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

Optimal. Leaf size=427 \[ \frac{3 \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt{3 x+2} \sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}+\sqrt{9 a+4 b}+\sqrt{b} (3 x+2)\right )}{2 \sqrt{2} b^{3/4} \sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}}-\frac{3 \log \left (\sqrt{2} \sqrt [4]{b} \sqrt{3 x+2} \sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}+\sqrt{9 a+4 b}+\sqrt{b} (3 x+2)\right )}{2 \sqrt{2} b^{3/4} \sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}-\sqrt{2} \sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}}\right )}{\sqrt{2} b^{3/4} \sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}+\sqrt{2} \sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}}\right )}{\sqrt{2} b^{3/4} \sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}} \]

[Out]

(3*ArcTanh[(Sqrt[2*Sqrt[b] + Sqrt[9*a + 4*b]] - Sqrt[2]*b^(1/4)*Sqrt[2 + 3*x])/S
qrt[2*Sqrt[b] - Sqrt[9*a + 4*b]]])/(Sqrt[2]*b^(3/4)*Sqrt[2*Sqrt[b] - Sqrt[9*a +
4*b]]) - (3*ArcTanh[(Sqrt[2*Sqrt[b] + Sqrt[9*a + 4*b]] + Sqrt[2]*b^(1/4)*Sqrt[2
+ 3*x])/Sqrt[2*Sqrt[b] - Sqrt[9*a + 4*b]]])/(Sqrt[2]*b^(3/4)*Sqrt[2*Sqrt[b] - Sq
rt[9*a + 4*b]]) + (3*Log[Sqrt[9*a + 4*b] - Sqrt[2]*b^(1/4)*Sqrt[2*Sqrt[b] + Sqrt
[9*a + 4*b]]*Sqrt[2 + 3*x] + Sqrt[b]*(2 + 3*x)])/(2*Sqrt[2]*b^(3/4)*Sqrt[2*Sqrt[
b] + Sqrt[9*a + 4*b]]) - (3*Log[Sqrt[9*a + 4*b] + Sqrt[2]*b^(1/4)*Sqrt[2*Sqrt[b]
+ Sqrt[9*a + 4*b]]*Sqrt[2 + 3*x] + Sqrt[b]*(2 + 3*x)])/(2*Sqrt[2]*b^(3/4)*Sqrt[
2*Sqrt[b] + Sqrt[9*a + 4*b]])

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Rubi [A]  time = 1.21866, antiderivative size = 427, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{3 \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt{3 x+2} \sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}+\sqrt{9 a+4 b}+\sqrt{b} (3 x+2)\right )}{2 \sqrt{2} b^{3/4} \sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}}-\frac{3 \log \left (\sqrt{2} \sqrt [4]{b} \sqrt{3 x+2} \sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}+\sqrt{9 a+4 b}+\sqrt{b} (3 x+2)\right )}{2 \sqrt{2} b^{3/4} \sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}-\sqrt{2} \sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}}\right )}{\sqrt{2} b^{3/4} \sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{\sqrt{9 a+4 b}+2 \sqrt{b}}+\sqrt{2} \sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}}\right )}{\sqrt{2} b^{3/4} \sqrt{2 \sqrt{b}-\sqrt{9 a+4 b}}} \]

Antiderivative was successfully verified.

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

[Out]

