\(\int \frac {\sqrt {2+3 x}}{a-b x^2} \, dx\) [653]

Optimal result

Integrand size = 20, antiderivative size = 132 \[ \int \frac {\sqrt {2+3 x}}{a-b x^2} \, dx=-\frac {\sqrt {3 \sqrt {a}-2 \sqrt {b}} \arctan \left (\frac {\sqrt [4]{b} \sqrt {2+3 x}}{\sqrt {3 \sqrt {a}-2 \sqrt {b}}}\right )}{\sqrt {a} b^{3/4}}+\frac {\sqrt {3 \sqrt {a}+2 \sqrt {b}} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {2+3 x}}{\sqrt {3 \sqrt {a}+2 \sqrt {b}}}\right )}{\sqrt {a} b^{3/4}} \]

[Out]

-arctan(b^(1/4)*(2+3*x)^(1/2)/(3*a^(1/2)-2*b^(1/2))^(1/2))*(3*a^(1/2)-2*b^(1/2))^(1/2)/b^(3/4)/a^(1/2)+arctanh
(b^(1/4)*(2+3*x)^(1/2)/(3*a^(1/2)+2*b^(1/2))^(1/2))*(3*a^(1/2)+2*b^(1/2))^(1/2)/b^(3/4)/a^(1/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {714, 1144, 214, 211} \[ \int \frac {\sqrt {2+3 x}}{a-b x^2} \, dx=\frac {\sqrt {3 \sqrt {a}+2 \sqrt {b}} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {3 x+2}}{\sqrt {3 \sqrt {a}+2 \sqrt {b}}}\right )}{\sqrt {a} b^{3/4}}-\frac {\sqrt {3 \sqrt {a}-2 \sqrt {b}} \arctan \left (\frac {\sqrt [4]{b} \sqrt {3 x+2}}{\sqrt {3 \sqrt {a}-2 \sqrt {b}}}\right )}{\sqrt {a} b^{3/4}} \]

[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 211

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

Rule 214

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

Rule 714

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 1144

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

Rubi steps \begin{align*} \text {integral}& = 6 \text {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 ) \text {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 ) \text {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] (verified)

Time = 0.32 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.14 \[ \int \frac {\sqrt {2+3 x}}{a-b x^2} \, dx=\frac {-\sqrt {-3 \sqrt {a} \sqrt {b}-2 b} \arctan \left (\frac {\sqrt {-3 \sqrt {a} \sqrt {b}-2 b} \sqrt {2+3 x}}{3 \sqrt {a}+2 \sqrt {b}}\right )-\sqrt {3 \sqrt {a} \sqrt {b}-2 b} \arctan \left (\frac {\sqrt {3 \sqrt {a} \sqrt {b}-2 b} \sqrt {2+3 x}}{3 \sqrt {a}-2 \sqrt {b}}\right )}{\sqrt {a} b} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 3.08 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.86

method	result	size
pseudoelliptic	\(-\frac {-\frac {\left (3 \sqrt {a b}+2 b \right ) \operatorname {arctanh}\left (\frac {b \sqrt {2+3 x}}{\sqrt {\left (3 \sqrt {a b}+2 b \right ) b}}\right )}{\sqrt {\left (3 \sqrt {a b}+2 b \right ) b}}+\frac {\left (3 \sqrt {a b}-2 b \right ) \arctan \left (\frac {b \sqrt {2+3 x}}{\sqrt {\left (3 \sqrt {a b}-2 b \right ) b}}\right )}{\sqrt {\left (3 \sqrt {a b}-2 b \right ) b}}}{\sqrt {a b}}\)	\(114\)
derivativedivides	\(-6 b \left (-\frac {\left (3 \sqrt {a b}+2 b \right ) \operatorname {arctanh}\left (\frac {b \sqrt {2+3 x}}{\sqrt {\left (3 \sqrt {a b}+2 b \right ) b}}\right )}{6 b \sqrt {a b}\, \sqrt {\left (3 \sqrt {a b}+2 b \right ) b}}+\frac {\left (3 \sqrt {a b}-2 b \right ) \arctan \left (\frac {b \sqrt {2+3 x}}{\sqrt {\left (3 \sqrt {a b}-2 b \right ) b}}\right )}{6 b \sqrt {a b}\, \sqrt {\left (3 \sqrt {a b}-2 b \right ) b}}\right )\)	\(127\)
default	\(-6 b \left (-\frac {\left (3 \sqrt {a b}+2 b \right ) \operatorname {arctanh}\left (\frac {b \sqrt {2+3 x}}{\sqrt {\left (3 \sqrt {a b}+2 b \right ) b}}\right )}{6 b \sqrt {a b}\, \sqrt {\left (3 \sqrt {a b}+2 b \right ) b}}+\frac {\left (3 \sqrt {a b}-2 b \right ) \arctan \left (\frac {b \sqrt {2+3 x}}{\sqrt {\left (3 \sqrt {a b}-2 b \right ) b}}\right )}{6 b \sqrt {a b}\, \sqrt {\left (3 \sqrt {a b}-2 b \right ) b}}\right )\)	\(127\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (92) = 184\).

