Integrand size = 24, antiderivative size = 109 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {26+41 x}{210 (3+2 x) \left (2+3 x^2\right )^{3/2}}+\frac {34+507 x}{1470 (3+2 x) \sqrt {2+3 x^2}}+\frac {277 \sqrt {2+3 x^2}}{5145 (3+2 x)}-\frac {176 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{1715 \sqrt {35}} \]
1/210*(26+41*x)/(3+2*x)/(3*x^2+2)^(3/2)-176/60025*arctanh(1/35*(4-9*x)*35^ (1/2)/(3*x^2+2)^(1/2))*35^(1/2)+1/1470*(34+507*x)/(3+2*x)/(3*x^2+2)^(1/2)+ 277/5145*(3*x^2+2)^(1/2)/(3+2*x)
Time = 0.61 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.93 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {35 \left (3966+9107 x+7362 x^2+10647 x^3+4986 x^4\right )-1056 \sqrt {35} \sqrt {2+3 x^2} \left (6+4 x+9 x^2+6 x^3\right ) \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{360150 (3+2 x) \left (2+3 x^2\right )^{3/2}} \]
(35*(3966 + 9107*x + 7362*x^2 + 10647*x^3 + 4986*x^4) - 1056*Sqrt[35]*Sqrt [2 + 3*x^2]*(6 + 4*x + 9*x^2 + 6*x^3)*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(360150*(3 + 2*x)*(2 + 3*x^2)^(3/2))
Time = 0.25 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {686, 27, 686, 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.
| \(\displaystyle \int \frac {5-x}{(2 x+3)^2 \left (3 x^2+2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {41 x+26}{210 (2 x+3) \left (3 x^2+2\right )^{3/2}}-\frac {1}{630} \int -\frac {6 (123 x+227)}{(2 x+3)^2 \left (3 x^2+2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{105} \int \frac {123 x+227}{(2 x+3)^2 \left (3 x^2+2\right )^{3/2}}dx+\frac {41 x+26}{210 (2 x+3) \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {1}{105} \left (\frac {507 x+34}{14 (2 x+3) \sqrt {3 x^2+2}}-\frac {1}{210} \int -\frac {30 (507 x+68)}{(2 x+3)^2 \sqrt {3 x^2+2}}dx\right )+\frac {41 x+26}{210 (2 x+3) \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{105} \left (\frac {1}{7} \int \frac {507 x+68}{(2 x+3)^2 \sqrt {3 x^2+2}}dx+\frac {507 x+34}{14 (2 x+3) \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 (2 x+3) \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 679 |
| \(\displaystyle \frac {1}{105} \left (\frac {1}{7} \left (\frac {528}{7} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx+\frac {277 \sqrt {3 x^2+2}}{7 (2 x+3)}\right )+\frac {507 x+34}{14 (2 x+3) \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 (2 x+3) \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {1}{105} \left (\frac {1}{7} \left (\frac {277 \sqrt {3 x^2+2}}{7 (2 x+3)}-\frac {528}{7} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}\right )+\frac {507 x+34}{14 (2 x+3) \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 (2 x+3) \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{105} \left (\frac {1}{7} \left (\frac {277 \sqrt {3 x^2+2}}{7 (2 x+3)}-\frac {528 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{7 \sqrt {35}}\right )+\frac {507 x+34}{14 (2 x+3) \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 (2 x+3) \left (3 x^2+2\right )^{3/2}}\) |
(26 + 41*x)/(210*(3 + 2*x)*(2 + 3*x^2)^(3/2)) + ((34 + 507*x)/(14*(3 + 2*x )*Sqrt[2 + 3*x^2]) + ((277*Sqrt[2 + 3*x^2])/(7*(3 + 2*x)) - (528*ArcTanh[( 4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(7*Sqrt[35]))/7)/105
3.15.26.3.1 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])
Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.64
| method | result | size |
| risch | \(\frac {4986 x^{4}+10647 x^{3}+7362 x^{2}+9107 x +3966}{10290 \left (3 x^{2}+2\right )^{\frac {3}{2}} \left (3+2 x \right )}-\frac {176 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{60025}\) | \(70\) |
| trager | \(\frac {4986 x^{4}+10647 x^{3}+7362 x^{2}+9107 x +3966}{10290 \left (3 x^{2}+2\right )^{\frac {3}{2}} \left (3+2 x \right )}+\frac {176 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x +35 \sqrt {3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )}{3+2 x}\right )}{60025}\) | \(86\) |
| default | \(\frac {22}{147 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}-\frac {17 x}{490 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {277 x}{3430 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}+\frac {88}{1715 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}-\frac {176 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{60025}-\frac {13}{70 \left (x +\frac {3}{2}\right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}\) | \(119\) |
1/10290*(4986*x^4+10647*x^3+7362*x^2+9107*x+3966)/(3*x^2+2)^(3/2)/(3+2*x)- 176/60025*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1 /2))
Time = 0.37 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.23 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {528 \, \sqrt {35} {\left (18 \, x^{5} + 27 \, x^{4} + 24 \, x^{3} + 36 \, x^{2} + 8 \, x + 12\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) + 35 \, {\left (4986 \, x^{4} + 10647 \, x^{3} + 7362 \, x^{2} + 9107 \, x + 3966\right )} \sqrt {3 \, x^{2} + 2}}{360150 \, {\left (18 \, x^{5} + 27 \, x^{4} + 24 \, x^{3} + 36 \, x^{2} + 8 \, x + 12\right )}} \]
1/360150*(528*sqrt(35)*(18*x^5 + 27*x^4 + 24*x^3 + 36*x^2 + 8*x + 12)*log( -(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) + 35*(4986*x^4 + 10647*x^3 + 7362*x^2 + 9107*x + 3966)*sqrt(3*x^2 + 2 ))/(18*x^5 + 27*x^4 + 24*x^3 + 36*x^2 + 8*x + 12)
Timed out. \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {176}{60025} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {277 \, x}{3430 \, \sqrt {3 \, x^{2} + 2}} + \frac {88}{1715 \, \sqrt {3 \, x^{2} + 2}} - \frac {17 \, x}{490 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {13}{35 \, {\left (2 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + 3 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}\right )}} + \frac {22}{147 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \]
176/60025*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2* x + 3)) + 277/3430*x/sqrt(3*x^2 + 2) + 88/1715/sqrt(3*x^2 + 2) - 17/490*x/ (3*x^2 + 2)^(3/2) - 13/35/(2*(3*x^2 + 2)^(3/2)*x + 3*(3*x^2 + 2)^(3/2)) + 22/147/(3*x^2 + 2)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (90) = 180\).
Time = 0.31 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.14 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx=-\frac {1}{360150} \, \sqrt {35} {\left (277 \, \sqrt {35} \sqrt {3} - 1056 \, \log \left (\sqrt {35} \sqrt {3} - 9\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {176 \, \sqrt {35} \log \left (\sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} - 9\right )}{60025 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} + \frac {\frac {\frac {\frac {7 \, {\left (\frac {4813}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} + \frac {4368}{{\left (2 \, x + 3\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{2 \, x + 3} - \frac {53523}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2 \, x + 3} + \frac {19269}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2 \, x + 3} - \frac {2493}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{10290 \, {\left (\frac {18}{2 \, x + 3} - \frac {35}{{\left (2 \, x + 3\right )}^{2}} - 3\right )} \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3}} \]
-1/360150*sqrt(35)*(277*sqrt(35)*sqrt(3) - 1056*log(sqrt(35)*sqrt(3) - 9)) *sgn(1/(2*x + 3)) - 176/60025*sqrt(35)*log(sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)) - 9)/sgn(1/(2*x + 3)) + 1/10290* (((7*(4813/sgn(1/(2*x + 3)) + 4368/((2*x + 3)*sgn(1/(2*x + 3))))/(2*x + 3) - 53523/sgn(1/(2*x + 3)))/(2*x + 3) + 19269/sgn(1/(2*x + 3)))/(2*x + 3) - 2493/sgn(1/(2*x + 3)))/((18/(2*x + 3) - 35/(2*x + 3)^2 - 3)*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3))
Time = 10.51 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.48 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {\sqrt {35}\,\left (3464\,\ln \left (x+\frac {3}{2}\right )-3464\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )\right )}{1500625}+\frac {\sqrt {35}\,\left (\frac {1872\,\ln \left (x+\frac {3}{2}\right )}{42875}-\frac {1872\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{42875}\right )}{70}-\frac {104\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{42875\,\left (x+\frac {3}{2}\right )}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {639}{19600}+\frac {\sqrt {6}\,597{}\mathrm {i}}{19600}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (-\frac {213}{9800}+\frac {\sqrt {6}\,199{}\mathrm {i}}{9800}\right )\,1{}\mathrm {i}}{2\,{\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {\frac {639}{19600}+\frac {\sqrt {6}\,597{}\mathrm {i}}{19600}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (\frac {213}{9800}+\frac {\sqrt {6}\,199{}\mathrm {i}}{9800}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-41568+\sqrt {6}\,27711{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{12348000\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (41568+\sqrt {6}\,27711{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{12348000\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \]
(35^(1/2)*(3464*log(x + 3/2) - 3464*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^ (1/2))/9 - 4/9)))/1500625 + (35^(1/2)*((1872*log(x + 3/2))/42875 - (1872*l og(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/42875))/70 - (104*3^ (1/2)*(x^2 + 2/3)^(1/2))/(42875*(x + 3/2)) - (3^(1/2)*(x^2 + 2/3)^(1/2)*(( (6^(1/2)*597i)/19600 - 639/19600)/(x - (6^(1/2)*1i)/3) - (6^(1/2)*((6^(1/2 )*199i)/9800 - 213/9800)*1i)/(2*(x - (6^(1/2)*1i)/3)^2)))/27 + (3^(1/2)*(x ^2 + 2/3)^(1/2)*(((6^(1/2)*597i)/19600 + 639/19600)/(x + (6^(1/2)*1i)/3) + (6^(1/2)*((6^(1/2)*199i)/9800 + 213/9800)*1i)/(2*(x + (6^(1/2)*1i)/3)^2)) )/27 - (3^(1/2)*6^(1/2)*(6^(1/2)*27711i - 41568)*(x^2 + 2/3)^(1/2)*1i)/(12 348000*(x + (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(6^(1/2)*27711i + 41568)*( x^2 + 2/3)^(1/2)*1i)/(12348000*(x - (6^(1/2)*1i)/3))