(3*ArcTanh[(Sqrt[2*Sqrt[b] + Sqrt[9*a + 4*b]] - Sqrt[2]*b^(1/4)*Sqrt[2 + 3*x])/S
qrt[2*Sqrt[b] - Sqrt[9*a + 4*b]]])/(Sqrt[2]*b^(3/4)*Sqrt[2*Sqrt[b] - Sqrt[9*a +
4*b]]) - (3*ArcTanh[(Sqrt[2*Sqrt[b] + Sqrt[9*a + 4*b]] + Sqrt[2]*b^(1/4)*Sqrt[2
+ 3*x])/Sqrt[2*Sqrt[b] - Sqrt[9*a + 4*b]]])/(Sqrt[2]*b^(3/4)*Sqrt[2*Sqrt[b] - Sq
rt[9*a + 4*b]]) + (3*Log[Sqrt[9*a + 4*b] - Sqrt[2]*b^(1/4)*Sqrt[2*Sqrt[b] + Sqrt
[9*a + 4*b]]*Sqrt[2 + 3*x] + Sqrt[b]*(2 + 3*x)])/(2*Sqrt[2]*b^(3/4)*Sqrt[2*Sqrt[
b] + Sqrt[9*a + 4*b]]) - (3*Log[Sqrt[9*a + 4*b] + Sqrt[2]*b^(1/4)*Sqrt[2*Sqrt[b]
+ Sqrt[9*a + 4*b]]*Sqrt[2 + 3*x] + Sqrt[b]*(2 + 3*x)])/(2*Sqrt[2]*b^(3/4)*Sqrt[
2*Sqrt[b] + Sqrt[9*a + 4*b]])

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Rubi in Sympy [A]  time = 99.6886, size = 386, normalized size = 0.9 \[ \frac{3 \sqrt{2} \log{\left (3 x + 2 + \frac{\sqrt{9 a + 4 b}}{\sqrt{b}} - \frac{\sqrt{2} \sqrt{2 \sqrt{b} + \sqrt{9 a + 4 b}} \sqrt{3 x + 2}}{\sqrt [4]{b}} \right )}}{4 b^{\frac{3}{4}} \sqrt{2 \sqrt{b} + \sqrt{9 a + 4 b}}} - \frac{3 \sqrt{2} \log{\left (3 x + 2 + \frac{\sqrt{9 a + 4 b}}{\sqrt{b}} + \frac{\sqrt{2} \sqrt{2 \sqrt{b} + \sqrt{9 a + 4 b}} \sqrt{3 x + 2}}{\sqrt [4]{b}} \right )}}{4 b^{\frac{3}{4}} \sqrt{2 \sqrt{b} + \sqrt{9 a + 4 b}}} - \frac{3 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (\sqrt [4]{b} \sqrt{3 x + 2} - \frac{\sqrt{4 \sqrt{b} + 2 \sqrt{9 a + 4 b}}}{2}\right )}{\sqrt{2 \sqrt{b} - \sqrt{9 a + 4 b}}} \right )}}{2 b^{\frac{3}{4}} \sqrt{2 \sqrt{b} - \sqrt{9 a + 4 b}}} - \frac{3 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (\sqrt [4]{b} \sqrt{3 x + 2} + \frac{\sqrt{4 \sqrt{b} + 2 \sqrt{9 a + 4 b}}}{2}\right )}{\sqrt{2 \sqrt{b} - \sqrt{9 a + 4 b}}} \right )}}{2 b^{\frac{3}{4}} \sqrt{2 \sqrt{b} - \sqrt{9 a + 4 b}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*sqrt(2)*log(3*x + 2 + sqrt(9*a + 4*b)/sqrt(b) - sqrt(2)*sqrt(2*sqrt(b) + sqrt(
9*a + 4*b))*sqrt(3*x + 2)/b**(1/4))/(4*b**(3/4)*sqrt(2*sqrt(b) + sqrt(9*a + 4*b)
)) - 3*sqrt(2)*log(3*x + 2 + sqrt(9*a + 4*b)/sqrt(b) + sqrt(2)*sqrt(2*sqrt(b) +
sqrt(9*a + 4*b))*sqrt(3*x + 2)/b**(1/4))/(4*b**(3/4)*sqrt(2*sqrt(b) + sqrt(9*a +
4*b))) - 3*sqrt(2)*atanh(sqrt(2)*(b**(1/4)*sqrt(3*x + 2) - sqrt(4*sqrt(b) + 2*s
qrt(9*a + 4*b))/2)/sqrt(2*sqrt(b) - sqrt(9*a + 4*b)))/(2*b**(3/4)*sqrt(2*sqrt(b)
- sqrt(9*a + 4*b))) - 3*sqrt(2)*atanh(sqrt(2)*(b**(1/4)*sqrt(3*x + 2) + sqrt(4*
sqrt(b) + 2*sqrt(9*a + 4*b))/2)/sqrt(2*sqrt(b) - sqrt(9*a + 4*b)))/(2*b**(3/4)*s
qrt(2*sqrt(b) - sqrt(9*a + 4*b)))

