3.53 \(\int \frac{a+b x}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx\)

Optimal. Leaf size=198 \[ \frac{a \tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt{3}}+\frac{a \tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}+\frac{a \tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3} \sqrt{3}}-\frac{a \tanh ^{-1}(x)}{6\ 2^{2/3}}-\frac{b \log \left (x^2+3\right )}{4\ 2^{2/3}}+\frac{3 b \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}}+\frac{\sqrt{3} b \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{2\ 2^{2/3}} \]

[Out]

(a*ArcTan[Sqrt[3]/x])/(2*2^(2/3)*Sqrt[3]) + (Sqrt[3]*b*ArcTan[(1 + (2 - 2*x^2)^(
1/3))/Sqrt[3]])/(2*2^(2/3)) + (a*ArcTan[(Sqrt[3]*(1 - 2^(1/3)*(1 - x^2)^(1/3)))/
x])/(2*2^(2/3)*Sqrt[3]) - (a*ArcTanh[x])/(6*2^(2/3)) + (a*ArcTanh[x/(1 + 2^(1/3)
*(1 - x^2)^(1/3))])/(2*2^(2/3)) - (b*Log[3 + x^2])/(4*2^(2/3)) + (3*b*Log[2^(2/3
) - (1 - x^2)^(1/3)])/(4*2^(2/3))

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Rubi [A]  time = 0.241473, antiderivative size = 198, 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 \[ \frac{a \tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt{3}}+\frac{a \tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}+\frac{a \tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3} \sqrt{3}}-\frac{a \tanh ^{-1}(x)}{6\ 2^{2/3}}-\frac{b \log \left (x^2+3\right )}{4\ 2^{2/3}}+\frac{3 b \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}}+\frac{\sqrt{3} b \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{2\ 2^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/((1 - x^2)^(1/3)*(3 + x^2)),x]

[Out]

(a*ArcTan[Sqrt[3]/x])/(2*2^(2/3)*Sqrt[3]) + (Sqrt[3]*b*ArcTan[(1 + (2 - 2*x^2)^(
1/3))/Sqrt[3]])/(2*2^(2/3)) + (a*ArcTan[(Sqrt[3]*(1 - 2^(1/3)*(1 - x^2)^(1/3)))/
x])/(2*2^(2/3)*Sqrt[3]) - (a*ArcTanh[x])/(6*2^(2/3)) + (a*ArcTanh[x/(1 + 2^(1/3)
*(1 - x^2)^(1/3))])/(2*2^(2/3)) - (b*Log[3 + x^2])/(4*2^(2/3)) + (3*b*Log[2^(2/3
) - (1 - x^2)^(1/3)])/(4*2^(2/3))

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Rubi in Sympy [A]  time = 13.6533, size = 231, normalized size = 1.17 \[ \frac{\sqrt [3]{2} a \log{\left (\sqrt [3]{2} \sqrt [3]{- x + 1} + \left (x + 1\right )^{\frac{2}{3}} \right )}}{8} - \frac{\sqrt [3]{2} a \log{\left (\left (- x + 1\right )^{\frac{2}{3}} + \sqrt [3]{2} \sqrt [3]{x + 1} \right )}}{8} - \frac{\sqrt [3]{2} \sqrt{3} a \operatorname{atan}{\left (\frac{\sqrt{3}}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} \left (x + 1\right )^{\frac{2}{3}}}{3 \sqrt [3]{- x + 1}} \right )}}{12} - \frac{\sqrt [3]{2} \sqrt{3} a \operatorname{atan}{\left (\frac{2^{\frac{2}{3}} \sqrt{3} \left (- x + 1\right )^{\frac{2}{3}}}{3 \sqrt [3]{x + 1}} - \frac{\sqrt{3}}{3} \right )}}{12} - \frac{\sqrt [3]{2} b \log{\left (x^{2} + 3 \right )}}{8} + \frac{3 \sqrt [3]{2} b \log{\left (- \sqrt [3]{- x^{2} + 1} + 2^{\frac{2}{3}} \right )}}{8} + \frac{\sqrt [3]{2} \sqrt{3} b \operatorname{atan}{\left (\sqrt{3} \left (\frac{\sqrt [3]{2} \sqrt [3]{- x^{2} + 1}}{3} + \frac{1}{3}\right ) \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)/(-x**2+1)**(1/3)/(x**2+3),x)

