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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.21155, 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 \[ \frac{a \tan ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}\right )}{2\ 2^{2/3}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{x^2+1}\right )}{x}\right )}{2\ 2^{2/3} \sqrt{3}}-\frac{a \tan ^{-1}(x)}{6\ 2^{2/3}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3} \sqrt{3}}+\frac{b \log \left (3-x^2\right )}{4\ 2^{2/3}}-\frac{3 b \log \left (2^{2/3}-\sqrt [3]{x^2+1}\right )}{4\ 2^{2/3}}-\frac{\sqrt{3} b \tan ^{-1}\left (\frac{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}{\sqrt{3}}\right )}{2\ 2^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.1202, size = 282, normalized size = 1.42 \[ \frac{\sqrt [3]{2} \sqrt{3} a \log{\left (- x + \sqrt{3} \right )}}{24} - \frac{\sqrt [3]{2} \sqrt{3} a \log{\left (x + \sqrt{3} \right )}}{24} + \frac{\sqrt [3]{2} \sqrt{3} a \log{\left (- x - \sqrt [3]{2} \sqrt{3} \sqrt [3]{x^{2} + 1} + \sqrt{3} \right )}}{24} - \frac{\sqrt [3]{2} \sqrt{3} a \log{\left (x - \sqrt [3]{2} \sqrt{3} \sqrt [3]{x^{2} + 1} + \sqrt{3} \right )}}{24} - \frac{\sqrt [3]{2} a \operatorname{atan}{\left (\frac{2^{\frac{2}{3}} \left (- x + \sqrt{3}\right )}{3 \sqrt [3]{x^{2} + 1}} + \frac{\sqrt{3}}{3} \right )}}{12} + \frac{\sqrt [3]{2} a \operatorname{atan}{\left (\frac{2^{\frac{2}{3}} \left (x + \sqrt{3}\right )}{3 \sqrt [3]{x^{2} + 1}} + \frac{\sqrt{3}}{3} \right )}}{12} + \frac{\sqrt [3]{2} b \log{\left (- x^{2} + 3 \right )}}{8} - \frac{3 \sqrt [3]{2} b \log{\left (- \sqrt [3]{x^{2} + 1} + 2^{\frac{2}{3}} \right )}}{8} - \frac{\sqrt [3]{2} \sqrt{3} b \operatorname{atan}{\left (\sqrt{3} \left (\frac{\sqrt [3]{2} \sqrt [3]{x^{2} + 1}}{3} + \frac{1}{3}\right ) \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(1/3)*sqrt(3)*a*log(-x + sqrt(3))/24 - 2**(1/3)*sqrt(3)*a*log(x + sqrt(3))/24
+ 2**(1/3)*sqrt(3)*a*log(-x - 2**(1/3)*sqrt(3)*(x**2 + 1)**(1/3) + sqrt(3))/24
- 2**(1/3)*sqrt(3)*a*log(x - 2**(1/3)*sqrt(3)*(x**2 + 1)**(1/3) + sqrt(3))/24 -
2**(1/3)*a*atan(2**(2/3)*(-x + sqrt(3))/(3*(x**2 + 1)**(1/3)) + sqrt(3)/3)/12 +
2**(1/3)*a*atan(2**(2/3)*(x + sqrt(3))/(3*(x**2 + 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

_______________________________________________________________________________________

Mathematica [C]  time = 0.441686, size = 220, normalized size = 1.11 \[ \frac{3 x \left (-\frac{3 a F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-x^2,\frac{x^2}{3}\right )}{2 x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-x^2,\frac{x^2}{3}\right )-F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-x^2,\frac{x^2}{3}\right )\right )+9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-x^2,\frac{x^2}{3}\right )}-\frac{b x F_1\left (1;\frac{1}{3},1;2;-x^2,\frac{x^2}{3}\right )}{x^2 \left (F_1\left (2;\frac{1}{3},2;3;-x^2,\frac{x^2}{3}\right )-F_1\left (2;\frac{4}{3},1;3;-x^2,\frac{x^2}{3}\right )\right )+6 F_1\left (1;\frac{1}{3},1;2;-x^2,\frac{x^2}{3}\right )}\right )}{\left (x^2-3\right ) \sqrt [3]{x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x)/((3 - x^2)*(1 + x^2)^(1/3)),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] - App
ellF1[2, 4/3, 1, 3, -x^2, x^2/3]))))/((-3 + x^2)*(1 + x^2)^(1/3))

_______________________________________________________________________________________

Maple [F]  time = 0.073, 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+3)/(x^2+1)^(1/3),x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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 + 1)^(1/3)*(x^2 - 3)),x, algorithm="fricas")

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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