Integrand size = 27, antiderivative size = 135 \[ \int (5-x) (3+2 x)^3 \sqrt {2+5 x+3 x^2} \, dx=\frac {6221 (5+6 x) \sqrt {2+5 x+3 x^2}}{5184}+\frac {11}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{18} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}+\frac {(27487+11538 x) \left (2+5 x+3 x^2\right )^{3/2}}{3240}-\frac {6221 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{10368 \sqrt {3}} \]
11/15*(3+2*x)^2*(3*x^2+5*x+2)^(3/2)-1/18*(3+2*x)^3*(3*x^2+5*x+2)^(3/2)+1/3 240*(27487+11538*x)*(3*x^2+5*x+2)^(3/2)-6221/31104*arctanh(1/6*(5+6*x)*3^( 1/2)/(3*x^2+5*x+2)^(1/2))*3^(1/2)+6221/5184*(5+6*x)*(3*x^2+5*x+2)^(1/2)
Time = 0.35 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.56 \[ \int (5-x) (3+2 x)^3 \sqrt {2+5 x+3 x^2} \, dx=\frac {-3 \sqrt {2+5 x+3 x^2} \left (-859701-2432350 x-2317848 x^2-825840 x^3-14976 x^4+34560 x^5\right )-31105 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{77760} \]
(-3*Sqrt[2 + 5*x + 3*x^2]*(-859701 - 2432350*x - 2317848*x^2 - 825840*x^3 - 14976*x^4 + 34560*x^5) - 31105*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2] /(1 + x)])/77760
Time = 0.31 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1236, 27, 1236, 27, 1225, 1087, 1092, 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.
\(\displaystyle \int (5-x) (2 x+3)^3 \sqrt {3 x^2+5 x+2} \, dx\) |
\(\Big \downarrow \) 1236 |
\(\displaystyle \frac {1}{18} \int \frac {3}{2} (2 x+3)^2 (132 x+203) \sqrt {3 x^2+5 x+2}dx-\frac {1}{18} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{12} \int (2 x+3)^2 (132 x+203) \sqrt {3 x^2+5 x+2}dx-\frac {1}{18} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}\) |
\(\Big \downarrow \) 1236 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{15} \int 3 (2 x+3) (1282 x+1703) \sqrt {3 x^2+5 x+2}dx+\frac {44}{5} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2\right )-\frac {1}{18} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{5} \int (2 x+3) (1282 x+1703) \sqrt {3 x^2+5 x+2}dx+\frac {44}{5} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2\right )-\frac {1}{18} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}\) |
\(\Big \downarrow \) 1225 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{5} \left (\frac {31105}{36} \int \sqrt {3 x^2+5 x+2}dx+\frac {1}{54} (11538 x+27487) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {44}{5} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2\right )-\frac {1}{18} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{5} \left (\frac {31105}{36} \left (\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}-\frac {1}{24} \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx\right )+\frac {1}{54} (11538 x+27487) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {44}{5} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2\right )-\frac {1}{18} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{5} \left (\frac {31105}{36} \left (\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}-\frac {1}{12} \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}\right )+\frac {1}{54} (11538 x+27487) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {44}{5} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2\right )-\frac {1}{18} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{12} \left (\frac {1}{5} \left (\frac {31105}{36} \left (\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}-\frac {\text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{24 \sqrt {3}}\right )+\frac {1}{54} (11538 x+27487) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {44}{5} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2\right )-\frac {1}{18} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}\) |
-1/18*((3 + 2*x)^3*(2 + 5*x + 3*x^2)^(3/2)) + ((44*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(3/2))/5 + (((27487 + 11538*x)*(2 + 5*x + 3*x^2)^(3/2))/54 + (31105 *(((5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/12 - ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt [2 + 5*x + 3*x^2])]/(24*Sqrt[3])))/36)/5)/12
3.25.7.3.1 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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c , d, e, f, g, p}, x] && !LeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1 )*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
Time = 0.33 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.52
method | result | size |
risch | \(-\frac {\left (34560 x^{5}-14976 x^{4}-825840 x^{3}-2317848 x^{2}-2432350 x -859701\right ) \sqrt {3 x^{2}+5 x +2}}{25920}-\frac {6221 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{31104}\) | \(70\) |
