Integrand size = 24, antiderivative size = 147 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^7} \, dx=\frac {43957 \sqrt {1-2 x}}{1333584 (2+3 x)^2}+\frac {43957 \sqrt {1-2 x}}{3111696 (2+3 x)}-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{945 (2+3 x)^5}-\frac {\sqrt {1-2 x} (3+5 x)^3}{18 (2+3 x)^6}-\frac {\sqrt {1-2 x} (98995+160029 x)}{476280 (2+3 x)^4}+\frac {43957 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1555848 \sqrt {21}} \]
43957/32672808*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+43957/1333584* (1-2*x)^(1/2)/(2+3*x)^2+43957/3111696*(1-2*x)^(1/2)/(2+3*x)-53/945*(3+5*x) ^2*(1-2*x)^(1/2)/(2+3*x)^5-1/18*(3+5*x)^3*(1-2*x)^(1/2)/(2+3*x)^6-1/476280 *(98995+160029*x)*(1-2*x)^(1/2)/(2+3*x)^4
Time = 0.36 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.51 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^7} \, dx=\frac {\frac {21 \sqrt {1-2 x} \left (-36741296-150340360 x-139462938 x^2+127601514 x^3+219565215 x^4+53407755 x^5\right )}{2 (2+3 x)^6}+219785 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{163364040} \]
((21*Sqrt[1 - 2*x]*(-36741296 - 150340360*x - 139462938*x^2 + 127601514*x^ 3 + 219565215*x^4 + 53407755*x^5))/(2*(2 + 3*x)^6) + 219785*Sqrt[21]*ArcTa nh[Sqrt[3/7]*Sqrt[1 - 2*x]])/163364040
Time = 0.23 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {108, 166, 162, 52, 52, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^7} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{18} \int \frac {(12-35 x) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)^6}dx-\frac {\sqrt {1-2 x} (5 x+3)^3}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{105} \int \frac {(247-3475 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)^5}dx-\frac {106 \sqrt {1-2 x} (5 x+3)^2}{105 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 162 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{105} \left (-\frac {219785}{252} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3}dx-\frac {\sqrt {1-2 x} (160029 x+98995)}{252 (3 x+2)^4}\right )-\frac {106 \sqrt {1-2 x} (5 x+3)^2}{105 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{105} \left (-\frac {219785}{252} \left (\frac {3}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (160029 x+98995)}{252 (3 x+2)^4}\right )-\frac {106 \sqrt {1-2 x} (5 x+3)^2}{105 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{105} \left (-\frac {219785}{252} \left (\frac {3}{14} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (160029 x+98995)}{252 (3 x+2)^4}\right )-\frac {106 \sqrt {1-2 x} (5 x+3)^2}{105 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{105} \left (-\frac {219785}{252} \left (\frac {3}{14} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (160029 x+98995)}{252 (3 x+2)^4}\right )-\frac {106 \sqrt {1-2 x} (5 x+3)^2}{105 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{105} \left (-\frac {219785}{252} \left (\frac {3}{14} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (160029 x+98995)}{252 (3 x+2)^4}\right )-\frac {106 \sqrt {1-2 x} (5 x+3)^2}{105 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{18 (3 x+2)^6}\) |
-1/18*(Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^6 + ((-106*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(105*(2 + 3*x)^5) + (-1/252*(Sqrt[1 - 2*x]*(98995 + 160029*x))/(2 + 3*x)^4 - (219785*(-1/14*Sqrt[1 - 2*x]/(2 + 3*x)^2 + (3*(-1/7*Sqrt[1 - 2* x]/(2 + 3*x) - (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21])))/14))/25 2)/105)/18
3.19.26.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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 = 1.00 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.45
| method | result | size |
| risch | \(-\frac {106815510 x^{6}+385722675 x^{5}+35637813 x^{4}-406527390 x^{3}-161217782 x^{2}+76857768 x +36741296}{15558480 \left (2+3 x \right )^{6} \sqrt {1-2 x}}+\frac {43957 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{32672808}\) | \(66\) |
| pseudoelliptic | \(\frac {439570 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{6} \sqrt {21}+21 \sqrt {1-2 x}\, \left (53407755 x^{5}+219565215 x^{4}+127601514 x^{3}-139462938 x^{2}-150340360 x -36741296\right )}{326728080 \left (2+3 x \right )^{6}}\) | \(70\) |
| derivativedivides | \(-\frac {11664 \left (\frac {43957 \left (1-2 x \right )^{\frac {11}{2}}}{74680704}-\frac {747269 \left (1-2 x \right )^{\frac {9}{2}}}{96018048}+\frac {1058581 \left (1-2 x \right )^{\frac {7}{2}}}{34292160}-\frac {1354639 \left (1-2 x \right )^{\frac {5}{2}}}{34292160}-\frac {630947 \left (1-2 x \right )^{\frac {3}{2}}}{52907904}+\frac {307699 \sqrt {1-2 x}}{7558272}\right )}{\left (-4-6 x \right )^{6}}+\frac {43957 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{32672808}\) | \(84\) |
