Integrand size = 24, antiderivative size = 133 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^5}{(3+5 x)^2} \, dx=-\frac {172 \sqrt {1-2 x} (2+3 x)^2}{3125}+\frac {64 \sqrt {1-2 x} (2+3 x)^3}{2625}+\frac {11}{75} \sqrt {1-2 x} (2+3 x)^4-\frac {\sqrt {1-2 x} (2+3 x)^5}{5 (3+5 x)}-\frac {4 \sqrt {1-2 x} (10998+3625 x)}{15625}-\frac {328 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625 \sqrt {55}} \]
-328/859375*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-172/3125*(2+3*x) ^2*(1-2*x)^(1/2)+64/2625*(2+3*x)^3*(1-2*x)^(1/2)+11/75*(2+3*x)^4*(1-2*x)^( 1/2)-1/5*(2+3*x)^5*(1-2*x)^(1/2)/(3+5*x)-4/15625*(10998+3625*x)*(1-2*x)^(1 /2)
Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {\frac {55 \sqrt {1-2 x} \left (-862072-1133340 x+2225760 x^2+4760100 x^3+3864375 x^4+1181250 x^5\right )}{3+5 x}-2296 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{6015625} \]
((55*Sqrt[1 - 2*x]*(-862072 - 1133340*x + 2225760*x^2 + 4760100*x^3 + 3864 375*x^4 + 1181250*x^5))/(3 + 5*x) - 2296*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[ 1 - 2*x]])/6015625
Time = 0.24 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.15, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {108, 170, 27, 170, 27, 170, 25, 164, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x} (3 x+2)^5}{(5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{5} \int \frac {(13-33 x) (3 x+2)^4}{\sqrt {1-2 x} (5 x+3)}dx-\frac {\sqrt {1-2 x} (3 x+2)^5}{5 (5 x+3)}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {1}{5} \left (\frac {11}{15} \sqrt {1-2 x} (3 x+2)^4-\frac {1}{45} \int -\frac {12 (15-16 x) (3 x+2)^3}{\sqrt {1-2 x} (5 x+3)}dx\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{5 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {4}{15} \int \frac {(15-16 x) (3 x+2)^3}{\sqrt {1-2 x} (5 x+3)}dx+\frac {11}{15} \sqrt {1-2 x} (3 x+2)^4\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{5 (5 x+3)}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {1}{5} \left (\frac {4}{15} \left (\frac {16}{35} \sqrt {1-2 x} (3 x+2)^3-\frac {1}{35} \int -\frac {21 (3 x+2)^2 (43 x+34)}{\sqrt {1-2 x} (5 x+3)}dx\right )+\frac {11}{15} \sqrt {1-2 x} (3 x+2)^4\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{5 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {4}{15} \left (\frac {3}{5} \int \frac {(3 x+2)^2 (43 x+34)}{\sqrt {1-2 x} (5 x+3)}dx+\frac {16}{35} \sqrt {1-2 x} (3 x+2)^3\right )+\frac {11}{15} \sqrt {1-2 x} (3 x+2)^4\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{5 (5 x+3)}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {1}{5} \left (\frac {4}{15} \left (\frac {3}{5} \left (-\frac {1}{25} \int -\frac {(3 x+2) (3625 x+2216)}{\sqrt {1-2 x} (5 x+3)}dx-\frac {43}{25} \sqrt {1-2 x} (3 x+2)^2\right )+\frac {16}{35} \sqrt {1-2 x} (3 x+2)^3\right )+\frac {11}{15} \sqrt {1-2 x} (3 x+2)^4\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{5 (5 x+3)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{5} \left (\frac {4}{15} \left (\frac {3}{5} \left (\frac {1}{25} \int \frac {(3 x+2) (3625 x+2216)}{\sqrt {1-2 x} (5 x+3)}dx-\frac {43}{25} \sqrt {1-2 x} (3 x+2)^2\right )+\frac {16}{35} \sqrt {1-2 x} (3 x+2)^3\right )+\frac {11}{15} \sqrt {1-2 x} (3 x+2)^4\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{5 (5 x+3)}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {1}{5} \left (\frac {4}{15} \left (\frac {3}{5} \left (\frac {1}{25} \left (\frac {41}{5} \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-\frac {1}{5} \sqrt {1-2 x} (3625 x+10998)\right )-\frac {43}{25} \sqrt {1-2 x} (3 x+2)^2\right )+\frac {16}{35} \sqrt {1-2 x} (3 x+2)^3\right )+\frac {11}{15} \sqrt {1-2 x} (3 x+2)^4\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{5 (5 x+3)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{5} \left (\frac {4}{15} \left (\frac {3}{5} \left (\frac {1}{25} \left (-\frac {41}{5} \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {1}{5} \sqrt {1-2 x} (3625 x+10998)\right )-\frac {43}{25} \sqrt {1-2 x} (3 x+2)^2\right )+\frac {16}{35} \sqrt {1-2 x} (3 x+2)^3\right )+\frac {11}{15} \sqrt {1-2 x} (3 x+2)^4\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{5 (5 x+3)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{5} \left (\frac {4}{15} \left (\frac {3}{5} \left (\frac {1}{25} \left (-\frac {82 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5 \sqrt {55}}-\frac {1}{5} \sqrt {1-2 x} (3625 x+10998)\right )-\frac {43}{25} \sqrt {1-2 x} (3 x+2)^2\right )+\frac {16}{35} \sqrt {1-2 x} (3 x+2)^3\right )+\frac {11}{15} \sqrt {1-2 x} (3 x+2)^4\right )-\frac {\sqrt {1-2 x} (3 x+2)^5}{5 (5 x+3)}\) |
-1/5*(Sqrt[1 - 2*x]*(2 + 3*x)^5)/(3 + 5*x) + ((11*Sqrt[1 - 2*x]*(2 + 3*x)^ 4)/15 + (4*((16*Sqrt[1 - 2*x]*(2 + 3*x)^3)/35 + (3*((-43*Sqrt[1 - 2*x]*(2 + 3*x)^2)/25 + (-1/5*(Sqrt[1 - 2*x]*(10998 + 3625*x)) - (82*ArcTanh[Sqrt[5 /11]*Sqrt[1 - 2*x]])/(5*Sqrt[55]))/25))/5))/15)/5
3.19.38.3.1 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[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
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 = 0.98 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.50
| method | result | size |
| risch | \(-\frac {2362500 x^{6}+6547500 x^{5}+5655825 x^{4}-308580 x^{3}-4492440 x^{2}-590804 x +862072}{109375 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {328 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{859375}\) | \(66\) |
| pseudoelliptic | \(\frac {-2296 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right ) \sqrt {55}+55 \sqrt {1-2 x}\, \left (1181250 x^{5}+3864375 x^{4}+4760100 x^{3}+2225760 x^{2}-1133340 x -862072\right )}{18046875+30078125 x}\) | \(67\) |
| derivativedivides | \(\frac {27 \left (1-2 x \right )^{\frac {9}{2}}}{200}-\frac {8829 \left (1-2 x \right )^{\frac {7}{2}}}{7000}+\frac {107109 \left (1-2 x \right )^{\frac {5}{2}}}{25000}-\frac {144681 \left (1-2 x \right )^{\frac {3}{2}}}{25000}+\frac {6 \sqrt {1-2 x}}{3125}+\frac {2 \sqrt {1-2 x}}{78125 \left (-\frac {6}{5}-2 x \right )}-\frac {328 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{859375}\) | \(81\) |
| default | \(\frac {27 \left (1-2 x \right )^{\frac {9}{2}}}{200}-\frac {8829 \left (1-2 x \right )^{\frac {7}{2}}}{7000}+\frac {107109 \left (1-2 x \right )^{\frac {5}{2}}}{25000}-\frac {144681 \left (1-2 x \right )^{\frac {3}{2}}}{25000}+\frac {6 \sqrt {1-2 x}}{3125}+\frac {2 \sqrt {1-2 x}}{78125 \left (-\frac {6}{5}-2 x \right )}-\frac {328 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{859375}\) | \(81\) |
| trager | \(\frac {\sqrt {1-2 x}\, \left (1181250 x^{5}+3864375 x^{4}+4760100 x^{3}+2225760 x^{2}-1133340 x -862072\right )}{328125+546875 x}+\frac {164 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{859375}\) | \(87\) |
-1/109375*(2362500*x^6+6547500*x^5+5655825*x^4-308580*x^3-4492440*x^2-5908 04*x+862072)/(3+5*x)/(1-2*x)^(1/2)-328/859375*arctanh(1/11*55^(1/2)*(1-2*x )^(1/2))*55^(1/2)