[Out]

2**(1/3)*a*log(2**(1/3)*(-x + 1)**(1/3) + (x + 1)**(2/3))/8 - 2**(1/3)*a*log((-x
+ 1)**(2/3) + 2**(1/3)*(x + 1)**(1/3))/8 - 2**(1/3)*sqrt(3)*a*atan(sqrt(3)/3 -
2**(2/3)*sqrt(3)*(x + 1)**(2/3)/(3*(-x + 1)**(1/3)))/12 - 2**(1/3)*sqrt(3)*a*ata
n(2**(2/3)*sqrt(3)*(-x + 1)**(2/3)/(3*(x + 1)**(1/3)) - sqrt(3)/3)/12 - 2**(1/3)
*b*log(x**2 + 3)/8 + 3*2**(1/3)*b*log(-(-x**2 + 1)**(1/3) + 2**(2/3))/8 + 2**(1/
3)*sqrt(3)*b*atan(sqrt(3)*(2**(1/3)*(-x**2 + 1)**(1/3)/3 + 1/3))/4

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Mathematica [C]  time = 0.468093, size = 205, normalized size = 1.04 \[ \frac{3 x \left (\frac{3 a F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )}{2 x^2 \left (F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )-F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right )\right )+9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )}+\frac{b x F_1\left (1;\frac{1}{3},1;2;x^2,-\frac{x^2}{3}\right )}{x^2 \left (F_1\left (2;\frac{4}{3},1;3;x^2,-\frac{x^2}{3}\right )-F_1\left (2;\frac{1}{3},2;3;x^2,-\frac{x^2}{3}\right )\right )+6 F_1\left (1;\frac{1}{3},1;2;x^2,-\frac{x^2}{3}\right )}\right )}{\sqrt [3]{1-x^2} \left (x^2+3\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x)/((1 - x^2)^(1/3)*(3 + x^2)),x]

[Out]

(3*x*((3*a*AppellF1[1/2, 1/3, 1, 3/2, x^2, -x^2/3])/(9*AppellF1[1/2, 1/3, 1, 3/2
, x^2, -x^2/3] + 2*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, x^2, -x^2/3] + AppellF1[3/2,
4/3, 1, 5/2, x^2, -x^2/3])) + (b*x*AppellF1[1, 1/3, 1, 2, x^2, -x^2/3])/(6*Appe
llF1[1, 1/3, 1, 2, x^2, -x^2/3] + x^2*(-AppellF1[2, 1/3, 2, 3, x^2, -x^2/3] + Ap
pellF1[2, 4/3, 1, 3, x^2, -x^2/3]))))/((1 - x^2)^(1/3)*(3 + x^2))

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int{\frac{bx+a}{{x}^{2}+3}{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(-x^2+1)^(1/3)/(x^2+3),x)

[Out]

int((b*x+a)/(-x^2+1)^(1/3)/(x^2+3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{b x + a}{{\left (x^{2} + 3\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/((x^2 + 3)*(-x^2 + 1)^(1/3)),x, algorithm="maxima")

[Out]

integrate((b*x + a)/((x^2 + 3)*(-x^2 + 1)^(1/3)), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/((x^2 + 3)*(-x^2 + 1)^(1/3)),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{a + b x}{\sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)/(-x**2+1)**(1/3)/(x**2+3),x)

[Out]

Integral((a + b*x)/((-(x - 1)*(x + 1))**(1/3)*(x**2 + 3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{b x + a}{{\left (x^{2} + 3\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/((x^2 + 3)*(-x^2 + 1)^(1/3)),x, algorithm="giac")

[Out]

integrate((b*x + a)/((x^2 + 3)*(-x^2 + 1)^(1/3)), x)