Integrand size = 24, antiderivative size = 107 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^5} \, dx=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{378 (2+3 x)^3}-\frac {\sqrt {1-2 x} (3+5 x)^3}{12 (2+3 x)^4}-\frac {5 \sqrt {1-2 x} (70429+110981 x)}{222264 (2+3 x)^2}+\frac {328715 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{111132 \sqrt {21}} \]
328715/2333772*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-53/378*(3+5*x) ^2*(1-2*x)^(1/2)/(2+3*x)^3-1/12*(3+5*x)^3*(1-2*x)^(1/2)/(2+3*x)^4-5/222264 *(70429+110981*x)*(1-2*x)^(1/2)/(2+3*x)^2
Time = 0.24 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^5} \, dx=\frac {-\frac {21 \sqrt {1-2 x} \left (2469626+11657098 x+18358575 x^2+9646695 x^3\right )}{2 (2+3 x)^4}+328715 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2333772} \]
((-21*Sqrt[1 - 2*x]*(2469626 + 11657098*x + 18358575*x^2 + 9646695*x^3))/( 2*(2 + 3*x)^4) + 328715*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/2333772
Time = 0.21 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {108, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^5} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{12} \int \frac {(12-35 x) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)^4}dx-\frac {\sqrt {1-2 x} (5 x+3)^3}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{63} \int \frac {5 (89-629 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)^3}dx-\frac {106 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{12} \left (\frac {5}{63} \int \frac {(89-629 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)^3}dx-\frac {106 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 162 |
| \(\displaystyle \frac {1}{12} \left (\frac {5}{63} \left (-\frac {65743}{294} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x} (110981 x+70429)}{294 (3 x+2)^2}\right )-\frac {106 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{12} \left (\frac {5}{63} \left (\frac {65743}{294} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x} (110981 x+70429)}{294 (3 x+2)^2}\right )-\frac {106 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{12} \left (\frac {5}{63} \left (\frac {65743 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}}-\frac {\sqrt {1-2 x} (110981 x+70429)}{294 (3 x+2)^2}\right )-\frac {106 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{12 (3 x+2)^4}\) |
-1/12*(Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^4 + ((-106*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(63*(2 + 3*x)^3) + (5*(-1/294*(Sqrt[1 - 2*x]*(70429 + 110981*x))/( 2 + 3*x)^2 + (65743*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(147*Sqrt[21])))/63) /12
3.19.24.3.1 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 = 0.99 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.52
| method | result | size |
| risch | \(\frac {19293390 x^{4}+27070455 x^{3}+4955621 x^{2}-6717846 x -2469626}{222264 \left (2+3 x \right )^{4} \sqrt {1-2 x}}+\frac {328715 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2333772}\) | \(56\) |
| pseudoelliptic | \(\frac {657430 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{4} \sqrt {21}-21 \sqrt {1-2 x}\, \left (9646695 x^{3}+18358575 x^{2}+11657098 x +2469626\right )}{4667544 \left (2+3 x \right )^{4}}\) | \(60\) |
| derivativedivides | \(-\frac {324 \left (-\frac {119095 \left (1-2 x \right )^{\frac {7}{2}}}{444528}+\frac {3126535 \left (1-2 x \right )^{\frac {5}{2}}}{1714608}-\frac {3040873 \left (1-2 x \right )^{\frac {3}{2}}}{734832}+\frac {328715 \sqrt {1-2 x}}{104976}\right )}{\left (-4-6 x \right )^{4}}+\frac {328715 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2333772}\) | \(66\) |
| default | \(-\frac {324 \left (-\frac {119095 \left (1-2 x \right )^{\frac {7}{2}}}{444528}+\frac {3126535 \left (1-2 x \right )^{\frac {5}{2}}}{1714608}-\frac {3040873 \left (1-2 x \right )^{\frac {3}{2}}}{734832}+\frac {328715 \sqrt {1-2 x}}{104976}\right )}{\left (-4-6 x \right )^{4}}+\frac {328715 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2333772}\) | \(66\) |
| trager | \(-\frac {\left (9646695 x^{3}+18358575 x^{2}+11657098 x +2469626\right ) \sqrt {1-2 x}}{222264 \left (2+3 x \right )^{4}}-\frac {328715 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{4667544}\) | \(77\) |
1/222264*(19293390*x^4+27070455*x^3+4955621*x^2-6717846*x-2469626)/(2+3*x) ^4/(1-2*x)^(1/2)+328715/2333772*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/ 2)
Time = 0.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^5} \, dx=\frac {328715 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (9646695 \, x^{3} + 18358575 \, x^{2} + 11657098 \, x + 2469626\right )} \sqrt {-2 \, x + 1}}{4667544 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
1/4667544*(328715*sqrt(21)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((3 *x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(9646695*x^3 + 18358575* x^2 + 11657098*x + 2469626)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^5} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^5} \, dx=-\frac {328715}{4667544} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {9646695 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 65657235 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 149002777 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 112749245 \, \sqrt {-2 \, x + 1}}{111132 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]
-328715/4667544*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3* sqrt(-2*x + 1))) + 1/111132*(9646695*(-2*x + 1)^(7/2) - 65657235*(-2*x + 1 )^(5/2) + 149002777*(-2*x + 1)^(3/2) - 112749245*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)
Time = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^5} \, dx=-\frac {328715}{4667544} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {9646695 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 65657235 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 149002777 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 112749245 \, \sqrt {-2 \, x + 1}}{1778112 \, {\left (3 \, x + 2\right )}^{4}} \]
-328715/4667544*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt (21) + 3*sqrt(-2*x + 1))) - 1/1778112*(9646695*(2*x - 1)^3*sqrt(-2*x + 1) + 65657235*(2*x - 1)^2*sqrt(-2*x + 1) - 149002777*(-2*x + 1)^(3/2) + 11274 9245*sqrt(-2*x + 1))/(3*x + 2)^4
Time = 1.34 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^5} \, dx=\frac {328715\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2333772}-\frac {\frac {328715\,\sqrt {1-2\,x}}{26244}-\frac {3040873\,{\left (1-2\,x\right )}^{3/2}}{183708}+\frac {3126535\,{\left (1-2\,x\right )}^{5/2}}{428652}-\frac {119095\,{\left (1-2\,x\right )}^{7/2}}{111132}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \]