Integrand size = 24, antiderivative size = 82 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {26+41 x}{70 (3+2 x) \sqrt {2+3 x^2}}+\frac {19 \sqrt {2+3 x^2}}{1225 (3+2 x)}-\frac {632 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{1225 \sqrt {35}} \]
-632/42875*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))*35^(1/2)+1/70*(2 6+41*x)/(3+2*x)/(3*x^2+2)^(1/2)+19/1225*(3*x^2+2)^(1/2)/(3+2*x)
Time = 0.52 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.79 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {\frac {35 \left (986+1435 x+114 x^2\right )}{(3+2 x) \sqrt {2+3 x^2}}-1264 \sqrt {35} \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{85750} \]
((35*(986 + 1435*x + 114*x^2))/((3 + 2*x)*Sqrt[2 + 3*x^2]) - 1264*Sqrt[35] *ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/85750
Time = 0.21 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {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.
\(\displaystyle \int \frac {5-x}{(2 x+3)^2 \left (3 x^2+2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 686 |
\(\displaystyle \frac {41 x+26}{70 (2 x+3) \sqrt {3 x^2+2}}-\frac {1}{210} \int -\frac {6 (41 x+52)}{(2 x+3)^2 \sqrt {3 x^2+2}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{35} \int \frac {41 x+52}{(2 x+3)^2 \sqrt {3 x^2+2}}dx+\frac {41 x+26}{70 (2 x+3) \sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 679 |
\(\displaystyle \frac {1}{35} \left (\frac {632}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx+\frac {19 \sqrt {3 x^2+2}}{35 (2 x+3)}\right )+\frac {41 x+26}{70 (2 x+3) \sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 488 |
\(\displaystyle \frac {1}{35} \left (\frac {19 \sqrt {3 x^2+2}}{35 (2 x+3)}-\frac {632}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}\right )+\frac {41 x+26}{70 (2 x+3) \sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{35} \left (\frac {19 \sqrt {3 x^2+2}}{35 (2 x+3)}-\frac {632 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}\right )+\frac {41 x+26}{70 (2 x+3) \sqrt {3 x^2+2}}\) |
(26 + 41*x)/(70*(3 + 2*x)*Sqrt[2 + 3*x^2]) + ((19*Sqrt[2 + 3*x^2])/(35*(3 + 2*x)) - (632*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(35*Sqrt[35] ))/35
3.15.14.3.1 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 = 60, normalized size of antiderivative = 0.73
method | result | size |
risch | \(\frac {114 x^{2}+1435 x +986}{2450 \left (3+2 x \right ) \sqrt {3 x^{2}+2}}-\frac {632 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{42875}\) | \(60\) |
default | \(\frac {316}{1225 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}+\frac {57 x}{2450 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}-\frac {632 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{42875}-\frac {13}{70 \left (x +\frac {3}{2}\right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}\) | \(86\) |
trager | \(\frac {\left (114 x^{2}+1435 x +986\right ) \sqrt {3 x^{2}+2}}{14700 x^{3}+22050 x^{2}+9800 x +14700}+\frac {632 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x +35 \sqrt {3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )}{3+2 x}\right )}{42875}\) | \(86\) |
1/2450*(114*x^2+1435*x+986)/(3+2*x)/(3*x^2+2)^(1/2)-632/42875*35^(1/2)*arc tanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))
Time = 0.31 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.27 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {632 \, \sqrt {35} {\left (6 \, x^{3} + 9 \, x^{2} + 4 \, x + 6\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) + 35 \, {\left (114 \, x^{2} + 1435 \, x + 986\right )} \sqrt {3 \, x^{2} + 2}}{85750 \, {\left (6 \, x^{3} + 9 \, x^{2} + 4 \, x + 6\right )}} \]
