Integrand size = 24, antiderivative size = 72 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{3+2 x} \, dx=\frac {1}{4} (13-x) \sqrt {2+3 x^2}-\frac {121 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{8 \sqrt {3}}-\frac {13}{8} \sqrt {35} \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right ) \]
-121/24*arcsinh(1/2*x*6^(1/2))*3^(1/2)-13/8*arctanh(1/35*(4-9*x)*35^(1/2)/ (3*x^2+2)^(1/2))*35^(1/2)+1/4*(13-x)*(3*x^2+2)^(1/2)
Time = 0.31 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.28 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{3+2 x} \, dx=\frac {1}{24} \left (-6 (-13+x) \sqrt {2+3 x^2}+78 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )+121 \sqrt {3} \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )\right ) \]
(-6*(-13 + x)*Sqrt[2 + 3*x^2] + 78*Sqrt[35]*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3] *x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]] + 121*Sqrt[3]*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/24
Time = 0.21 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {682, 27, 719, 222, 488, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(5-x) \sqrt {3 x^2+2}}{2 x+3} \, dx\) |
\(\Big \downarrow \) 682 |
\(\displaystyle \frac {1}{24} \int \frac {6 (46-121 x)}{(2 x+3) \sqrt {3 x^2+2}}dx+\frac {1}{4} \sqrt {3 x^2+2} (13-x)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \int \frac {46-121 x}{(2 x+3) \sqrt {3 x^2+2}}dx+\frac {1}{4} \sqrt {3 x^2+2} (13-x)\) |
\(\Big \downarrow \) 719 |
\(\displaystyle \frac {1}{4} \left (\frac {455}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {121}{2} \int \frac {1}{\sqrt {3 x^2+2}}dx\right )+\frac {1}{4} \sqrt {3 x^2+2} (13-x)\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {1}{4} \left (\frac {455}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {121 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}\right )+\frac {1}{4} \sqrt {3 x^2+2} (13-x)\) |
\(\Big \downarrow \) 488 |
\(\displaystyle \frac {1}{4} \left (-\frac {455}{2} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {121 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}\right )+\frac {1}{4} \sqrt {3 x^2+2} (13-x)\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{4} \left (-\frac {121 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}-\frac {13}{2} \sqrt {35} \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )\right )+\frac {1}{4} \sqrt {3 x^2+2} (13-x)\) |
((13 - x)*Sqrt[2 + 3*x^2])/4 + ((-121*ArcSinh[Sqrt[3/2]*x])/(2*Sqrt[3]) - (13*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/2)/4
3.14.61.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.81
method | result | size |
risch | \(-\frac {\left (-13+x \right ) \sqrt {3 x^{2}+2}}{4}-\frac {121 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{24}-\frac {13 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{8}\) | \(58\) |
default | \(-\frac {x \sqrt {3 x^{2}+2}}{4}-\frac {121 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{24}+\frac {13 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{8}-\frac {13 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{8}\) | \(72\) |
trager | \(\left (\frac {13}{4}-\frac {x}{4}\right ) \sqrt {3 x^{2}+2}+\frac {13 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x +35 \sqrt {3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )}{3+2 x}\right )}{8}+\frac {121 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{24}\) | \(93\) |
-1/4*(-13+x)*(3*x^2+2)^(1/2)-121/24*arcsinh(1/2*x*6^(1/2))*3^(1/2)-13/8*35 ^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))
Time = 0.30 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.25 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{3+2 x} \, dx=-\frac {1}{4} \, \sqrt {3 \, x^{2} + 2} {\left (x - 13\right )} + \frac {121}{48} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + \frac {13}{16} \, \sqrt {35} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) \]
-1/4*sqrt(3*x^2 + 2)*(x - 13) + 121/48*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 2) *x - 3*x^2 - 1) + 13/16*sqrt(35)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9))
\[ \int \frac {(5-x) \sqrt {2+3 x^2}}{3+2 x} \, dx=- \int \left (- \frac {5 \sqrt {3 x^{2} + 2}}{2 x + 3}\right )\, dx - \int \frac {x \sqrt {3 x^{2} + 2}}{2 x + 3}\, dx \]
Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.97 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{3+2 x} \, dx=-\frac {1}{4} \, \sqrt {3 \, x^{2} + 2} x - \frac {121}{24} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {13}{8} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {13}{4} \, \sqrt {3 \, x^{2} + 2} \]
-1/4*sqrt(3*x^2 + 2)*x - 121/24*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 13/8*sqrt (35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 13/4 *sqrt(3*x^2 + 2)
Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.44 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{3+2 x} \, dx=-\frac {1}{4} \, \sqrt {3 \, x^{2} + 2} {\left (x - 13\right )} + \frac {121}{24} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {13}{8} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) \]
-1/4*sqrt(3*x^2 + 2)*(x - 13) + 121/24*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 13/8*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sq rt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2)))
Time = 0.19 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.92 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{3+2 x} \, dx=\frac {\sqrt {35}\,\left (910\,\ln \left (x+\frac {3}{2}\right )-910\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )\right )}{560}-\frac {121\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{24}-\frac {\sqrt {3}\,\left (\frac {3\,x}{4}-\frac {39}{4}\right )\,\sqrt {x^2+\frac {2}{3}}}{3} \]