| default | \(-\frac {11664 \left (\frac {43957 \left (1-2 x \right )^{\frac {11}{2}}}{74680704}-\frac {747269 \left (1-2 x \right )^{\frac {9}{2}}}{96018048}+\frac {1058581 \left (1-2 x \right )^{\frac {7}{2}}}{34292160}-\frac {1354639 \left (1-2 x \right )^{\frac {5}{2}}}{34292160}-\frac {630947 \left (1-2 x \right )^{\frac {3}{2}}}{52907904}+\frac {307699 \sqrt {1-2 x}}{7558272}\right )}{\left (-4-6 x \right )^{6}}+\frac {43957 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{32672808}\) | \(84\) |
| trager | \(\frac {\left (53407755 x^{5}+219565215 x^{4}+127601514 x^{3}-139462938 x^{2}-150340360 x -36741296\right ) \sqrt {1-2 x}}{15558480 \left (2+3 x \right )^{6}}+\frac {43957 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{65345616}\) | \(87\) |
-1/15558480*(106815510*x^6+385722675*x^5+35637813*x^4-406527390*x^3-161217 782*x^2+76857768*x+36741296)/(2+3*x)^6/(1-2*x)^(1/2)+43957/32672808*arctan h(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^7} \, dx=\frac {219785 \, \sqrt {21} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (53407755 \, x^{5} + 219565215 \, x^{4} + 127601514 \, x^{3} - 139462938 \, x^{2} - 150340360 \, x - 36741296\right )} \sqrt {-2 \, x + 1}}{326728080 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
1/326728080*(219785*sqrt(21)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2 160*x^2 + 576*x + 64)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(53407755*x^5 + 219565215*x^4 + 127601514*x^3 - 139462938*x^2 - 150340 360*x - 36741296)*sqrt(-2*x + 1))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^ 3 + 2160*x^2 + 576*x + 64)
Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^7} \, dx=\text {Timed out} \]
Time = 0.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^7} \, dx=-\frac {43957}{65345616} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {53407755 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 706169205 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 2801005326 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 3584374794 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 1082074105 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 3693926495 \, \sqrt {-2 \, x + 1}}{7779240 \, {\left (729 \, {\left (2 \, x - 1\right )}^{6} + 10206 \, {\left (2 \, x - 1\right )}^{5} + 59535 \, {\left (2 \, x - 1\right )}^{4} + 185220 \, {\left (2 \, x - 1\right )}^{3} + 324135 \, {\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \]
-43957/65345616*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3* sqrt(-2*x + 1))) - 1/7779240*(53407755*(-2*x + 1)^(11/2) - 706169205*(-2*x + 1)^(9/2) + 2801005326*(-2*x + 1)^(7/2) - 3584374794*(-2*x + 1)^(5/2) - 1082074105*(-2*x + 1)^(3/2) + 3693926495*sqrt(-2*x + 1))/(729*(2*x - 1)^6 + 10206*(2*x - 1)^5 + 59535*(2*x - 1)^4 + 185220*(2*x - 1)^3 + 324135*(2*x - 1)^2 + 605052*x - 184877)
Time = 0.29 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^7} \, dx=-\frac {43957}{65345616} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {53407755 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + 706169205 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 2801005326 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 3584374794 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 1082074105 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 3693926495 \, \sqrt {-2 \, x + 1}}{497871360 \, {\left (3 \, x + 2\right )}^{6}} \]
-43957/65345616*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt (21) + 3*sqrt(-2*x + 1))) + 1/497871360*(53407755*(2*x - 1)^5*sqrt(-2*x + 1) + 706169205*(2*x - 1)^4*sqrt(-2*x + 1) + 2801005326*(2*x - 1)^3*sqrt(-2 *x + 1) + 3584374794*(2*x - 1)^2*sqrt(-2*x + 1) + 1082074105*(-2*x + 1)^(3 /2) - 3693926495*sqrt(-2*x + 1))/(3*x + 2)^6
Time = 0.10 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^7} \, dx=\frac {43957\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{32672808}-\frac {\frac {307699\,\sqrt {1-2\,x}}{472392}-\frac {630947\,{\left (1-2\,x\right )}^{3/2}}{3306744}-\frac {1354639\,{\left (1-2\,x\right )}^{5/2}}{2143260}+\frac {1058581\,{\left (1-2\,x\right )}^{7/2}}{2143260}-\frac {747269\,{\left (1-2\,x\right )}^{9/2}}{6001128}+\frac {43957\,{\left (1-2\,x\right )}^{11/2}}{4667544}}{\frac {67228\,x}{81}+\frac {12005\,{\left (2\,x-1\right )}^2}{27}+\frac {6860\,{\left (2\,x-1\right )}^3}{27}+\frac {245\,{\left (2\,x-1\right )}^4}{3}+14\,{\left (2\,x-1\right )}^5+{\left (2\,x-1\right )}^6-\frac {184877}{729}} \]
(43957*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/32672808 - ((307699*( 1 - 2*x)^(1/2))/472392 - (630947*(1 - 2*x)^(3/2))/3306744 - (1354639*(1 - 2*x)^(5/2))/2143260 + (1058581*(1 - 2*x)^(7/2))/2143260 - (747269*(1 - 2*x )^(9/2))/6001128 + (43957*(1 - 2*x)^(11/2))/4667544)/((67228*x)/81 + (1200 5*(2*x - 1)^2)/27 + (6860*(2*x - 1)^3)/27 + (245*(2*x - 1)^4)/3 + 14*(2*x - 1)^5 + (2*x - 1)^6 - 184877/729)