trager | \(\left (-\frac {4}{3} x^{5}+\frac {26}{45} x^{4}+\frac {1147}{36} x^{3}+\frac {96577}{1080} x^{2}+\frac {243235}{2592} x +\frac {286567}{8640}\right ) \sqrt {3 x^{2}+5 x +2}-\frac {6221 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{31104}\) | \(81\) |
default | \(\frac {6221 \left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{5184}-\frac {6221 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{31104}+\frac {44011 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{3240}-\frac {4 x^{3} \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{9}+\frac {14 x^{2} \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{15}+\frac {337 x \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{36}\) | \(113\) |
-1/25920*(34560*x^5-14976*x^4-825840*x^3-2317848*x^2-2432350*x-859701)*(3* x^2+5*x+2)^(1/2)-6221/31104*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))* 3^(1/2)
Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.58 \[ \int (5-x) (3+2 x)^3 \sqrt {2+5 x+3 x^2} \, dx=-\frac {1}{25920} \, {\left (34560 \, x^{5} - 14976 \, x^{4} - 825840 \, x^{3} - 2317848 \, x^{2} - 2432350 \, x - 859701\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {6221}{62208} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \]
-1/25920*(34560*x^5 - 14976*x^4 - 825840*x^3 - 2317848*x^2 - 2432350*x - 8 59701)*sqrt(3*x^2 + 5*x + 2) + 6221/62208*sqrt(3)*log(-4*sqrt(3)*sqrt(3*x^ 2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49)
Time = 0.54 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.61 \[ \int (5-x) (3+2 x)^3 \sqrt {2+5 x+3 x^2} \, dx=\sqrt {3 x^{2} + 5 x + 2} \left (- \frac {4 x^{5}}{3} + \frac {26 x^{4}}{45} + \frac {1147 x^{3}}{36} + \frac {96577 x^{2}}{1080} + \frac {243235 x}{2592} + \frac {286567}{8640}\right ) - \frac {6221 \sqrt {3} \log {\left (6 x + 2 \sqrt {3} \sqrt {3 x^{2} + 5 x + 2} + 5 \right )}}{31104} \]
sqrt(3*x**2 + 5*x + 2)*(-4*x**5/3 + 26*x**4/45 + 1147*x**3/36 + 96577*x**2 /1080 + 243235*x/2592 + 286567/8640) - 6221*sqrt(3)*log(6*x + 2*sqrt(3)*sq rt(3*x**2 + 5*x + 2) + 5)/31104
Time = 0.29 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.90 \[ \int (5-x) (3+2 x)^3 \sqrt {2+5 x+3 x^2} \, dx=-\frac {4}{9} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{3} + \frac {14}{15} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{2} + \frac {337}{36} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {44011}{3240} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {6221}{864} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {6221}{31104} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {31105}{5184} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
-4/9*(3*x^2 + 5*x + 2)^(3/2)*x^3 + 14/15*(3*x^2 + 5*x + 2)^(3/2)*x^2 + 337 /36*(3*x^2 + 5*x + 2)^(3/2)*x + 44011/3240*(3*x^2 + 5*x + 2)^(3/2) + 6221/ 864*sqrt(3*x^2 + 5*x + 2)*x - 6221/31104*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + 31105/5184*sqrt(3*x^2 + 5*x + 2)
Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.55 \[ \int (5-x) (3+2 x)^3 \sqrt {2+5 x+3 x^2} \, dx=-\frac {1}{25920} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (30 \, x - 13\right )} x - 5735\right )} x - 96577\right )} x - 1216175\right )} x - 859701\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {6221}{31104} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]
-1/25920*(2*(12*(6*(8*(30*x - 13)*x - 5735)*x - 96577)*x - 1216175)*x - 85 9701)*sqrt(3*x^2 + 5*x + 2) + 6221/31104*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt( 3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))
Time = 12.47 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.13 \[ \int (5-x) (3+2 x)^3 \sqrt {2+5 x+3 x^2} \, dx=\frac {14\,x^2\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{15}-\frac {4\,x^3\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{9}-\frac {2093\,\sqrt {3}\,\ln \left (\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (3\,x+\frac {5}{2}\right )}{3}\right )}{1296}+\frac {2093\,\left (\frac {x}{2}+\frac {5}{12}\right )\,\sqrt {3\,x^2+5\,x+2}}{18}+\frac {44011\,\sqrt {3\,x^2+5\,x+2}\,\left (72\,x^2+30\,x-27\right )}{77760}+\frac {337\,x\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{36}+\frac {44011\,\sqrt {3}\,\ln \left (2\,\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (6\,x+5\right )}{3}\right )}{31104} \]
(14*x^2*(5*x + 3*x^2 + 2)^(3/2))/15 - (4*x^3*(5*x + 3*x^2 + 2)^(3/2))/9 - (2093*3^(1/2)*log((5*x + 3*x^2 + 2)^(1/2) + (3^(1/2)*(3*x + 5/2))/3))/1296 + (2093*(x/2 + 5/12)*(5*x + 3*x^2 + 2)^(1/2))/18 + (44011*(5*x + 3*x^2 + 2)^(1/2)*(30*x + 72*x^2 - 27))/77760 + (337*x*(5*x + 3*x^2 + 2)^(3/2))/36 + (44011*3^(1/2)*log(2*(5*x + 3*x^2 + 2)^(1/2) + (3^(1/2)*(6*x + 5))/3))/3 1104