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Mathematica [C]  time = 0.34457, size = 186, normalized size = 0.44 \[ \frac{\sqrt [4]{-1} \left (\frac{\left (3 \sqrt{a}-2 i \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{a} \sqrt{b}-2 i b}}\right )}{\sqrt{3 \sqrt{a} \sqrt{b}-2 i b}}+\frac{i \left (3 \sqrt{a}+2 i \sqrt{b}\right ) \tan ^{-1}\left (\frac{(-1)^{3/4} \sqrt{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{a} \sqrt{b}+2 i b}}\right )}{\sqrt{3 \sqrt{a} \sqrt{b}+2 i b}}\right )}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.142, size = 944, normalized size = 2.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^(1/2)/(b*x^2+a),x)

[Out]

-1/12*(2*(9*a*b+4*b^2)^(1/2)+4*b)^(1/2)/a/b^(3/2)*(9*a*b+4*b^2)^(1/2)*ln((2+3*x)
*b^(1/2)+(2*(b*(9*a+4*b))^(1/2)+4*b)^(1/2)*(2+3*x)^(1/2)+(9*a+4*b)^(1/2))+1/6*(2
*(9*a*b+4*b^2)^(1/2)+4*b)^(1/2)/a/b^(3/2)*(9*a*b+4*b^2)^(1/2)*(2*(b*(9*a+4*b))^(
1/2)+4*b)^(1/2)/(4*(9*a+4*b)^(1/2)*b^(1/2)-2*(b*(9*a+4*b))^(1/2)-4*b)^(1/2)*arct
an((2*(2+3*x)^(1/2)*b^(1/2)+(2*(b*(9*a+4*b))^(1/2)+4*b)^(1/2))/(4*(9*a+4*b)^(1/2
)*b^(1/2)-2*(b*(9*a+4*b))^(1/2)-4*b)^(1/2))+1/6*(2*(9*a*b+4*b^2)^(1/2)+4*b)^(1/2
)/a/b^(1/2)*ln((2+3*x)*b^(1/2)+(2*(b*(9*a+4*b))^(1/2)+4*b)^(1/2)*(2+3*x)^(1/2)+(
9*a+4*b)^(1/2))-1/3*(2*(9*a*b+4*b^2)^(1/2)+4*b)^(1/2)/a/b^(1/2)*(2*(b*(9*a+4*b))
^(1/2)+4*b)^(1/2)/(4*(9*a+4*b)^(1/2)*b^(1/2)-2*(b*(9*a+4*b))^(1/2)-4*b)^(1/2)*ar
ctan((2*(2+3*x)^(1/2)*b^(1/2)+(2*(b*(9*a+4*b))^(1/2)+4*b)^(1/2))/(4*(9*a+4*b)^(1
/2)*b^(1/2)-2*(b*(9*a+4*b))^(1/2)-4*b)^(1/2))+1/12*(2*(9*a*b+4*b^2)^(1/2)+4*b)^(
1/2)/a/b^(3/2)*(9*a*b+4*b^2)^(1/2)*ln(-(2+3*x)*b^(1/2)+(2*(b*(9*a+4*b))^(1/2)+4*
b)^(1/2)*(2+3*x)^(1/2)-(9*a+4*b)^(1/2))-1/6*(2*(9*a*b+4*b^2)^(1/2)+4*b)^(1/2)/a/
b^(3/2)*(9*a*b+4*b^2)^(1/2)*(2*(b*(9*a+4*b))^(1/2)+4*b)^(1/2)/(4*(9*a+4*b)^(1/2)
*b^(1/2)-2*(b*(9*a+4*b))^(1/2)-4*b)^(1/2)*arctan((-2*(2+3*x)^(1/2)*b^(1/2)+(2*(b
*(9*a+4*b))^(1/2)+4*b)^(1/2))/(4*(9*a+4*b)^(1/2)*b^(1/2)-2*(b*(9*a+4*b))^(1/2)-4
