Integrand size = 24, antiderivative size = 134 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^6} \, dx=-\frac {67 \sqrt {1-2 x} (3+5 x)^2}{315 (2+3 x)^3}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {3 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^4}-\frac {2 \sqrt {1-2 x} (9529+15074 x)}{9261 (2+3 x)^2}-\frac {13892 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9261 \sqrt {21}} \]
-1/15*(1-2*x)^(3/2)*(3+5*x)^3/(2+3*x)^5-13892/194481*arctanh(1/7*21^(1/2)* (1-2*x)^(1/2))*21^(1/2)-67/315*(3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^3+3/5*(3+5* x)^3*(1-2*x)^(1/2)/(2+3*x)^4-2/9261*(9529+15074*x)*(1-2*x)^(1/2)/(2+3*x)^2
Time = 0.30 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.52 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {4 \left (\frac {21 \sqrt {1-2 x} \left (300049+2619854 x+7992771 x^2+10375830 x^3+4904370 x^4\right )}{4 (2+3 x)^5}-17365 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )}{972405} \]
(4*((21*Sqrt[1 - 2*x]*(300049 + 2619854*x + 7992771*x^2 + 10375830*x^3 + 4 904370*x^4))/(4*(2 + 3*x)^5) - 17365*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2 *x]]))/972405
Time = 0.23 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {108, 27, 166, 27, 166, 27, 162, 73, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)^3}{(3 x+2)^6} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{15} \int \frac {3 (2-15 x) \sqrt {1-2 x} (5 x+3)^2}{(3 x+2)^5}dx-\frac {(1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \frac {(2-15 x) \sqrt {1-2 x} (5 x+3)^2}{(3 x+2)^5}dx-\frac {(1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{5} \left (\frac {3 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^4}-\frac {1}{12} \int -\frac {12 (19-5 x) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)^4}dx\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\int \frac {(19-5 x) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)^4}dx+\frac {3 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^4}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{63} \int \frac {10 (136-19 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)^3}dx+\frac {3 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^4}-\frac {67 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {10}{63} \int \frac {(136-19 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)^3}dx+\frac {3 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^4}-\frac {67 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 162 |
| \(\displaystyle \frac {1}{5} \left (\frac {10}{63} \left (\frac {3473}{147} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x} (15074 x+9529)}{147 (3 x+2)^2}\right )+\frac {3 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^4}-\frac {67 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{5} \left (\frac {10}{63} \left (-\frac {3473}{147} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x} (15074 x+9529)}{147 (3 x+2)^2}\right )+\frac {3 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^4}-\frac {67 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{5} \left (\frac {10}{63} \left (-\frac {6946 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}}-\frac {\sqrt {1-2 x} (15074 x+9529)}{147 (3 x+2)^2}\right )+\frac {3 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^4}-\frac {67 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}\) |
-1/15*((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(2 + 3*x)^5 + ((-67*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(63*(2 + 3*x)^3) + (3*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^4 + (1 0*(-1/147*(Sqrt[1 - 2*x]*(9529 + 15074*x))/(2 + 3*x)^2 - (6946*ArcTanh[Sqr t[3/7]*Sqrt[1 - 2*x]])/(147*Sqrt[21])))/63)/5
3.19.92.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) )*((g_.) + (h_.)*(x_)), x_] :> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e *g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b *c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] + Sim p[(f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d *(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/( b^2*(b*c - a*d)^2*(m + 1)*(m + 2))) Int[(a + b*x)^(m + 2)*(c + d*x)^n, x] , x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] && !LtQ[n, -2]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
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])
Time = 1.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.46
| method | result | size |
| risch | \(-\frac {9808740 x^{5}+15847290 x^{4}+5609712 x^{3}-2753063 x^{2}-2019756 x -300049}{46305 \left (2+3 x \right )^{5} \sqrt {1-2 x}}-\frac {13892 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{194481}\) | \(61\) |
