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

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

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Rubi [A]  time = 0.53, antiderivative size = 427, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {700, 1129, 634, 618, 206, 628} \begin {gather*} \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}}} \end {gather*}

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])/Sqrt[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]] + Sqr
t[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] - 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]]) - (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]])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

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 1129

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/
c, 2]}, Dist[1/(2*c*r), Int[x^(m - 1)/(q - r*x + x^2), x], x] - Dist[1/(2*c*r), Int[x^(m - 1)/(q + r*x + x^2),
x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 1] && LtQ[m, 3] && NegQ[b^2 - 4*a*c]

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

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Mathematica [A]  time = 0.14, size = 133, normalized size = 0.31 \begin {gather*} \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 {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}} \end {gather*}

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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IntegrateAlgebraic [C]  time = 0.36, size = 241, normalized size = 0.56 \begin {gather*} \frac {\left (3 \sqrt [4]{-1} \sqrt {a}-2 (-1)^{3/4} \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {3 x+2} \sqrt {3 \sqrt {a} \sqrt {b}-2 i b}}{3 \sqrt {a}-2 i \sqrt {b}}\right )}{\sqrt {a} \sqrt {b} \sqrt {\sqrt {b} \left (3 \sqrt {a}-2 i \sqrt {b}\right )}}+\frac {\left (3 (-1)^{3/4} \sqrt {a}-2 \sqrt [4]{-1} \sqrt {b}\right ) \tan ^{-1}\left (\frac {(-1)^{3/4} \sqrt {3 x+2} \sqrt {3 \sqrt {a} \sqrt {b}+2 i b}}{3 \sqrt {a}+2 i \sqrt {b}}\right )}{\sqrt {a} \sqrt {b} \sqrt {\sqrt {b} \left (3 \sqrt {a}+2 i \sqrt {b}\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 311, normalized size = 0.73 \begin {gather*} -\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 {gather*}

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*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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giac [A]  time = 0.41, size = 234, normalized size = 0.55 \begin {gather*} \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 {2 \, \sqrt {\frac {1}{2}} \sqrt {3 \, x + 2}}{\sqrt {-\frac {4 \, b + \sqrt {-4 \, {\left (9 \, a + 4 \, b\right )} b + 16 \, 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 {2 \, \sqrt {\frac {1}{2}} \sqrt {3 \, x + 2}}{\sqrt {-\frac {4 \, b - \sqrt {-4 \, {\left (9 \, a + 4 \, b\right )} b + 16 \, b^{2}}}{b}}}\right )}{4 \, a^{2} b^{3} - 17 \, a b^{4}} \end {gather*}

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]

(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(2
*sqrt(1/2)*sqrt(3*x + 2)/sqrt(-(4*b + sqrt(-4*(9*a + 4*b)*b + 16*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(2*sqrt(1/2)
*sqrt(3*x + 2)/sqrt(-(4*b - sqrt(-4*(9*a + 4*b)*b + 16*b^2))/b))/(4*a^2*b^3 - 17*a*b^4)

