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

Optimal. Leaf size=135 \[ -\frac{1}{2} a \tan ^{-1}\left (\frac{\sqrt{x^2+1}+1}{x \sqrt [4]{x^2+1}}\right )-\frac{1}{2} a \tanh ^{-1}\left (\frac{1-\sqrt{x^2+1}}{x \sqrt [4]{x^2+1}}\right )-\frac{b \tan ^{-1}\left (\frac{1-\sqrt{x^2+1}}{\sqrt{2} \sqrt [4]{x^2+1}}\right )}{\sqrt{2}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{x^2+1}+1}{\sqrt{2} \sqrt [4]{x^2+1}}\right )}{\sqrt{2}} \]

[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 - (a*ArcTanh[(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]

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Rubi [A]  time = 0.108503, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{1}{2} a \tan ^{-1}\left (\frac{\sqrt{x^2+1}+1}{x \sqrt [4]{x^2+1}}\right )-\frac{1}{2} a \tanh ^{-1}\left (\frac{1-\sqrt{x^2+1}}{x \sqrt [4]{x^2+1}}\right )-\frac{b \tan ^{-1}\left (\frac{1-\sqrt{x^2+1}}{\sqrt{2} \sqrt [4]{x^2+1}}\right )}{\sqrt{2}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{x^2+1}+1}{\sqrt{2} \sqrt [4]{x^2+1}}\right )}{\sqrt{2}} \]

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 - (a*ArcTanh[(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]

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Rubi in Sympy [A]  time = 36.0933, size = 173, normalized size = 1.28 \[ \frac{i a \sqrt{- x^{2}} \Pi \left (- i; \operatorname{asin}{\left (\sqrt [4]{x^{2} + 1} \right )}\middle | -1\right )}{x} - \frac{i a \sqrt{- x^{2}} \Pi \left (i; \operatorname{asin}{\left (\sqrt [4]{x^{2} + 1} \right )}\middle | -1\right )}{x} + \frac{\sqrt{2} b \log{\left (- \sqrt{2} \sqrt [4]{x^{2} + 1} + \sqrt{x^{2} + 1} + 1 \right )}}{4} - \frac{\sqrt{2} b \log{\left (\sqrt{2} \sqrt [4]{x^{2} + 1} + \sqrt{x^{2} + 1} + 1 \right )}}{4} + \frac{\sqrt{2} b \operatorname{atan}{\left (\sqrt{2} \sqrt [4]{x^{2} + 1} - 1 \right )}}{2} + \frac{\sqrt{2} b \operatorname{atan}{\left (\sqrt{2} \sqrt [4]{x^{2} + 1} + 1 \right )}}{2} \]

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) + sq
rt(x**2 + 1) + 1)/4 + sqrt(2)*b*atan(sqrt(2)*(x**2 + 1)**(1/4) - 1)/2 + sqrt(2)*
b*atan(sqrt(2)*(x**2 + 1)**(1/4) + 1)/2

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Mathematica [C]  time = 0.425114, size = 219, normalized size = 1.62 \[ \frac{2 x \left (-\frac{3 a F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-x^2,-\frac{x^2}{2}\right )}{x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-x^2,-\frac{x^2}{2}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-x^2,-\frac{x^2}{2}\right )\right )-6 F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-x^2,-\frac{x^2}{2}\right )}-\frac{2 b x F_1\left (1;\frac{1}{4},1;2;-x^2,-\frac{x^2}{2}\right )}{x^2 \left (2 F_1\left (2;\frac{1}{4},2;3;-x^2,-\frac{x^2}{2}\right )+F_1\left (2;\frac{5}{4},1;3;-x^2,-\frac{x^2}{2}\right )\right )-8 F_1\left (1;\frac{1}{4},1;2;-x^2,-\frac{x^2}{2}\right )}\right )}{\sqrt [4]{x^2+1} \left (x^2+2\right )} \]

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 = 0.045, 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)