| pseudoelliptic | \(\frac {-69460 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{5} \sqrt {21}+21 \sqrt {1-2 x}\, \left (4904370 x^{4}+10375830 x^{3}+7992771 x^{2}+2619854 x +300049\right )}{972405 \left (2+3 x \right )^{5}}\) | \(65\) |
| derivativedivides | \(\frac {-\frac {217972 \left (1-2 x \right )^{\frac {9}{2}}}{1029}+\frac {36616 \left (1-2 x \right )^{\frac {7}{2}}}{21}-\frac {1682344 \left (1-2 x \right )^{\frac {5}{2}}}{315}+\frac {194488 \left (1-2 x \right )^{\frac {3}{2}}}{27}-\frac {97244 \sqrt {1-2 x}}{27}}{\left (-4-6 x \right )^{5}}-\frac {13892 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{194481}\) | \(75\) |
| default | \(\frac {-\frac {217972 \left (1-2 x \right )^{\frac {9}{2}}}{1029}+\frac {36616 \left (1-2 x \right )^{\frac {7}{2}}}{21}-\frac {1682344 \left (1-2 x \right )^{\frac {5}{2}}}{315}+\frac {194488 \left (1-2 x \right )^{\frac {3}{2}}}{27}-\frac {97244 \sqrt {1-2 x}}{27}}{\left (-4-6 x \right )^{5}}-\frac {13892 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{194481}\) | \(75\) |
| trager | \(\frac {\left (4904370 x^{4}+10375830 x^{3}+7992771 x^{2}+2619854 x +300049\right ) \sqrt {1-2 x}}{46305 \left (2+3 x \right )^{5}}-\frac {6946 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{194481}\) | \(82\) |
-1/46305*(9808740*x^5+15847290*x^4+5609712*x^3-2753063*x^2-2019756*x-30004 9)/(2+3*x)^5/(1-2*x)^(1/2)-13892/194481*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2) )*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {34730 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (4904370 \, x^{4} + 10375830 \, x^{3} + 7992771 \, x^{2} + 2619854 \, x + 300049\right )} \sqrt {-2 \, x + 1}}{972405 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
1/972405*(34730*sqrt(21)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(4904370*x^4 + 10375830*x^3 + 7992771*x^2 + 2619854*x + 300049)*sqrt(-2*x + 1))/(243*x^ 5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)
Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^6} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.96 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {6946}{194481} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4 \, {\left (2452185 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 20184570 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 61826142 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 83386730 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 41693365 \, \sqrt {-2 \, x + 1}\right )}}{46305 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]
6946/194481*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt (-2*x + 1))) + 4/46305*(2452185*(-2*x + 1)^(9/2) - 20184570*(-2*x + 1)^(7/ 2) + 61826142*(-2*x + 1)^(5/2) - 83386730*(-2*x + 1)^(3/2) + 41693365*sqrt (-2*x + 1))/(243*(2*x - 1)^5 + 2835*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 3087 0*(2*x - 1)^2 + 72030*x - 19208)
Time = 0.29 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {6946}{194481} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2452185 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 20184570 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 61826142 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 83386730 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 41693365 \, \sqrt {-2 \, x + 1}}{370440 \, {\left (3 \, x + 2\right )}^{5}} \]
6946/194481*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/370440*(2452185*(2*x - 1)^4*sqrt(-2*x + 1) + 201 84570*(2*x - 1)^3*sqrt(-2*x + 1) + 61826142*(2*x - 1)^2*sqrt(-2*x + 1) - 8 3386730*(-2*x + 1)^(3/2) + 41693365*sqrt(-2*x + 1))/(3*x + 2)^5
Time = 0.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.80 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {\frac {97244\,\sqrt {1-2\,x}}{6561}-\frac {194488\,{\left (1-2\,x\right )}^{3/2}}{6561}+\frac {1682344\,{\left (1-2\,x\right )}^{5/2}}{76545}-\frac {36616\,{\left (1-2\,x\right )}^{7/2}}{5103}+\frac {217972\,{\left (1-2\,x\right )}^{9/2}}{250047}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}}-\frac {13892\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{194481} \]
((97244*(1 - 2*x)^(1/2))/6561 - (194488*(1 - 2*x)^(3/2))/6561 + (1682344*( 1 - 2*x)^(5/2))/76545 - (36616*(1 - 2*x)^(7/2))/5103 + (217972*(1 - 2*x)^( 9/2))/250047)/((24010*x)/81 + (3430*(2*x - 1)^2)/27 + (490*(2*x - 1)^3)/9 + (35*(2*x - 1)^4)/3 + (2*x - 1)^5 - 19208/243) - (13892*21^(1/2)*atanh((2 1^(1/2)*(1 - 2*x)^(1/2))/7))/194481