Time = 0.51 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.27 \[ \int \frac {\sqrt {2+3 x}}{a-b x^2} \, dx=\frac {1}{2} \, \sqrt {\frac {3 \, a b \sqrt {\frac {1}{a b^{3}}} + 2}{a b}} \log \left (a b^{2} \sqrt {\frac {3 \, a b \sqrt {\frac {1}{a b^{3}}} + 2}{a b}} \sqrt {\frac {1}{a b^{3}}} + \sqrt {3 \, x + 2}\right ) - \frac {1}{2} \, \sqrt {\frac {3 \, a b \sqrt {\frac {1}{a b^{3}}} + 2}{a b}} \log \left (-a b^{2} \sqrt {\frac {3 \, a b \sqrt {\frac {1}{a b^{3}}} + 2}{a b}} \sqrt {\frac {1}{a b^{3}}} + \sqrt {3 \, x + 2}\right ) - \frac {1}{2} \, \sqrt {-\frac {3 \, a b \sqrt {\frac {1}{a b^{3}}} - 2}{a b}} \log \left (a b^{2} \sqrt {-\frac {3 \, a b \sqrt {\frac {1}{a b^{3}}} - 2}{a b}} \sqrt {\frac {1}{a b^{3}}} + \sqrt {3 \, x + 2}\right ) + \frac {1}{2} \, \sqrt {-\frac {3 \, a b \sqrt {\frac {1}{a b^{3}}} - 2}{a b}} \log \left (-a b^{2} \sqrt {-\frac {3 \, a b \sqrt {\frac {1}{a b^{3}}} - 2}{a b}} \sqrt {\frac {1}{a b^{3}}} + \sqrt {3 \, x + 2}\right ) \]

[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))

Sympy [F]

\[ \int \frac {\sqrt {2+3 x}}{a-b x^2} \, dx=- \int \frac {\sqrt {3 x + 2}}{- a + b x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {2+3 x}}{a-b x^2} \, dx=\int { -\frac {\sqrt {3 \, x + 2}}{b x^{2} - a} \,d x } \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (92) = 184\).

Time = 0.40 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.64 \[ \int \frac {\sqrt {2+3 x}}{a-b x^2} \, dx=-\frac {{\left (4 \, \sqrt {a b} \sqrt {-2 \, b^{2} - 3 \, \sqrt {a b} b} a + 17 \, \sqrt {a b} \sqrt {-2 \, b^{2} - 3 \, \sqrt {a b} b} b\right )} {\left | b \right |} \arctan \left (\frac {\sqrt {3 \, x + 2}}{\sqrt {-\frac {2 \, b + \sqrt {{\left (9 \, a - 4 \, b\right )} b + 4 \, b^{2}}}{b}}}\right )}{4 \, a^{2} b^{3} + 17 \, a b^{4}} + \frac {{\left (4 \, \sqrt {a b} \sqrt {-2 \, b^{2} + 3 \, \sqrt {a b} b} a + 17 \, \sqrt {a b} \sqrt {-2 \, b^{2} + 3 \, \sqrt {a b} b} b\right )} {\left | b \right |} \arctan \left (\frac {\sqrt {3 \, x + 2}}{\sqrt {-\frac {2 \, b - \sqrt {{\left (9 \, a - 4 \, b\right )} b + 4 \, b^{2}}}{b}}}\right )}{4 \, a^{2} b^{3} + 17 \, a b^{4}} \]

