Integrand size = 24, antiderivative size = 82 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^3}{3+5 x} \, dx=\frac {2}{625} \sqrt {1-2 x}-\frac {1299}{500} (1-2 x)^{3/2}+\frac {162}{125} (1-2 x)^{5/2}-\frac {27}{140} (1-2 x)^{7/2}-\frac {2}{625} \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Output:
2/625*(1-2*x)^(1/2)-1299/500*(1-2*x)^(3/2)+162/125*(1-2*x)^(5/2)-27/140*(1 -2*x)^(7/2)-2/3125*55^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^3}{3+5 x} \, dx=\frac {5 \sqrt {1-2 x} \left (-6526+5115 x+12555 x^2+6750 x^3\right )-14 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{21875} \] Input:
Integrate[(Sqrt[1 - 2*x]*(2 + 3*x)^3)/(3 + 5*x),x]
Output:
(5*Sqrt[1 - 2*x]*(-6526 + 5115*x + 12555*x^2 + 6750*x^3) - 14*Sqrt[55]*Arc Tanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/21875
Time = 0.20 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {99, 2009}
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.
\(\displaystyle \int \frac {\sqrt {1-2 x} (3 x+2)^3}{5 x+3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {27}{20} (1-2 x)^{5/2}-\frac {162}{25} (1-2 x)^{3/2}+\frac {\sqrt {1-2 x}}{125 (5 x+3)}+\frac {3897}{500} \sqrt {1-2 x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2}{625} \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )-\frac {27}{140} (1-2 x)^{7/2}+\frac {162}{125} (1-2 x)^{5/2}-\frac {1299}{500} (1-2 x)^{3/2}+\frac {2}{625} \sqrt {1-2 x}\) |
Input:
Int[(Sqrt[1 - 2*x]*(2 + 3*x)^3)/(3 + 5*x),x]
Output:
(2*Sqrt[1 - 2*x])/625 - (1299*(1 - 2*x)^(3/2))/500 + (162*(1 - 2*x)^(5/2)) /125 - (27*(1 - 2*x)^(7/2))/140 - (2*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/625
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Time = 0.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.54
method | result | size |
pseudoelliptic | \(-\frac {2 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{3125}+\frac {\sqrt {1-2 x}\, \left (6750 x^{3}+12555 x^{2}+5115 x -6526\right )}{4375}\) | \(44\) |
risch | \(-\frac {\left (6750 x^{3}+12555 x^{2}+5115 x -6526\right ) \left (-1+2 x \right )}{4375 \sqrt {1-2 x}}-\frac {2 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{3125}\) | \(49\) |
derivativedivides | \(\frac {2 \sqrt {1-2 x}}{625}-\frac {1299 \left (1-2 x \right )^{\frac {3}{2}}}{500}+\frac {162 \left (1-2 x \right )^{\frac {5}{2}}}{125}-\frac {27 \left (1-2 x \right )^{\frac {7}{2}}}{140}-\frac {2 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{3125}\) | \(56\) |
default | \(\frac {2 \sqrt {1-2 x}}{625}-\frac {1299 \left (1-2 x \right )^{\frac {3}{2}}}{500}+\frac {162 \left (1-2 x \right )^{\frac {5}{2}}}{125}-\frac {27 \left (1-2 x \right )^{\frac {7}{2}}}{140}-\frac {2 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{3125}\) | \(56\) |
trager | \(\left (\frac {54}{35} x^{3}+\frac {2511}{875} x^{2}+\frac {1023}{875} x -\frac {6526}{4375}\right ) \sqrt {1-2 x}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x -8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{3125}\) | \(70\) |
Input:
int((1-2*x)^(1/2)*(2+3*x)^3/(3+5*x),x,method=_RETURNVERBOSE)
Output:
-2/3125*55^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))+1/4375*(1-2*x)^(1/2) *(6750*x^3+12555*x^2+5115*x-6526)
Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^3}{3+5 x} \, dx=\frac {1}{4375} \, {\left (6750 \, x^{3} + 12555 \, x^{2} + 5115 \, x - 6526\right )} \sqrt {-2 \, x + 1} + \frac {1}{625} \, \sqrt {\frac {11}{5}} \log \left (\frac {5 \, x + 5 \, \sqrt {\frac {11}{5}} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) \] Input:
