### 3.653 $$\int \frac{\sqrt{2+3 x}}{a-b x^2} \, dx$$

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)*Sqrt[2 + 3*x])/Sqrt[3*Sqrt[a] + 2*Sqrt[b]]])/(Sqrt[a]*b^(3/4)
)

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Rubi [A]  time = 0.132167, 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, Rules used = {700, 1130, 208, 205} $\frac{\sqrt{3 \sqrt{a}+2 \sqrt{b}} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{a}+2 \sqrt{b}}}\right )}{\sqrt{a} b^{3/4}}-\frac{\sqrt{3 \sqrt{a}-2 \sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{a}-2 \sqrt{b}}}\right )}{\sqrt{a} b^{3/4}}$

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)*Sqrt[2 + 3*x])/Sqrt[3*Sqrt[a] + 2*Sqrt[b]]])/(Sqrt[a]*b^(3/4)
)

Rule 700

Int[Sqrt[(d_) + (e_.)*(x_)]/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[2*e, Subst[Int[x^2/(c*d^2 + a*e^2 - 2*c*d
*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]

Rule 1130

Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(
d^2*(b/q + 1))/2, Int[(d*x)^(m - 2)/(b/2 + q/2 + c*x^2), x], x] - Dist[(d^2*(b/q - 1))/2, Int[(d*x)^(m - 2)/(b
/2 - q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\sqrt{2+3 x}}{a-b x^2} \, dx &=6 \operatorname{Subst}\left (\int \frac{x^2}{9 a-4 b+4 b x^2-b x^4} \, dx,x,\sqrt{2+3 x}\right )\\ &=\left (3-\frac{2 \sqrt{b}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{-3 \sqrt{a} \sqrt{b}+2 b-b x^2} \, dx,x,\sqrt{2+3 x}\right )+\left (3+\frac{2 \sqrt{b}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{3 \sqrt{a} \sqrt{b}+2 b-b x^2} \, dx,x,\sqrt{2+3 x}\right )\\ &=-\frac{\sqrt{3 \sqrt{a}-2 \sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{2+3 x}}{\sqrt{3 \sqrt{a}-2 \sqrt{b}}}\right )}{\sqrt{a} b^{3/4}}+\frac{\sqrt{3 \sqrt{a}+2 \sqrt{b}} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{2+3 x}}{\sqrt{3 \sqrt{a}+2 \sqrt{b}}}\right )}{\sqrt{a} b^{3/4}}\\ \end{align*}

Mathematica [A]  time = 0.0932695, size = 123, normalized size = 0.93 $\frac{\sqrt{3 \sqrt{a}+2 \sqrt{b}} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{a}+2 \sqrt{b}}}\right )-\sqrt{3 \sqrt{a}-2 \sqrt{b}} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{3 x+2}}{\sqrt{3 \sqrt{a}-2 \sqrt{b}}}\right )}{\sqrt{a} b^{3/4}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[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[3*Sqrt[a] +
2*Sqrt[b]]*ArcTanh[(b^(1/4)*Sqrt[2 + 3*x])/Sqrt[3*Sqrt[a] + 2*Sqrt[b]]])/(Sqrt[a]*b^(3/4))

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Maple [A]  time = 0.158, size = 182, normalized size = 1.4 \begin{align*} 3\,{\frac{1}{\sqrt{ \left ( 3\,\sqrt{ab}+2\,b \right ) b}}{\it Artanh} \left ({\frac{b\sqrt{2+3\,x}}{\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{b\sqrt{2+3\,x}}{\sqrt{ \left ( 3\,\sqrt{ab}+2\,b \right ) b}}} \right ) }-3\,{\frac{1}{\sqrt{ \left ( 3\,\sqrt{ab}-2\,b \right ) b}}\arctan \left ({\frac{b\sqrt{2+3\,x}}{\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{b\sqrt{2+3\,x}}{\sqrt{ \left ( 3\,\sqrt{ab}-2\,b \right ) b}}} \right ) } \end{align*}

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.2393, size = 703, normalized size = 5.33 \begin{align*} \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 ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(1/2)/(-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*sqr
t(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 = 4.56618, size = 58, normalized size = 0.44 \begin{align*} - 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 )} \end{align*}

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*log(-576*_t**3*a*b**2 + 8*_t*b
+ sqrt(3*x + 2))))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError