Integrand size = 24, antiderivative size = 104 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {26+41 x}{70 (3+2 x)^2 \sqrt {2+3 x^2}}+\frac {9 \sqrt {2+3 x^2}}{245 (3+2 x)^2}-\frac {331 \sqrt {2+3 x^2}}{8575 (3+2 x)}-\frac {1962 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{8575 \sqrt {35}} \]
-1962/300125*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)^2/(3*x^2+2)^(1/2)+9/245*(3*x^2+2)^(1/2)/(3+2*x)^2-331/85 75*(3*x^2+2)^(1/2)/(3+2*x)
Time = 0.58 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.80 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {\frac {35 \left (3658+7397 x-4068 x^2-3972 x^3\right )}{(3+2 x)^2 \sqrt {2+3 x^2}}+7848 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{600250} \]
((35*(3658 + 7397*x - 4068*x^2 - 3972*x^3))/((3 + 2*x)^2*Sqrt[2 + 3*x^2]) + 7848*Sqrt[35]*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt [35]])/600250
Time = 0.24 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {686, 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.
\(\displaystyle \int \frac {5-x}{(2 x+3)^3 \left (3 x^2+2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 686 |
\(\displaystyle \frac {41 x+26}{70 (2 x+3)^2 \sqrt {3 x^2+2}}-\frac {1}{210} \int -\frac {12 (41 x+39)}{(2 x+3)^3 \sqrt {3 x^2+2}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2}{35} \int \frac {41 x+39}{(2 x+3)^3 \sqrt {3 x^2+2}}dx+\frac {41 x+26}{70 (2 x+3)^2 \sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 688 |
\(\displaystyle \frac {2}{35} \left (\frac {9 \sqrt {3 x^2+2}}{14 (2 x+3)^2}-\frac {1}{70} \int -\frac {5 (27 x+206)}{(2 x+3)^2 \sqrt {3 x^2+2}}dx\right )+\frac {41 x+26}{70 (2 x+3)^2 \sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2}{35} \left (\frac {1}{14} \int \frac {27 x+206}{(2 x+3)^2 \sqrt {3 x^2+2}}dx+\frac {9 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )+\frac {41 x+26}{70 (2 x+3)^2 \sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 679 |
\(\displaystyle \frac {2}{35} \left (\frac {1}{14} \left (\frac {1962}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {331 \sqrt {3 x^2+2}}{35 (2 x+3)}\right )+\frac {9 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )+\frac {41 x+26}{70 (2 x+3)^2 \sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 488 |
\(\displaystyle \frac {2}{35} \left (\frac {1}{14} \left (-\frac {1962}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {331 \sqrt {3 x^2+2}}{35 (2 x+3)}\right )+\frac {9 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )+\frac {41 x+26}{70 (2 x+3)^2 \sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2}{35} \left (\frac {1}{14} \left (-\frac {1962 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}-\frac {331 \sqrt {3 x^2+2}}{35 (2 x+3)}\right )+\frac {9 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )+\frac {41 x+26}{70 (2 x+3)^2 \sqrt {3 x^2+2}}\) |
(26 + 41*x)/(70*(3 + 2*x)^2*Sqrt[2 + 3*x^2]) + (2*((9*Sqrt[2 + 3*x^2])/(14 *(3 + 2*x)^2) + ((-331*Sqrt[2 + 3*x^2])/(35*(3 + 2*x)) - (1962*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(35*Sqrt[35]))/14))/35
3.15.15.3.1 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 = 65, normalized size of antiderivative = 0.62
method | result | size |
risch | \(-\frac {3972 x^{3}+4068 x^{2}-7397 x -3658}{17150 \left (3+2 x \right )^{2} \sqrt {3 x^{2}+2}}-\frac {1962 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{300125}\) | \(65\) |
trager | \(-\frac {3972 x^{3}+4068 x^{2}-7397 x -3658}{17150 \left (3+2 x \right )^{2} \sqrt {3 x^{2}+2}}+\frac {1962 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x +35 \sqrt {3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )}{3+2 x}\right )}{300125}\) | \(81\) |
default | \(-\frac {103}{980 \left (x +\frac {3}{2}\right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}+\frac {981}{8575 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}-\frac {993 x}{17150 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}-\frac {1962 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{300125}-\frac {13}{280 \left (x +\frac {3}{2}\right )^{2} \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}\) | \(107\) |
-1/17150*(3972*x^3+4068*x^2-7397*x-3658)/(3+2*x)^2/(3*x^2+2)^(1/2)-1962/30 0125*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 = 119, normalized size of antiderivative = 1.14 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {1962 \, \sqrt {35} {\left (12 \, x^{4} + 36 \, x^{3} + 35 \, x^{2} + 24 \, x + 18\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \, {\left (3972 \, x^{3} + 4068 \, x^{2} - 7397 \, x - 3658\right )} \sqrt {3 \, x^{2} + 2}}{600250 \, {\left (12 \, x^{4} + 36 \, x^{3} + 35 \, x^{2} + 24 \, x + 18\right )}} \]
1/600250*(1962*sqrt(35)*(12*x^4 + 36*x^3 + 35*x^2 + 24*x + 18)*log(-(sqrt( 35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(3972*x^3 + 4068*x^2 - 7397*x - 3658)*sqrt(3*x^2 + 2))/(12*x^4 + 36*x^3 + 35*x^2 + 24*x + 18)
Timed out. \[ \int \frac {5-x}{(3+2 x)^3 \left (2+3 x^2\right )^{3/2}} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.23 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {1962}{300125} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) - \frac {993 \, x}{17150 \, \sqrt {3 \, x^{2} + 2}} + \frac {981}{8575 \, \sqrt {3 \, x^{2} + 2}} - \frac {13}{70 \, {\left (4 \, \sqrt {3 \, x^{2} + 2} x^{2} + 12 \, \sqrt {3 \, x^{2} + 2} x + 9 \, \sqrt {3 \, x^{2} + 2}\right )}} - \frac {103}{490 \, {\left (2 \, \sqrt {3 \, x^{2} + 2} x + 3 \, \sqrt {3 \, x^{2} + 2}\right )}} \]
1962/300125*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs( 2*x + 3)) - 993/17150*x/sqrt(3*x^2 + 2) + 981/8575/sqrt(3*x^2 + 2) - 13/70 /(4*sqrt(3*x^2 + 2)*x^2 + 12*sqrt(3*x^2 + 2)*x + 9*sqrt(3*x^2 + 2)) - 103/ 490/(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. 199 vs. \(2 (85) = 170\).
Time = 0.30 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.91 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {1962}{300125} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) - \frac {3 \, {\left (157 \, x - 1478\right )}}{85750 \, \sqrt {3 \, x^{2} + 2}} - \frac {768 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} + 2461 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 6168 \, \sqrt {3} x + 856 \, \sqrt {3} + 6168 \, \sqrt {3 \, x^{2} + 2}}{6125 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{2}} \]
1962/300125*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt (3*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 3 /85750*(157*x - 1478)/sqrt(3*x^2 + 2) - 1/6125*(768*(sqrt(3)*x - sqrt(3*x^ 2 + 2))^3 + 2461*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 6168*sqrt(3)*x + 856*sqrt(3) + 6168*sqrt(3*x^2 + 2))/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3 *sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^2
Time = 10.78 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.74 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {1962\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{300125}-\frac {1962\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{300125}-\frac {157\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{171500\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {157\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{171500\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {107\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{6125\,\left (x+\frac {3}{2}\right )}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{2450\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,739{}\mathrm {i}}{171500\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,739{}\mathrm {i}}{171500\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \]
(1962*35^(1/2)*log(x + 3/2))/300125 - (1962*35^(1/2)*log(x - (3^(1/2)*35^( 1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/300125 - (157*3^(1/2)*(x^2 + 2/3)^(1/2)) /(171500*(x - (6^(1/2)*1i)/3)) - (157*3^(1/2)*(x^2 + 2/3)^(1/2))/(171500*( x + (6^(1/2)*1i)/3)) - (107*3^(1/2)*(x^2 + 2/3)^(1/2))/(6125*(x + 3/2)) - (13*3^(1/2)*(x^2 + 2/3)^(1/2))/(2450*(3*x + x^2 + 9/4)) - (3^(1/2)*6^(1/2) *(x^2 + 2/3)^(1/2)*739i)/(171500*(x - (6^(1/2)*1i)/3)) + (3^(1/2)*6^(1/2)* (x^2 + 2/3)^(1/2)*739i)/(171500*(x + (6^(1/2)*1i)/3))