3.642 \(\int \frac{\sqrt{2+3 x}}{a-b x^2} \, dx\)

Optimal. Leaf size=132 \[ \frac{\sqrt{3 \sqrt{a}+2 \sqrt{b}} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{a}+2 \sqrt{b}}}\right )}{\sqrt{a} b^{3/4}}-\frac{\sqrt{3 \sqrt{a}-2 \sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{a}-2 \sqrt{b}}}\right )}{\sqrt{a} b^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.293787, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\sqrt{3 \sqrt{a}+2 \sqrt{b}} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{a}+2 \sqrt{b}}}\right )}{\sqrt{a} b^{3/4}}-\frac{\sqrt{3 \sqrt{a}-2 \sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{a}-2 \sqrt{b}}}\right )}{\sqrt{a} b^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 37.5344, size = 117, normalized size = 0.89 \[ - \frac{\sqrt{3 \sqrt{a} - 2 \sqrt{b}} \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{3 x + 2}}{\sqrt{3 \sqrt{a} - 2 \sqrt{b}}} \right )}}{\sqrt{a} b^{\frac{3}{4}}} + \frac{\sqrt{3 \sqrt{a} + 2 \sqrt{b}} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt{3 x + 2}}{\sqrt{3 \sqrt{a} + 2 \sqrt{b}}} \right )}}{\sqrt{a} b^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(3*sqrt(a) - 2*sqrt(b))*atan(b**(1/4)*sqrt(3*x + 2)/sqrt(3*sqrt(a) - 2*sqrt
(b)))/(sqrt(a)*b**(3/4)) + sqrt(3*sqrt(a) + 2*sqrt(b))*atanh(b**(1/4)*sqrt(3*x +
2)/sqrt(3*sqrt(a) + 2*sqrt(b)))/(sqrt(a)*b**(3/4))

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Mathematica [A]  time = 0.247226, size = 158, normalized size = 1.2 \[ \frac{-\frac{\left (3 \sqrt{a}+2 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{3 x+2}}{\sqrt{-3 \sqrt{a} \sqrt{b}-2 b}}\right )}{\sqrt{-3 \sqrt{a} \sqrt{b}-2 b}}-\frac{\left (3 \sqrt{a}-2 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{a} \sqrt{b}-2 b}}\right )}{\sqrt{3 \sqrt{a} \sqrt{b}-2 b}}}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

(-(((3*Sqrt[a] + 2*Sqrt[b])*ArcTan[(Sqrt[b]*Sqrt[2 + 3*x])/Sqrt[-3*Sqrt[a]*Sqrt[
b] - 2*b]])/Sqrt[-3*Sqrt[a]*Sqrt[b] - 2*b]) - ((3*Sqrt[a] - 2*Sqrt[b])*ArcTan[(S
qrt[b]*Sqrt[2 + 3*x])/Sqrt[3*Sqrt[a]*Sqrt[b] - 2*b]])/Sqrt[3*Sqrt[a]*Sqrt[b] - 2
*b])/(Sqrt[a]*Sqrt[b])

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Maple [A]  time = 0.045, size = 182, normalized size = 1.4 \[ -3\,{\frac{1}{\sqrt{ \left ( 3\,\sqrt{ab}-2\,b \right ) b}}\arctan \left ({\frac{\sqrt{2+3\,x}b}{\sqrt{ \left ( 3\,\sqrt{ab}-2\,b \right ) b}}} \right ) }+2\,{\frac{b}{\sqrt{ab}\sqrt{ \left ( 3\,\sqrt{ab}-2\,b \right ) b}}\arctan \left ({\frac{\sqrt{2+3\,x}b}{\sqrt{ \left ( 3\,\sqrt{ab}-2\,b \right ) b}}} \right ) }+3\,{\frac{1}{\sqrt{ \left ( 3\,\sqrt{ab}+2\,b \right ) b}}{\it Artanh} \left ({\frac{\sqrt{2+3\,x}b}{\sqrt{ \left ( 3\,\sqrt{ab}+2\,b \right ) b}}} \right ) }+2\,{\frac{b}{\sqrt{ab}\sqrt{ \left ( 3\,\sqrt{ab}+2\,b \right ) b}}{\it Artanh} \left ({\frac{\sqrt{2+3\,x}b}{\sqrt{ \left ( 3\,\sqrt{ab}+2\,b \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^(1/2)/(-b*x^2+a),x)

