3.7.52 \(\int \frac {\sqrt {2+3 x}}{a+b x^2} \, dx\) [652]

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

[Out]

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

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Rubi [A]
time = 0.37, 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 = {714, 1143, 648, 632, 212, 642} \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 212

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

Rule 632

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 642

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 648

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 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 1143

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 \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 )\\ &=\frac {3 \text {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 \text {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 \text {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 \text {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 \text {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 \text {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 \text {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 \text {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 [C] Result contains complex when optimal does not.
time = 0.24, size = 180, normalized size = 0.42 \begin {gather*} \frac {\sqrt [4]{-1} \sqrt {3 \sqrt {a} \sqrt {b}-2 i b} \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {3 \sqrt {a} \sqrt {b}-2 i b} \sqrt {2+3 x}}{3 \sqrt {a}-2 i \sqrt {b}}\right )+(-1)^{3/4} \sqrt {3 \sqrt {a} \sqrt {b}+2 i b} \tan ^{-1}\left (\frac {(-1)^{3/4} \sqrt {3 \sqrt {a} \sqrt {b}+2 i b} \sqrt {2+3 x}}{3 \sqrt {a}+2 i \sqrt {b}}\right )}{\sqrt {a} b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.45, size = 447, normalized size = 1.05

method	result	size
derivativedivides	\(-\frac {\sqrt {2 \sqrt {9 a b +4 b^{2}}+4 b}\, \left (\sqrt {9 a b +4 b^{2}}-2 b \right ) \left (\frac {\ln \left (\left (2+3 x \right ) \sqrt {b}+\sqrt {2+3 x}\, \sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}+\sqrt {9 a +4 b}\right )}{2 \sqrt {b}}-\frac {\sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}\, \arctan \left (\frac {2 \sqrt {b}\, \sqrt {2+3 x}+\sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}}{\sqrt {4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {b \left (9 a +4 b \right )}-4 b}}\right )}{\sqrt {b}\, \sqrt {4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {b \left (9 a +4 b \right )}-4 b}}\right )}{6 a b}+\frac {\sqrt {2 \sqrt {9 a b +4 b^{2}}+4 b}\, \left (\sqrt {9 a b +4 b^{2}}-2 b \right ) \left (\frac {\ln \left (-\left (2+3 x \right ) \sqrt {b}+\sqrt {2+3 x}\, \sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}-\sqrt {9 a +4 b}\right )}{2 \sqrt {b}}-\frac {\sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}\, \arctan \left (\frac {-2 \sqrt {b}\, \sqrt {2+3 x}+\sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}}{\sqrt {4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {b \left (9 a +4 b \right )}-4 b}}\right )}{\sqrt {b}\, \sqrt {4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {b \left (9 a +4 b \right )}-4 b}}\right )}{6 a b}\)	\(447\)
default	\(-\frac {\sqrt {2 \sqrt {9 a b +4 b^{2}}+4 b}\, \left (\sqrt {9 a b +4 b^{2}}-2 b \right ) \left (\frac {\ln \left (\left (2+3 x \right ) \sqrt {b}+\sqrt {2+3 x}\, \sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}+\sqrt {9 a +4 b}\right )}{2 \sqrt {b}}-\frac {\sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}\, \arctan \left (\frac {2 \sqrt {b}\, \sqrt {2+3 x}+\sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}}{\sqrt {4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {b \left (9 a +4 b \right )}-4 b}}\right )}{\sqrt {b}\, \sqrt {4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {b \left (9 a +4 b \right )}-4 b}}\right )}{6 a b}+\frac {\sqrt {2 \sqrt {9 a b +4 b^{2}}+4 b}\, \left (\sqrt {9 a b +4 b^{2}}-2 b \right ) \left (\frac {\ln \left (-\left (2+3 x \right ) \sqrt {b}+\sqrt {2+3 x}\, \sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}-\sqrt {9 a +4 b}\right )}{2 \sqrt {b}}-\frac {\sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}\, \arctan \left (\frac {-2 \sqrt {b}\, \sqrt {2+3 x}+\sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}}{\sqrt {4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {b \left (9 a +4 b \right )}-4 b}}\right )}{\sqrt {b}\, \sqrt {4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {b \left (9 a +4 b \right )}-4 b}}\right )}{6 a b}\)	\(447\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*(9*a*b+4*b^2)^(1/2)+4*b)^(1/2)*((9*a*b+4*b^2)^(1/2)-2*b)/a/b*(1/2/b^(1/2)*ln((2+3*x)*b^(1/2)+(2+3*x)^(
1/2)*(2*(b*(9*a+4*b))^(1/2)+4*b)^(1/2)+(9*a+4*b)^(1/2))-1/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)*(2+3*x)^(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)*((9
*a*b+4*b^2)^(1/2)-2*b)/a/b*(1/2/b^(1/2)*ln(-(2+3*x)*b^(1/2)+(2+3*x)^(1/2)*(2*(b*(9*a+4*b))^(1/2)+4*b)^(1/2)-(9
*a+4*b)^(1/2))-1/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)*(2+3*x)^(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*} \text {Failed to integrate} \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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Fricas [A]
time = 2.57, 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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Sympy [A]
time = 2.47, 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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Giac [A]
time = 2.71, 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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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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