Integrand size = 24, antiderivative size = 126 \[ \int \frac {5-x}{(3+2 x)^4 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {26+41 x}{70 (3+2 x)^3 \sqrt {2+3 x^2}}+\frac {23 \sqrt {2+3 x^2}}{525 (3+2 x)^3}-\frac {27 \sqrt {2+3 x^2}}{1225 (3+2 x)^2}-\frac {1051 \sqrt {2+3 x^2}}{42875 (3+2 x)}-\frac {3312 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{42875 \sqrt {35}} \]
-3312/1500625*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))*35^(1/2)+1/70 *(26+41*x)/(3+2*x)^3/(3*x^2+2)^(1/2)+23/525*(3*x^2+2)^(1/2)/(3+2*x)^3-27/1 225*(3*x^2+2)^(1/2)/(3+2*x)^2-1051/42875*(3*x^2+2)^(1/2)/(3+2*x)
Time = 0.80 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.70 \[ \int \frac {5-x}{(3+2 x)^4 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {-\frac {35 \left (29438+23349 x+237930 x^2+261036 x^3+75672 x^4\right )}{(3+2 x)^3 \sqrt {2+3 x^2}}+39744 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{9003750} \]
((-35*(29438 + 23349*x + 237930*x^2 + 261036*x^3 + 75672*x^4))/((3 + 2*x)^ 3*Sqrt[2 + 3*x^2]) + 39744*Sqrt[35]*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*S qrt[2 + 3*x^2])/Sqrt[35]])/9003750
Time = 0.28 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {686, 27, 688, 27, 688, 27, 679, 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.
| \(\displaystyle \int \frac {5-x}{(2 x+3)^4 \left (3 x^2+2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {41 x+26}{70 (2 x+3)^3 \sqrt {3 x^2+2}}-\frac {1}{210} \int -\frac {6 (123 x+104)}{(2 x+3)^4 \sqrt {3 x^2+2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{35} \int \frac {123 x+104}{(2 x+3)^4 \sqrt {3 x^2+2}}dx+\frac {41 x+26}{70 (2 x+3)^3 \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {1}{35} \left (\frac {23 \sqrt {3 x^2+2}}{15 (2 x+3)^3}-\frac {1}{105} \int -\frac {42 (23 x+102)}{(2 x+3)^3 \sqrt {3 x^2+2}}dx\right )+\frac {41 x+26}{70 (2 x+3)^3 \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{35} \left (\frac {2}{5} \int \frac {23 x+102}{(2 x+3)^3 \sqrt {3 x^2+2}}dx+\frac {23 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )+\frac {41 x+26}{70 (2 x+3)^3 \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {1}{35} \left (\frac {2}{5} \left (-\frac {1}{70} \int -\frac {5 (404-81 x)}{(2 x+3)^2 \sqrt {3 x^2+2}}dx-\frac {27 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )+\frac {23 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )+\frac {41 x+26}{70 (2 x+3)^3 \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{35} \left (\frac {2}{5} \left (\frac {1}{14} \int \frac {404-81 x}{(2 x+3)^2 \sqrt {3 x^2+2}}dx-\frac {27 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )+\frac {23 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )+\frac {41 x+26}{70 (2 x+3)^3 \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 679 |
| \(\displaystyle \frac {1}{35} \left (\frac {2}{5} \left (\frac {1}{14} \left (\frac {3312}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {1051 \sqrt {3 x^2+2}}{35 (2 x+3)}\right )-\frac {27 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )+\frac {23 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )+\frac {41 x+26}{70 (2 x+3)^3 \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {1}{35} \left (\frac {2}{5} \left (\frac {1}{14} \left (-\frac {3312}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {1051 \sqrt {3 x^2+2}}{35 (2 x+3)}\right )-\frac {27 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )+\frac {23 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )+\frac {41 x+26}{70 (2 x+3)^3 \sqrt {3 x^2+2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{35} \left (\frac {2}{5} \left (\frac {1}{14} \left (-\frac {3312 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}-\frac {1051 \sqrt {3 x^2+2}}{35 (2 x+3)}\right )-\frac {27 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )+\frac {23 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )+\frac {41 x+26}{70 (2 x+3)^3 \sqrt {3 x^2+2}}\) |
(26 + 41*x)/(70*(3 + 2*x)^3*Sqrt[2 + 3*x^2]) + ((23*Sqrt[2 + 3*x^2])/(15*( 3 + 2*x)^3) + (2*((-27*Sqrt[2 + 3*x^2])/(14*(3 + 2*x)^2) + ((-1051*Sqrt[2 + 3*x^2])/(35*(3 + 2*x)) - (3312*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^ 2])])/(35*Sqrt[35]))/14))/5)/35
3.15.16.3.1 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/(((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[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 )/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ [p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( (m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Time = 0.30 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.56
| method | result | size |
| risch | \(-\frac {75672 x^{4}+261036 x^{3}+237930 x^{2}+23349 x +29438}{257250 \left (3+2 x \right )^{3} \sqrt {3 x^{2}+2}}-\frac {3312 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{1500625}\) | \(70\) |
| trager | \(-\frac {75672 x^{4}+261036 x^{3}+237930 x^{2}+23349 x +29438}{257250 \left (3+2 x \right )^{3} \sqrt {3 x^{2}+2}}+\frac {3312 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x +35 \sqrt {3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )}{3+2 x}\right )}{1500625}\) | \(86\) |