1/85750*(632*sqrt(35)*(6*x^3 + 9*x^2 + 4*x + 6)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) + 35*(114*x^2 + 1 435*x + 986)*sqrt(3*x^2 + 2))/(6*x^3 + 9*x^2 + 4*x + 6)
\[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=- \int \frac {x}{12 x^{4} \sqrt {3 x^{2} + 2} + 36 x^{3} \sqrt {3 x^{2} + 2} + 35 x^{2} \sqrt {3 x^{2} + 2} + 24 x \sqrt {3 x^{2} + 2} + 18 \sqrt {3 x^{2} + 2}}\, dx - \int \left (- \frac {5}{12 x^{4} \sqrt {3 x^{2} + 2} + 36 x^{3} \sqrt {3 x^{2} + 2} + 35 x^{2} \sqrt {3 x^{2} + 2} + 24 x \sqrt {3 x^{2} + 2} + 18 \sqrt {3 x^{2} + 2}}\right )\, dx \]
-Integral(x/(12*x**4*sqrt(3*x**2 + 2) + 36*x**3*sqrt(3*x**2 + 2) + 35*x**2 *sqrt(3*x**2 + 2) + 24*x*sqrt(3*x**2 + 2) + 18*sqrt(3*x**2 + 2)), x) - Int egral(-5/(12*x**4*sqrt(3*x**2 + 2) + 36*x**3*sqrt(3*x**2 + 2) + 35*x**2*sq rt(3*x**2 + 2) + 24*x*sqrt(3*x**2 + 2) + 18*sqrt(3*x**2 + 2)), x)
Time = 0.31 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.05 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {632}{42875} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {57 \, x}{2450 \, \sqrt {3 \, x^{2} + 2}} + \frac {316}{1225 \, \sqrt {3 \, x^{2} + 2}} - \frac {13}{35 \, {\left (2 \, \sqrt {3 \, x^{2} + 2} x + 3 \, \sqrt {3 \, x^{2} + 2}\right )}} \]
632/42875*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2* x + 3)) + 57/2450*x/sqrt(3*x^2 + 2) + 316/1225/sqrt(3*x^2 + 2) - 13/35/(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. 168 vs. \(2 (67) = 134\).
Time = 0.31 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.05 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=-\frac {1}{85750} \, \sqrt {35} {\left (19 \, \sqrt {35} \sqrt {3} - 1264 \, \log \left (\sqrt {35} \sqrt {3} - 9\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + \frac {\frac {\frac {1093}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} - \frac {1820}{{\left (2 \, x + 3\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2 \, x + 3} + \frac {57}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2450 \, \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3}} - \frac {632 \, \sqrt {35} \log \left (\sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} - 9\right )}{42875 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} \]
-1/85750*sqrt(35)*(19*sqrt(35)*sqrt(3) - 1264*log(sqrt(35)*sqrt(3) - 9))*s gn(1/(2*x + 3)) + 1/2450*((1093/sgn(1/(2*x + 3)) - 1820/((2*x + 3)*sgn(1/( 2*x + 3))))/(2*x + 3) + 57/sgn(1/(2*x + 3)))/sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) - 632/42875*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))
Time = 0.12 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.91 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {632\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{42875}-\frac {632\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{42875}+\frac {71\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{4900\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {71\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{4900\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {26\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1225\,\left (x+\frac {3}{2}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,199{}\mathrm {i}}{14700\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,199{}\mathrm {i}}{14700\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \]
(632*35^(1/2)*log(x + 3/2))/42875 - (632*35^(1/2)*log(x - (3^(1/2)*35^(1/2 )*(x^2 + 2/3)^(1/2))/9 - 4/9))/42875 + (71*3^(1/2)*(x^2 + 2/3)^(1/2))/(490 0*(x - (6^(1/2)*1i)/3)) + (71*3^(1/2)*(x^2 + 2/3)^(1/2))/(4900*(x + (6^(1/ 2)*1i)/3)) - (26*3^(1/2)*(x^2 + 2/3)^(1/2))/(1225*(x + 3/2)) - (3^(1/2)*6^ (1/2)*(x^2 + 2/3)^(1/2)*199i)/(14700*(x - (6^(1/2)*1i)/3)) + (3^(1/2)*6^(1 /2)*(x^2 + 2/3)^(1/2)*199i)/(14700*(x + (6^(1/2)*1i)/3))