 1} + 1 \right )}}{4} - \frac{\sqrt{2} b \operatorname{atan}{\left (\sqrt{2} \sqrt [4]{- x^{2} + 1} - 1 \right )}}{2} - \frac{\sqrt{2} b \operatorname{atan}{\left (\sqrt{2} \sqrt [4]{- x^{2} + 1} + 1 \right )}}{2} \]

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]

-I*a*sqrt(x**2)*elliptic_pi(-I, asin((-x**2 + 1)**(1/4)), -1)/x + I*a*sqrt(x**2)
*elliptic_pi(I, asin((-x**2 + 1)**(1/4)), -1)/x - sqrt(2)*b*log(-sqrt(2)*(-x**2
+ 1)**(1/4) + sqrt(-x**2 + 1) + 1)/4 + sqrt(2)*b*log(sqrt(2)*(-x**2 + 1)**(1/4)
+ sqrt(-x**2 + 1) + 1)/4 - sqrt(2)*b*atan(sqrt(2)*(-x**2 + 1)**(1/4) - 1)/2 - sq
rt(2)*b*atan(sqrt(2)*(-x**2 + 1)**(1/4) + 1)/2

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Mathematica [C]  time = 0.445522, size = 205, normalized size = 1.38 \[ \frac{2 x \left (-\frac{3 a F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};x^2,\frac{x^2}{2}\right )}{x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};x^2,\frac{x^2}{2}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};x^2,\frac{x^2}{2}\right )\right )+6 F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};x^2,\frac{x^2}{2}\right )}-\frac{2 b x F_1\left (1;\frac{1}{4},1;2;x^2,\frac{x^2}{2}\right )}{x^2 \left (2 F_1\left (2;\frac{1}{4},2;3;x^2,\frac{x^2}{2}\right )+F_1\left (2;\frac{5}{4},1;3;x^2,\frac{x^2}{2}\right )\right )+8 F_1\left (1;\frac{1}{4},1;2;x^2,\frac{x^2}{2}\right )}\right )}{\sqrt [4]{1-x^2} \left (x^2-2\right )} \]

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*AppellF
1[1, 1/4, 1, 2, x^2, x^2/2] + x^2*(2*AppellF1[2, 1/4, 2, 3, x^2, x^2/2] + Appell
F1[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 = 0.07, 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. \[ - \int \frac{a}{x^{2} \sqrt [4]{- x^{2} + 1} - 2 \sqrt [4]{- x^{2} + 1}}\, dx - \int \frac{b x}{x^{2} \sqrt [4]{- x^{2} + 1} - 2 \sqrt [4]{- x^{2} + 1}}\, dx \]

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