*b)^(1/2))-1/6*(2*(9*a*b+4*b^2)^(1/2)+4*b)^(1/2)/a/b^(1/2)*ln(-(2+3*x)*b^(1/2)+(
2*(b*(9*a+4*b))^(1/2)+4*b)^(1/2)*(2+3*x)^(1/2)-(9*a+4*b)^(1/2))+1/3*(2*(9*a*b+4*
b^2)^(1/2)+4*b)^(1/2)/a/b^(1/2)*(2*(b*(9*a+4*b))^(1/2)+4*b)^(1/2)/(4*(9*a+4*b)^(
1/2)*b^(1/2)-2*(b*(9*a+4*b))^(1/2)-4*b)^(1/2)*arctan((-2*(2+3*x)^(1/2)*b^(1/2)+(
2*(b*(9*a+4*b))^(1/2)+4*b)^(1/2))/(4*(9*a+4*b)^(1/2)*b^(1/2)-2*(b*(9*a+4*b))^(1/
2)-4*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 = 0.219834, size = 420, normalized size = 0.98 \[ -\frac{1}{2} \, \sqrt{-\frac{3 \, a b \sqrt{-\frac{1}{a b^{3}}} + 2}{a b}} \log \left (a b^{2} \sqrt{-\frac{3 \, a b \sqrt{-\frac{1}{a b^{3}}} + 2}{a b}} \sqrt{-\frac{1}{a b^{3}}} + \sqrt{3 \, x + 2}\right ) + \frac{1}{2} \, \sqrt{-\frac{3 \, a b \sqrt{-\frac{1}{a b^{3}}} + 2}{a b}} \log \left (-a b^{2} \sqrt{-\frac{3 \, a b \sqrt{-\frac{1}{a b^{3}}} + 2}{a b}} \sqrt{-\frac{1}{a b^{3}}} + \sqrt{3 \, x + 2}\right ) + \frac{1}{2} \, \sqrt{\frac{3 \, a b \sqrt{-\frac{1}{a b^{3}}} - 2}{a b}} \log \left (a b^{2} \sqrt{\frac{3 \, a b \sqrt{-\frac{1}{a b^{3}}} - 2}{a b}} \sqrt{-\frac{1}{a b^{3}}} + \sqrt{3 \, x + 2}\right ) - \frac{1}{2} \, \sqrt{\frac{3 \, a b \sqrt{-\frac{1}{a b^{3}}} - 2}{a b}} \log \left (-a b^{2} \sqrt{\frac{3 \, a b \sqrt{-\frac{1}{a b^{3}}} - 2}{a b}} \sqrt{-\frac{1}{a b^{3}}} + \sqrt{3 \, x + 2}\right ) \]

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 = 10.7644, size = 56, normalized size = 0.13 \[ 6 \operatorname{RootSum}{\left (20736 t^{4} a^{2} b^{3} + 576 t^{2} a b^{2} + 9 a + 4 b, \left ( t \mapsto t \log{\left (576 t^{3} a b^{2} + 8 t b + \sqrt{3 x + 2} \right )} \right )\right )} \]

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*lo
g(576*_t**3*a*b**2 + 8*_t*b + sqrt(3*x + 2))))

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GIAC/XCAS [A]  time = 23.7987, size = 1, normalized size = 0. \[ \mathit{Done} \]

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