| default | \(-\frac {17}{700 \left (x +\frac {3}{2}\right )^{2} \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}-\frac {101}{2450 \left (x +\frac {3}{2}\right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}+\frac {1656}{42875 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}-\frac {3153 x}{85750 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}-\frac {3312 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{1500625}-\frac {13}{840 \left (x +\frac {3}{2}\right )^{3} \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}\) | \(128\) |
-1/257250*(75672*x^4+261036*x^3+237930*x^2+23349*x+29438)/(3+2*x)^3/(3*x^2 +2)^(1/2)-3312/1500625*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^ 2-36*x-19)^(1/2))
Time = 0.32 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.06 \[ \int \frac {5-x}{(3+2 x)^4 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {9936 \, \sqrt {35} {\left (24 \, x^{5} + 108 \, x^{4} + 178 \, x^{3} + 153 \, x^{2} + 108 \, x + 54\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \, {\left (75672 \, x^{4} + 261036 \, x^{3} + 237930 \, x^{2} + 23349 \, x + 29438\right )} \sqrt {3 \, x^{2} + 2}}{9003750 \, {\left (24 \, x^{5} + 108 \, x^{4} + 178 \, x^{3} + 153 \, x^{2} + 108 \, x + 54\right )}} \]
1/9003750*(9936*sqrt(35)*(24*x^5 + 108*x^4 + 178*x^3 + 153*x^2 + 108*x + 5 4)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(75672*x^4 + 261036*x^3 + 237930*x^2 + 23349*x + 29438)*s qrt(3*x^2 + 2))/(24*x^5 + 108*x^4 + 178*x^3 + 153*x^2 + 108*x + 54)
Timed out. \[ \int \frac {5-x}{(3+2 x)^4 \left (2+3 x^2\right )^{3/2}} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.46 \[ \int \frac {5-x}{(3+2 x)^4 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {3312}{1500625} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) - \frac {3153 \, x}{85750 \, \sqrt {3 \, x^{2} + 2}} + \frac {1656}{42875 \, \sqrt {3 \, x^{2} + 2}} - \frac {13}{105 \, {\left (8 \, \sqrt {3 \, x^{2} + 2} x^{3} + 36 \, \sqrt {3 \, x^{2} + 2} x^{2} + 54 \, \sqrt {3 \, x^{2} + 2} x + 27 \, \sqrt {3 \, x^{2} + 2}\right )}} - \frac {17}{175 \, {\left (4 \, \sqrt {3 \, x^{2} + 2} x^{2} + 12 \, \sqrt {3 \, x^{2} + 2} x + 9 \, \sqrt {3 \, x^{2} + 2}\right )}} - \frac {101}{1225 \, {\left (2 \, \sqrt {3 \, x^{2} + 2} x + 3 \, \sqrt {3 \, x^{2} + 2}\right )}} \]
3312/1500625*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs (2*x + 3)) - 3153/85750*x/sqrt(3*x^2 + 2) + 1656/42875/sqrt(3*x^2 + 2) - 1 3/105/(8*sqrt(3*x^2 + 2)*x^3 + 36*sqrt(3*x^2 + 2)*x^2 + 54*sqrt(3*x^2 + 2) *x + 27*sqrt(3*x^2 + 2)) - 17/175/(4*sqrt(3*x^2 + 2)*x^2 + 12*sqrt(3*x^2 + 2)*x + 9*sqrt(3*x^2 + 2)) - 101/1225/(2*sqrt(3*x^2 + 2)*x + 3*sqrt(3*x^2 + 2))
Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (103) = 206\).
Time = 0.30 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.97 \[ \int \frac {5-x}{(3+2 x)^4 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {3312}{1500625} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) - \frac {3 \, {\left (10281 \, x - 12674\right )}}{3001250 \, \sqrt {3 \, x^{2} + 2}} - \frac {2 \, \sqrt {3} {\left (12983 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} + 253320 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 298170 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 1481160 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 425140 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 106016\right )}}{1500625 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{3}} \]
3312/1500625*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqr t(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 3/3001250*(10281*x - 12674)/sqrt(3*x^2 + 2) - 2/1500625*sqrt(3)*(12983*sqr t(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 253320*(sqrt(3)*x - sqrt(3*x^2 + 2) )^4 + 298170*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 1481160*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 425140*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 1060 16)/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^3
Time = 10.72 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.67 \[ \int \frac {5-x}{(3+2 x)^4 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {3312\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{1500625}-\frac {3312\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{1500625}-\frac {10281\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{6002500\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {10281\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{6002500\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {13252\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1500625\,\left (x+\frac {3}{2}\right )}-\frac {197\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{42875\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{7350\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,6337{}\mathrm {i}}{6002500\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,6337{}\mathrm {i}}{6002500\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \]
(3312*35^(1/2)*log(x + 3/2))/1500625 - (3312*35^(1/2)*log(x - (3^(1/2)*35^ (1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/1500625 - (10281*3^(1/2)*(x^2 + 2/3)^(1 /2))/(6002500*(x - (6^(1/2)*1i)/3)) - (10281*3^(1/2)*(x^2 + 2/3)^(1/2))/(6 002500*(x + (6^(1/2)*1i)/3)) - (13252*3^(1/2)*(x^2 + 2/3)^(1/2))/(1500625* (x + 3/2)) - (197*3^(1/2)*(x^2 + 2/3)^(1/2))/(42875*(3*x + x^2 + 9/4)) - ( 13*3^(1/2)*(x^2 + 2/3)^(1/2))/(7350*((27*x)/4 + (9*x^2)/2 + x^3 + 27/8)) - (3^(1/2)*6^(1/2)*(x^2 + 2/3)^(1/2)*6337i)/(6002500*(x - (6^(1/2)*1i)/3)) + (3^(1/2)*6^(1/2)*(x^2 + 2/3)^(1/2)*6337i)/(6002500*(x + (6^(1/2)*1i)/3))