[Out]

-3/((3*(a*b)^(1/2)-2*b)*b)^(1/2)*arctan((2+3*x)^(1/2)*b/((3*(a*b)^(1/2)-2*b)*b)^
(1/2))+2*b/(a*b)^(1/2)/((3*(a*b)^(1/2)-2*b)*b)^(1/2)*arctan((2+3*x)^(1/2)*b/((3*
(a*b)^(1/2)-2*b)*b)^(1/2))+3/((3*(a*b)^(1/2)+2*b)*b)^(1/2)*arctanh((2+3*x)^(1/2)
*b/((3*(a*b)^(1/2)+2*b)*b)^(1/2))+2*b/(a*b)^(1/2)/((3*(a*b)^(1/2)+2*b)*b)^(1/2)*
arctanh((2+3*x)^(1/2)*b/((3*(a*b)^(1/2)+2*b)*b)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.220618, size = 404, normalized size = 3.06 \[ \frac{1}{2} \, \sqrt{\frac{3 \, a b \sqrt{\frac{1}{a b^{3}}} + 2}{a b}} \log \left (a b^{2} \sqrt{\frac{3 \, a b \sqrt{\frac{1}{a b^{3}}} + 2}{a b}} \sqrt{\frac{1}{a b^{3}}} + \sqrt{3 \, x + 2}\right ) - \frac{1}{2} \, \sqrt{\frac{3 \, a b \sqrt{\frac{1}{a b^{3}}} + 2}{a b}} \log \left (-a b^{2} \sqrt{\frac{3 \, a b \sqrt{\frac{1}{a b^{3}}} + 2}{a b}} \sqrt{\frac{1}{a b^{3}}} + \sqrt{3 \, x + 2}\right ) - \frac{1}{2} \, \sqrt{-\frac{3 \, a b \sqrt{\frac{1}{a b^{3}}} - 2}{a b}} \log \left (a b^{2} \sqrt{-\frac{3 \, a b \sqrt{\frac{1}{a b^{3}}} - 2}{a b}} \sqrt{\frac{1}{a b^{3}}} + \sqrt{3 \, x + 2}\right ) + \frac{1}{2} \, \sqrt{-\frac{3 \, a b \sqrt{\frac{1}{a b^{3}}} - 2}{a b}} \log \left (-a b^{2} \sqrt{-\frac{3 \, a b \sqrt{\frac{1}{a b^{3}}} - 2}{a b}} \sqrt{\frac{1}{a b^{3}}} + \sqrt{3 \, x + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt((3*a*b*sqrt(1/(a*b^3)) + 2)/(a*b))*log(a*b^2*sqrt((3*a*b*sqrt(1/(a*b^3)
) + 2)/(a*b))*sqrt(1/(a*b^3)) + sqrt(3*x + 2)) - 1/2*sqrt((3*a*b*sqrt(1/(a*b^3))
+ 2)/(a*b))*log(-a*b^2*sqrt((3*a*b*sqrt(1/(a*b^3)) + 2)/(a*b))*sqrt(1/(a*b^3))
+ sqrt(3*x + 2)) - 1/2*sqrt(-(3*a*b*sqrt(1/(a*b^3)) - 2)/(a*b))*log(a*b^2*sqrt(-
(3*a*b*sqrt(1/(a*b^3)) - 2)/(a*b))*sqrt(1/(a*b^3)) + sqrt(3*x + 2)) + 1/2*sqrt(-
(3*a*b*sqrt(1/(a*b^3)) - 2)/(a*b))*log(-a*b^2*sqrt(-(3*a*b*sqrt(1/(a*b^3)) - 2)/
(a*b))*sqrt(1/(a*b^3)) + sqrt(3*x + 2))

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Sympy [A]  time = 11.3319, size = 58, normalized size = 0.44 \[ - 6 \operatorname{RootSum}{\left (20736 t^{4} a^{2} b^{3} - 576 t^{2} a b^{2} - 9 a + 4 b, \left ( t \mapsto t \log{\left (- 576 t^{3} a b^{2} + 8 t b + \sqrt{3 x + 2} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6*RootSum(20736*_t**4*a**2*b**3 - 576*_t**2*a*b**2 - 9*a + 4*b, Lambda(_t, _t*l
og(-576*_t**3*a*b**2 + 8*_t*b + sqrt(3*x + 2))))

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GIAC/XCAS [A]  time = 44.0252, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done