Integrand size = 27, antiderivative size = 144 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^6} \, dx=\frac {3159 (7+8 x) \sqrt {2+5 x+3 x^2}}{20000 (3+2 x)^2}-\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{25 (3+2 x)^5}-\frac {339 \left (2+5 x+3 x^2\right )^{3/2}}{500 (3+2 x)^4}-\frac {87 \left (2+5 x+3 x^2\right )^{3/2}}{125 (3+2 x)^3}-\frac {3159 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{40000 \sqrt {5}} \]
-13/25*(3*x^2+5*x+2)^(3/2)/(3+2*x)^5-339/500*(3*x^2+5*x+2)^(3/2)/(3+2*x)^4 -87/125*(3*x^2+5*x+2)^(3/2)/(3+2*x)^3-3159/200000*arctanh(1/10*(7+8*x)*5^( 1/2)/(3*x^2+5*x+2)^(1/2))*5^(1/2)+3159/20000*(7+8*x)*(3*x^2+5*x+2)^(1/2)/( 3+2*x)^2
Time = 0.44 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.54 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^6} \, dx=\frac {\frac {5 \sqrt {2+5 x+3 x^2} \left (244331+606326 x+549516 x^2+225816 x^3+35136 x^4\right )}{(3+2 x)^5}-3159 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )}{100000} \]
((5*Sqrt[2 + 5*x + 3*x^2]*(244331 + 606326*x + 549516*x^2 + 225816*x^3 + 3 5136*x^4))/(3 + 2*x)^5 - 3159*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)])/100000
Time = 0.33 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1237, 27, 1237, 27, 1228, 1152, 1154, 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.
| \(\displaystyle \int \frac {(5-x) \sqrt {3 x^2+5 x+2}}{(2 x+3)^6} \, dx\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {1}{25} \int -\frac {3 (35-52 x) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)^5}dx-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{25 (2 x+3)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{50} \int \frac {(35-52 x) \sqrt {3 x^2+5 x+2}}{(2 x+3)^5}dx-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{25 (2 x+3)^5}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {3}{50} \left (-\frac {1}{20} \int -\frac {3 (241-226 x) \sqrt {3 x^2+5 x+2}}{(2 x+3)^4}dx-\frac {113 \left (3 x^2+5 x+2\right )^{3/2}}{10 (2 x+3)^4}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{25 (2 x+3)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{50} \left (\frac {3}{20} \int \frac {(241-226 x) \sqrt {3 x^2+5 x+2}}{(2 x+3)^4}dx-\frac {113 \left (3 x^2+5 x+2\right )^{3/2}}{10 (2 x+3)^4}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{25 (2 x+3)^5}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {3}{50} \left (\frac {3}{20} \left (351 \int \frac {\sqrt {3 x^2+5 x+2}}{(2 x+3)^3}dx-\frac {232 \left (3 x^2+5 x+2\right )^{3/2}}{3 (2 x+3)^3}\right )-\frac {113 \left (3 x^2+5 x+2\right )^{3/2}}{10 (2 x+3)^4}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{25 (2 x+3)^5}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {3}{50} \left (\frac {3}{20} \left (351 \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {1}{40} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {232 \left (3 x^2+5 x+2\right )^{3/2}}{3 (2 x+3)^3}\right )-\frac {113 \left (3 x^2+5 x+2\right )^{3/2}}{10 (2 x+3)^4}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{25 (2 x+3)^5}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {3}{50} \left (\frac {3}{20} \left (351 \left (\frac {1}{20} \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (8 x+7)}{20 (2 x+3)^2}\right )-\frac {232 \left (3 x^2+5 x+2\right )^{3/2}}{3 (2 x+3)^3}\right )-\frac {113 \left (3 x^2+5 x+2\right )^{3/2}}{10 (2 x+3)^4}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{25 (2 x+3)^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{50} \left (\frac {3}{20} \left (351 \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {\text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40 \sqrt {5}}\right )-\frac {232 \left (3 x^2+5 x+2\right )^{3/2}}{3 (2 x+3)^3}\right )-\frac {113 \left (3 x^2+5 x+2\right )^{3/2}}{10 (2 x+3)^4}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{25 (2 x+3)^5}\) |
(-13*(2 + 5*x + 3*x^2)^(3/2))/(25*(3 + 2*x)^5) + (3*((-113*(2 + 5*x + 3*x^ 2)^(3/2))/(10*(3 + 2*x)^4) + (3*((-232*(2 + 5*x + 3*x^2)^(3/2))/(3*(3 + 2* x)^3) + 351*(((7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(20*(3 + 2*x)^2) - ArcTanh[ (7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])]/(40*Sqrt[5]))))/20))/50
3.25.16.3.1 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[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b *x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a *c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] && GtQ[p, 0]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^ (m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ] && EqQ[Simplify[m + 2*p + 3], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) *(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Time = 0.34 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.58
| method | result | size |
| risch | \(\frac {105408 x^{6}+853128 x^{5}+2847900 x^{4}+5018190 x^{3}+4863655 x^{2}+2434307 x +488662}{20000 \left (3+2 x \right )^{5} \sqrt {3 x^{2}+5 x +2}}+\frac {3159 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{200000}\) | \(83\) |
| trager | \(\frac {\left (35136 x^{4}+225816 x^{3}+549516 x^{2}+606326 x +244331\right ) \sqrt {3 x^{2}+5 x +2}}{20000 \left (3+2 x \right )^{5}}-\frac {3159 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{200000}\) | \(92\) |
