Integrand size = 27, antiderivative size = 183 \[ \int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {454969 (5+6 x) \sqrt {2+5 x+3 x^2}}{4478976}+\frac {454969 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{559872}+\frac {487}{486} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac {299}{648} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}+\frac {(420721+188910 x) \left (2+5 x+3 x^2\right )^{5/2}}{58320}+\frac {454969 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{8957952 \sqrt {3}} \]
454969/559872*(5+6*x)*(3*x^2+5*x+2)^(3/2)+487/486*(3+2*x)^2*(3*x^2+5*x+2)^ (5/2)+299/648*(3+2*x)^3*(3*x^2+5*x+2)^(5/2)-1/27*(3+2*x)^4*(3*x^2+5*x+2)^( 5/2)+1/58320*(420721+188910*x)*(3*x^2+5*x+2)^(5/2)+454969/26873856*arctanh (1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x+2)^(1/2))*3^(1/2)-454969/4478976*(5+6*x)*( 3*x^2+5*x+2)^(1/2)
Time = 0.64 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.50 \[ \int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2} \, dx=\frac {-3 \sqrt {2+5 x+3 x^2} \left (-2471988351-15049298650 x-37650690888 x^2-49917376080 x^3-37262745216 x^4-14811482880 x^5-2143687680 x^6+370759680 x^7+119439360 x^8\right )+2274845 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{67184640} \]
(-3*Sqrt[2 + 5*x + 3*x^2]*(-2471988351 - 15049298650*x - 37650690888*x^2 - 49917376080*x^3 - 37262745216*x^4 - 14811482880*x^5 - 2143687680*x^6 + 37 0759680*x^7 + 119439360*x^8) + 2274845*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/67184640
Time = 0.39 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {1236, 27, 1236, 1236, 27, 1225, 1087, 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.
| \(\displaystyle \int (5-x) (2 x+3)^4 \left (3 x^2+5 x+2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {1}{27} \int \frac {1}{2} (2 x+3)^3 (598 x+917) \left (3 x^2+5 x+2\right )^{3/2}dx-\frac {1}{27} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{54} \int (2 x+3)^3 (598 x+917) \left (3 x^2+5 x+2\right )^{3/2}dx-\frac {1}{27} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {1}{54} \left (\frac {1}{24} \int (2 x+3)^2 (27272 x+36423) \left (3 x^2+5 x+2\right )^{3/2}dx+\frac {299}{12} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^3\right )-\frac {1}{27} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {1}{54} \left (\frac {1}{24} \left (\frac {1}{21} \int 7 (2 x+3) (113346 x+150539) \left (3 x^2+5 x+2\right )^{3/2}dx+\frac {3896}{3} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {299}{12} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^3\right )-\frac {1}{27} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{54} \left (\frac {1}{24} \left (\frac {1}{3} \int (2 x+3) (113346 x+150539) \left (3 x^2+5 x+2\right )^{3/2}dx+\frac {3896}{3} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {299}{12} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^3\right )-\frac {1}{27} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {1}{54} \left (\frac {1}{24} \left (\frac {1}{3} \left (\frac {454969}{6} \int \left (3 x^2+5 x+2\right )^{3/2}dx+\frac {1}{15} (188910 x+420721) \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {3896}{3} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {299}{12} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^3\right )-\frac {1}{27} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{54} \left (\frac {1}{24} \left (\frac {1}{3} \left (\frac {454969}{6} \left (\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{16} \int \sqrt {3 x^2+5 x+2}dx\right )+\frac {1}{15} (188910 x+420721) \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {3896}{3} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {299}{12} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^3\right )-\frac {1}{27} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{54} \left (\frac {1}{24} \left (\frac {1}{3} \left (\frac {454969}{6} \left (\frac {1}{16} \left (\frac {1}{24} \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1}{15} (188910 x+420721) \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {3896}{3} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {299}{12} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^3\right )-\frac {1}{27} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{54} \left (\frac {1}{24} \left (\frac {1}{3} \left (\frac {454969}{6} \left (\frac {1}{16} \left (\frac {1}{12} \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1}{15} (188910 x+420721) \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {3896}{3} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {299}{12} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^3\right )-\frac {1}{27} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{54} \left (\frac {1}{24} \left (\frac {1}{3} \left (\frac {454969}{6} \left (\frac {1}{16} \left (\frac {\text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{24 \sqrt {3}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1}{15} (188910 x+420721) \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {3896}{3} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {299}{12} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^3\right )-\frac {1}{27} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{5/2}\) |
-1/27*((3 + 2*x)^4*(2 + 5*x + 3*x^2)^(5/2)) + ((299*(3 + 2*x)^3*(2 + 5*x + 3*x^2)^(5/2))/12 + ((3896*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(5/2))/3 + (((420 721 + 188910*x)*(2 + 5*x + 3*x^2)^(5/2))/15 + (454969*(((5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/24 + (-1/12*((5 + 6*x)*Sqrt[2 + 5*x + 3*x^2]) + ArcTanh[( 5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])]/(24*Sqrt[3]))/16))/6)/3)/24)/5 4