integrate((1-2*x)^(1/2)*(2+3*x)^3/(3+5*x),x, algorithm="fricas")
Output:
1/4375*(6750*x^3 + 12555*x^2 + 5115*x - 6526)*sqrt(-2*x + 1) + 1/625*sqrt( 11/5)*log((5*x + 5*sqrt(11/5)*sqrt(-2*x + 1) - 8)/(5*x + 3))
Time = 2.24 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^3}{3+5 x} \, dx=- \frac {27 \left (1 - 2 x\right )^{\frac {7}{2}}}{140} + \frac {162 \left (1 - 2 x\right )^{\frac {5}{2}}}{125} - \frac {1299 \left (1 - 2 x\right )^{\frac {3}{2}}}{500} + \frac {2 \sqrt {1 - 2 x}}{625} + \frac {\sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{3125} \] Input:
integrate((1-2*x)**(1/2)*(2+3*x)**3/(3+5*x),x)
Output:
-27*(1 - 2*x)**(7/2)/140 + 162*(1 - 2*x)**(5/2)/125 - 1299*(1 - 2*x)**(3/2 )/500 + 2*sqrt(1 - 2*x)/625 + sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5))/3125
Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^3}{3+5 x} \, dx=-\frac {27}{140} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {162}{125} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {1299}{500} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{3125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2}{625} \, \sqrt {-2 \, x + 1} \] Input:
integrate((1-2*x)^(1/2)*(2+3*x)^3/(3+5*x),x, algorithm="maxima")
Output:
-27/140*(-2*x + 1)^(7/2) + 162/125*(-2*x + 1)^(5/2) - 1299/500*(-2*x + 1)^ (3/2) + 1/3125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*s qrt(-2*x + 1))) + 2/625*sqrt(-2*x + 1)
Time = 0.13 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^3}{3+5 x} \, dx=\frac {27}{140} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {162}{125} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {1299}{500} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{3125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2}{625} \, \sqrt {-2 \, x + 1} \] Input:
integrate((1-2*x)^(1/2)*(2+3*x)^3/(3+5*x),x, algorithm="giac")
Output:
27/140*(2*x - 1)^3*sqrt(-2*x + 1) + 162/125*(2*x - 1)^2*sqrt(-2*x + 1) - 1 299/500*(-2*x + 1)^(3/2) + 1/3125*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sq rt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2/625*sqrt(-2*x + 1)
Time = 1.06 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^3}{3+5 x} \, dx=\frac {2\,\sqrt {1-2\,x}}{625}-\frac {1299\,{\left (1-2\,x\right )}^{3/2}}{500}+\frac {162\,{\left (1-2\,x\right )}^{5/2}}{125}-\frac {27\,{\left (1-2\,x\right )}^{7/2}}{140}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2{}\mathrm {i}}{3125} \] Input:
int(((1 - 2*x)^(1/2)*(3*x + 2)^3)/(5*x + 3),x)
Output:
(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*2i)/3125 + (2*(1 - 2*x)^( 1/2))/625 - (1299*(1 - 2*x)^(3/2))/500 + (162*(1 - 2*x)^(5/2))/125 - (27*( 1 - 2*x)^(7/2))/140
Time = 0.16 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^3}{3+5 x} \, dx=\frac {54 \sqrt {-2 x +1}\, x^{3}}{35}+\frac {2511 \sqrt {-2 x +1}\, x^{2}}{875}+\frac {1023 \sqrt {-2 x +1}\, x}{875}-\frac {6526 \sqrt {-2 x +1}}{4375}+\frac {\sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right )}{3125}-\frac {\sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right )}{3125} \] Input:
int((1-2*x)^(1/2)*(2+3*x)^3/(3+5*x),x)
Output:
(33750*sqrt( - 2*x + 1)*x**3 + 62775*sqrt( - 2*x + 1)*x**2 + 25575*sqrt( - 2*x + 1)*x - 32630*sqrt( - 2*x + 1) + 7*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55)) - 7*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55)))/21875