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maple [B]  time = 0.37, size = 944, normalized size = 2.21 \begin {gather*} -\frac {\sqrt {4 b +2 \sqrt {9 a b +4 b^{2}}}\, \sqrt {4 b +2 \sqrt {\left (9 a +4 b \right ) b}}\, \arctan \left (\frac {2 \sqrt {3 x +2}\, \sqrt {b}-\sqrt {4 b +2 \sqrt {\left (9 a +4 b \right ) b}}}{\sqrt {-4 b +4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {\left (9 a +4 b \right ) b}}}\right )}{3 \sqrt {-4 b +4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {\left (9 a +4 b \right ) b}}\, a \sqrt {b}}-\frac {\sqrt {4 b +2 \sqrt {9 a b +4 b^{2}}}\, \sqrt {4 b +2 \sqrt {\left (9 a +4 b \right ) b}}\, \arctan \left (\frac {2 \sqrt {3 x +2}\, \sqrt {b}+\sqrt {4 b +2 \sqrt {\left (9 a +4 b \right ) b}}}{\sqrt {-4 b +4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {\left (9 a +4 b \right ) b}}}\right )}{3 \sqrt {-4 b +4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {\left (9 a +4 b \right ) b}}\, a \sqrt {b}}-\frac {\sqrt {4 b +2 \sqrt {9 a b +4 b^{2}}}\, \ln \left (\left (3 x +2\right ) \sqrt {b}-\sqrt {3 x +2}\, \sqrt {4 b +2 \sqrt {\left (9 a +4 b \right ) b}}+\sqrt {9 a +4 b}\right )}{6 a \sqrt {b}}+\frac {\sqrt {4 b +2 \sqrt {9 a b +4 b^{2}}}\, \ln \left (\left (3 x +2\right ) \sqrt {b}+\sqrt {3 x +2}\, \sqrt {4 b +2 \sqrt {\left (9 a +4 b \right ) b}}+\sqrt {9 a +4 b}\right )}{6 a \sqrt {b}}+\frac {\sqrt {4 b +2 \sqrt {9 a b +4 b^{2}}}\, \sqrt {9 a b +4 b^{2}}\, \sqrt {4 b +2 \sqrt {\left (9 a +4 b \right ) b}}\, \arctan \left (\frac {2 \sqrt {3 x +2}\, \sqrt {b}-\sqrt {4 b +2 \sqrt {\left (9 a +4 b \right ) b}}}{\sqrt {-4 b +4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {\left (9 a +4 b \right ) b}}}\right )}{6 \sqrt {-4 b +4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {\left (9 a +4 b \right ) b}}\, a \,b^{\frac {3}{2}}}+\frac {\sqrt {4 b +2 \sqrt {9 a b +4 b^{2}}}\, \sqrt {9 a b +4 b^{2}}\, \sqrt {4 b +2 \sqrt {\left (9 a +4 b \right ) b}}\, \arctan \left (\frac {2 \sqrt {3 x +2}\, \sqrt {b}+\sqrt {4 b +2 \sqrt {\left (9 a +4 b \right ) b}}}{\sqrt {-4 b +4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {\left (9 a +4 b \right ) b}}}\right )}{6 \sqrt {-4 b +4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {\left (9 a +4 b \right ) b}}\, a \,b^{\frac {3}{2}}}+\frac {\sqrt {4 b +2 \sqrt {9 a b +4 b^{2}}}\, \sqrt {9 a b +4 b^{2}}\, \ln \left (\left (3 x +2\right ) \sqrt {b}-\sqrt {3 x +2}\, \sqrt {4 b +2 \sqrt {\left (9 a +4 b \right ) b}}+\sqrt {9 a +4 b}\right )}{12 a \,b^{\frac {3}{2}}}-\frac {\sqrt {4 b +2 \sqrt {9 a b +4 b^{2}}}\, \sqrt {9 a b +4 b^{2}}\, \ln \left (\left (3 x +2\right ) \sqrt {b}+\sqrt {3 x +2}\, \sqrt {4 b +2 \sqrt {\left (9 a +4 b \right ) b}}+\sqrt {9 a +4 b}\right )}{12 a \,b^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+2)^(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((3*x+2)*b^(1/2)-(3*x+2)^(1/2)*(2*(b*(9
*a+4*b))^(1/2)+4*b)^(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*b^(1/
2)*(3*x+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
))-1/6*(2*(9*a*b+4*b^2)^(1/2)+4*b)^(1/2)/a/b^(1/2)*ln((3*x+2)*b^(1/2)-(3*x+2)^(1/2)*(2*(b*(9*a+4*b))^(1/2)+4*b
)^(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*b^(1/2)*(3*x+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))-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((3*x+2)*b^(1/2)+(3*x+2)^(1/2)*(2*(b*(9*a+4*b))^(1/2)+4*b)^(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*b^(1/2)*(3*x+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))+1/6*(2*(9*a*b+4*b^2)^(1/2)+4*b)^
(1/2)/a/b^(1/2)*ln((3*x+2)*b^(1/2)+(3*x+2)^(1/2)*(2*(b*(9*a+4*b))^(1/2)+4*b)^(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*b^(1/2)*(3*x+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))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {3 \, x + 2}}{b x^{2} + a}\,{d x} \end {gather*}

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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mupad [B]  time = 0.94, size = 261, normalized size = 0.61 \begin {gather*} -2\,\mathrm {atanh}\left (\frac {2\,\left (\left (1296\,a\,b^2-576\,b^3\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}}}{1728\,b^2+3888\,a\,b}\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 (1296\,a\,b^2-576\,b^3\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}}}{1728\,b^2+3888\,a\,b}\right )\,\sqrt {-\frac {3\,\sqrt {-a^3\,b^3}+2\,a\,b^2}{4\,a^2\,b^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 6.02, size = 56, normalized size = 0.13 \begin {gather*} 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 {gather*}

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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