3.25.18.3.1 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 = 0.34 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.46
| method | result | size |
| risch | \(-\frac {\left (119439360 x^{8}+370759680 x^{7}-2143687680 x^{6}-14811482880 x^{5}-37262745216 x^{4}-49917376080 x^{3}-37650690888 x^{2}-15049298650 x -2471988351\right ) \sqrt {3 x^{2}+5 x +2}}{22394880}+\frac {454969 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{26873856}\) | \(85\) |
| trager | \(\left (-\frac {16}{3} x^{8}-\frac {149}{9} x^{7}+\frac {1723}{18} x^{6}+\frac {428573}{648} x^{5}+\frac {32346133}{19440} x^{4}+\frac {69329689}{31104} x^{3}+\frac {1568778787}{933120} x^{2}+\frac {1504929865}{2239488} x +\frac {823996117}{7464960}\right ) \sqrt {3 x^{2}+5 x +2}-\frac {454969 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{26873856}\) | \(96\) |
| default | \(\frac {454969 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{559872}-\frac {454969 \left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{4478976}+\frac {454969 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{26873856}+\frac {1498291 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{58320}-\frac {16 x^{4} \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{27}+\frac {11 x^{3} \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{81}+\frac {6133 x^{2} \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{486}+\frac {2317 x \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{72}\) | \(149\) |
-1/22394880*(119439360*x^8+370759680*x^7-2143687680*x^6-14811482880*x^5-37 262745216*x^4-49917376080*x^3-37650690888*x^2-15049298650*x-2471988351)*(3 *x^2+5*x+2)^(1/2)+454969/26873856*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^( 1/2))*3^(1/2)
Time = 0.30 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.51 \[ \int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {1}{22394880} \, {\left (119439360 \, x^{8} + 370759680 \, x^{7} - 2143687680 \, x^{6} - 14811482880 \, x^{5} - 37262745216 \, x^{4} - 49917376080 \, x^{3} - 37650690888 \, x^{2} - 15049298650 \, x - 2471988351\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {454969}{53747712} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \]
-1/22394880*(119439360*x^8 + 370759680*x^7 - 2143687680*x^6 - 14811482880* x^5 - 37262745216*x^4 - 49917376080*x^3 - 37650690888*x^2 - 15049298650*x - 2471988351)*sqrt(3*x^2 + 5*x + 2) + 454969/53747712*sqrt(3)*log(4*sqrt(3 )*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49)
Time = 0.64 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.57 \[ \int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2} \, dx=\sqrt {3 x^{2} + 5 x + 2} \left (- \frac {16 x^{8}}{3} - \frac {149 x^{7}}{9} + \frac {1723 x^{6}}{18} + \frac {428573 x^{5}}{648} + \frac {32346133 x^{4}}{19440} + \frac {69329689 x^{3}}{31104} + \frac {1568778787 x^{2}}{933120} + \frac {1504929865 x}{2239488} + \frac {823996117}{7464960}\right ) + \frac {454969 \sqrt {3} \log {\left (6 x + 2 \sqrt {3} \sqrt {3 x^{2} + 5 x + 2} + 5 \right )}}{26873856} \]
sqrt(3*x**2 + 5*x + 2)*(-16*x**8/3 - 149*x**7/9 + 1723*x**6/18 + 428573*x* *5/648 + 32346133*x**4/19440 + 69329689*x**3/31104 + 1568778787*x**2/93312 0 + 1504929865*x/2239488 + 823996117/7464960) + 454969*sqrt(3)*log(6*x + 2 *sqrt(3)*sqrt(3*x**2 + 5*x + 2) + 5)/26873856
Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.91 \[ \int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {16}{27} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x^{4} + \frac {11}{81} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x^{3} + \frac {6133}{486} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x^{2} + \frac {2317}{72} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {1498291}{58320} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {454969}{93312} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {2274845}{559872} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {454969}{746496} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {454969}{26873856} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {2274845}{4478976} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
-16/27*(3*x^2 + 5*x + 2)^(5/2)*x^4 + 11/81*(3*x^2 + 5*x + 2)^(5/2)*x^3 + 6 133/486*(3*x^2 + 5*x + 2)^(5/2)*x^2 + 2317/72*(3*x^2 + 5*x + 2)^(5/2)*x + 1498291/58320*(3*x^2 + 5*x + 2)^(5/2) + 454969/93312*(3*x^2 + 5*x + 2)^(3/ 2)*x + 2274845/559872*(3*x^2 + 5*x + 2)^(3/2) - 454969/746496*sqrt(3*x^2 + 5*x + 2)*x + 454969/26873856*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 2274845/4478976*sqrt(3*x^2 + 5*x + 2)
Time = 0.30 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.49 \[ \int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {1}{22394880} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (30 \, {\left (36 \, {\left (2 \, {\left (48 \, x + 149\right )} x - 1723\right )} x - 428573\right )} x - 32346133\right )} x - 346648445\right )} x - 1568778787\right )} x - 7524649325\right )} x - 2471988351\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {454969}{26873856} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]
-1/22394880*(2*(12*(6*(8*(30*(36*(2*(48*x + 149)*x - 1723)*x - 428573)*x - 32346133)*x - 346648445)*x - 1568778787)*x - 7524649325)*x - 2471988351)* sqrt(3*x^2 + 5*x + 2) - 454969/26873856*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3 )*x - sqrt(3*x^2 + 5*x + 2)) - 5))
Timed out. \[ \int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\int {\left (2\,x+3\right )}^4\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2} \,d x \]