[In]

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

[Out]

-(4*sqrt(a*b)*sqrt(-2*b^2 - 3*sqrt(a*b)*b)*a + 17*sqrt(a*b)*sqrt(-2*b^2 - 3*sqrt(a*b)*b)*b)*abs(b)*arctan(sqrt
(3*x + 2)/sqrt(-(2*b + sqrt((9*a - 4*b)*b + 4*b^2))/b))/(4*a^2*b^3 + 17*a*b^4) + (4*sqrt(a*b)*sqrt(-2*b^2 + 3*
sqrt(a*b)*b)*a + 17*sqrt(a*b)*sqrt(-2*b^2 + 3*sqrt(a*b)*b)*b)*abs(b)*arctan(sqrt(3*x + 2)/sqrt(-(2*b - sqrt((9
*a - 4*b)*b + 4*b^2))/b))/(4*a^2*b^3 + 17*a*b^4)

Mupad [B] (verification not implemented)

Time = 0.76 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.93 \[ \int \frac {\sqrt {2+3 x}}{a-b x^2} \, dx=2\,\mathrm {atanh}\left (\frac {2\,\left (\left (576\,b^3+1296\,a\,b^2\right )\,\sqrt {3\,x+2}+\frac {288\,b\,\sqrt {3\,x+2}\,\left (3\,\sqrt {a^3\,b^3}-2\,a\,b^2\right )}{a}\right )\,\sqrt {-\frac {3\,\sqrt {a^3\,b^3}-2\,a\,b^2}{4\,a^2\,b^3}}}{3888\,a\,b-1728\,b^2}\right )\,\sqrt {-\frac {3\,\sqrt {a^3\,b^3}-2\,a\,b^2}{4\,a^2\,b^3}}+2\,\mathrm {atanh}\left (\frac {2\,\left (\left (576\,b^3+1296\,a\,b^2\right )\,\sqrt {3\,x+2}-\frac {288\,b\,\sqrt {3\,x+2}\,\left (3\,\sqrt {a^3\,b^3}+2\,a\,b^2\right )}{a}\right )\,\sqrt {\frac {3\,\sqrt {a^3\,b^3}+2\,a\,b^2}{4\,a^2\,b^3}}}{3888\,a\,b-1728\,b^2}\right )\,\sqrt {\frac {3\,\sqrt {a^3\,b^3}+2\,a\,b^2}{4\,a^2\,b^3}} \]

[In]

int((3*x + 2)^(1/2)/(a - b*x^2),x)

[Out]

2*atanh((2*((1296*a*b^2 + 576*b^3)*(3*x + 2)^(1/2) + (288*b*(3*x + 2)^(1/2)*(3*(a^3*b^3)^(1/2) - 2*a*b^2))/a)*
(-(3*(a^3*b^3)^(1/2) - 2*a*b^2)/(4*a^2*b^3))^(1/2))/(3888*a*b - 1728*b^2))*(-(3*(a^3*b^3)^(1/2) - 2*a*b^2)/(4*
a^2*b^3))^(1/2) + 2*atanh((2*((1296*a*b^2 + 576*b^3)*(3*x + 2)^(1/2) - (288*b*(3*x + 2)^(1/2)*(3*(a^3*b^3)^(1/
2) + 2*a*b^2))/a)*((3*(a^3*b^3)^(1/2) + 2*a*b^2)/(4*a^2*b^3))^(1/2))/(3888*a*b - 1728*b^2))*((3*(a^3*b^3)^(1/2
) + 2*a*b^2)/(4*a^2*b^3))^(1/2)