| default | \(-\frac {339 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{8000 \left (x +\frac {3}{2}\right )^{4}}-\frac {87 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{1000 \left (x +\frac {3}{2}\right )^{3}}-\frac {3159 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{20000 \left (x +\frac {3}{2}\right )^{2}}-\frac {3159 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{12500 \left (x +\frac {3}{2}\right )}-\frac {3159 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{200000}+\frac {3159 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{200000}+\frac {3159 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{25000}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{800 \left (x +\frac {3}{2}\right )^{5}}\) | \(174\) |
1/20000*(105408*x^6+853128*x^5+2847900*x^4+5018190*x^3+4863655*x^2+2434307 *x+488662)/(3+2*x)^5/(3*x^2+5*x+2)^(1/2)+3159/200000*5^(1/2)*arctanh(2/5*( -7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))
Time = 0.29 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.98 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^6} \, dx=\frac {3159 \, \sqrt {5} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (-\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (35136 \, x^{4} + 225816 \, x^{3} + 549516 \, x^{2} + 606326 \, x + 244331\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{400000 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \]
1/400000*(3159*sqrt(5)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 24 3)*log(-(4*sqrt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) - 124*x^2 - 212*x - 89) /(4*x^2 + 12*x + 9)) + 20*(35136*x^4 + 225816*x^3 + 549516*x^2 + 606326*x + 244331)*sqrt(3*x^2 + 5*x + 2))/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)
\[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^6} \, dx=- \int \left (- \frac {5 \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \frac {x \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\, dx \]
-Integral(-5*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320 *x**3 + 4860*x**2 + 2916*x + 729), x) - Integral(x*sqrt(3*x**2 + 5*x + 2)/ (64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860*x**2 + 2916*x + 729), x )
Time = 0.29 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.47 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^6} \, dx=\frac {3159}{200000} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {9477}{20000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{25 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {339 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{500 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {87 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{125 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {3159 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{5000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {3159 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{5000 \, {\left (2 \, x + 3\right )}} \]
3159/200000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/a bs(2*x + 3) - 2) + 9477/20000*sqrt(3*x^2 + 5*x + 2) - 13/25*(3*x^2 + 5*x + 2)^(3/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 339/500* (3*x^2 + 5*x + 2)^(3/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 87/125* (3*x^2 + 5*x + 2)^(3/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 3159/5000*(3*x^2 + 5*x + 2)^(3/2)/(4*x^2 + 12*x + 9) - 3159/5000*sqrt(3*x^2 + 5*x + 2)/(2*x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 363 vs. \(2 (118) = 236\).
Time = 0.32 (sec) , antiderivative size = 363, normalized size of antiderivative = 2.52 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^6} \, dx=-\frac {3159}{200000} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac {\sqrt {3} {\left (50544 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 2047032 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 11747352 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 121295556 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 269183136 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 1164571962 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 1077361162 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 1845838971 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 592102521 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 244862928\right )}}{60000 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{5}} \]
-3159/200000*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt (3*x^2 + 5*x + 2))/abs(-4*sqrt(3)*x + 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) + 1/60000*sqrt(3)*(50544*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5 *x + 2))^9 + 2047032*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 11747352*sqrt (3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 + 121295556*(sqrt(3)*x - sqrt(3* x^2 + 5*x + 2))^6 + 269183136*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^ 5 + 1164571962*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 1077361162*sqrt(3)* (sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 1845838971*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 592102521*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 2 44862928)/(2*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 6*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^5
Timed out. \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^6} \, dx=-\int \frac {\left (x-5\right )\,\sqrt {3\,x^2+5\,x+2}}{{\left (2\,x+3\right )}^6} \,d x \]