Time = 0.23 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {1148 \, \sqrt {55} {\left (5 \, x + 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (1181250 \, x^{5} + 3864375 \, x^{4} + 4760100 \, x^{3} + 2225760 \, x^{2} - 1133340 \, x - 862072\right )} \sqrt {-2 \, x + 1}}{6015625 \, {\left (5 \, x + 3\right )}} \]
1/6015625*(1148*sqrt(55)*(5*x + 3)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8) /(5*x + 3)) + 55*(1181250*x^5 + 3864375*x^4 + 4760100*x^3 + 2225760*x^2 - 1133340*x - 862072)*sqrt(-2*x + 1))/(5*x + 3)
Time = 48.91 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.66 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {27 \left (1 - 2 x\right )^{\frac {9}{2}}}{200} - \frac {8829 \left (1 - 2 x\right )^{\frac {7}{2}}}{7000} + \frac {107109 \left (1 - 2 x\right )^{\frac {5}{2}}}{25000} - \frac {144681 \left (1 - 2 x\right )^{\frac {3}{2}}}{25000} + \frac {6 \sqrt {1 - 2 x}}{3125} + \frac {163 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{859375} - \frac {44 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{15625} \]
27*(1 - 2*x)**(9/2)/200 - 8829*(1 - 2*x)**(7/2)/7000 + 107109*(1 - 2*x)**( 5/2)/25000 - 144681*(1 - 2*x)**(3/2)/25000 + 6*sqrt(1 - 2*x)/3125 + 163*sq rt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5)) /859375 - 44*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/4 + log(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/4 - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 + 1 )) - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)))/605, (sqrt(1 - 2*x) > -sqrt(55 )/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/15625
Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {27}{200} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {8829}{7000} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {107109}{25000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {144681}{25000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {164}{859375} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {6}{3125} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{15625 \, {\left (5 \, x + 3\right )}} \]
27/200*(-2*x + 1)^(9/2) - 8829/7000*(-2*x + 1)^(7/2) + 107109/25000*(-2*x + 1)^(5/2) - 144681/25000*(-2*x + 1)^(3/2) + 164/859375*sqrt(55)*log(-(sqr t(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 6/3125*sqrt(-2* x + 1) - 1/15625*sqrt(-2*x + 1)/(5*x + 3)
Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {27}{200} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {8829}{7000} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {107109}{25000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {144681}{25000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {164}{859375} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {6}{3125} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{15625 \, {\left (5 \, x + 3\right )}} \]
27/200*(2*x - 1)^4*sqrt(-2*x + 1) + 8829/7000*(2*x - 1)^3*sqrt(-2*x + 1) + 107109/25000*(2*x - 1)^2*sqrt(-2*x + 1) - 144681/25000*(-2*x + 1)^(3/2) + 164/859375*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55 ) + 5*sqrt(-2*x + 1))) + 6/3125*sqrt(-2*x + 1) - 1/15625*sqrt(-2*x + 1)/(5 *x + 3)
Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {6\,\sqrt {1-2\,x}}{3125}-\frac {2\,\sqrt {1-2\,x}}{78125\,\left (2\,x+\frac {6}{5}\right )}-\frac {144681\,{\left (1-2\,x\right )}^{3/2}}{25000}+\frac {107109\,{\left (1-2\,x\right )}^{5/2}}{25000}-\frac {8829\,{\left (1-2\,x\right )}^{7/2}}{7000}+\frac {27\,{\left (1-2\,x\right )}^{9/2}}{200}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,328{}\mathrm {i}}{859375} \]