Integrand size = 22, antiderivative size = 89 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^2} \, dx=\frac {560}{81} \sqrt {1-2 x}+\frac {80}{81} (1-2 x)^{3/2}+\frac {16}{63} (1-2 x)^{5/2}+\frac {(1-2 x)^{7/2}}{21 (2+3 x)}-\frac {560}{81} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
80/81*(1-2*x)^(3/2)+16/63*(1-2*x)^(5/2)+1/21*(1-2*x)^(7/2)/(2+3*x)-560/243 *arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+560/81*(1-2*x)^(1/2)
Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^2} \, dx=\frac {1}{243} \left (\frac {3 \sqrt {1-2 x} \left (1325+1474 x-516 x^2+216 x^3\right )}{2+3 x}-560 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right ) \]
((3*Sqrt[1 - 2*x]*(1325 + 1474*x - 516*x^2 + 216*x^3))/(2 + 3*x) - 560*Sqr t[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/243
Time = 0.19 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {87, 60, 60, 60, 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.
\(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)}{(3 x+2)^2} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {40}{21} \int \frac {(1-2 x)^{5/2}}{3 x+2}dx+\frac {(1-2 x)^{7/2}}{21 (3 x+2)}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {40}{21} \left (\frac {7}{3} \int \frac {(1-2 x)^{3/2}}{3 x+2}dx+\frac {2}{15} (1-2 x)^{5/2}\right )+\frac {(1-2 x)^{7/2}}{21 (3 x+2)}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {40}{21} \left (\frac {7}{3} \left (\frac {7}{3} \int \frac {\sqrt {1-2 x}}{3 x+2}dx+\frac {2}{9} (1-2 x)^{3/2}\right )+\frac {2}{15} (1-2 x)^{5/2}\right )+\frac {(1-2 x)^{7/2}}{21 (3 x+2)}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {40}{21} \left (\frac {7}{3} \left (\frac {7}{3} \left (\frac {7}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {2}{3} \sqrt {1-2 x}\right )+\frac {2}{9} (1-2 x)^{3/2}\right )+\frac {2}{15} (1-2 x)^{5/2}\right )+\frac {(1-2 x)^{7/2}}{21 (3 x+2)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {40}{21} \left (\frac {7}{3} \left (\frac {7}{3} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {7}{3} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {2}{9} (1-2 x)^{3/2}\right )+\frac {2}{15} (1-2 x)^{5/2}\right )+\frac {(1-2 x)^{7/2}}{21 (3 x+2)}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {40}{21} \left (\frac {7}{3} \left (\frac {7}{3} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {2}{3} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )+\frac {2}{9} (1-2 x)^{3/2}\right )+\frac {2}{15} (1-2 x)^{5/2}\right )+\frac {(1-2 x)^{7/2}}{21 (3 x+2)}\) |
(1 - 2*x)^(7/2)/(21*(2 + 3*x)) + (40*((2*(1 - 2*x)^(5/2))/15 + (7*((2*(1 - 2*x)^(3/2))/9 + (7*((2*Sqrt[1 - 2*x])/3 - (2*Sqrt[7/3]*ArcTanh[Sqrt[3/7]* Sqrt[1 - 2*x]])/3))/3))/3))/21
3.20.36.3.1 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/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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 = 1.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.63
method | result | size |
risch | \(-\frac {432 x^{4}-1248 x^{3}+3464 x^{2}+1176 x -1325}{81 \left (2+3 x \right ) \sqrt {1-2 x}}-\frac {560 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(56\) |
pseudoelliptic | \(\frac {-560 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right ) \sqrt {21}+3 \sqrt {1-2 x}\, \left (216 x^{3}-516 x^{2}+1474 x +1325\right )}{486+729 x}\) | \(57\) |
derivativedivides | \(\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{9}+\frac {74 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {182 \sqrt {1-2 x}}{27}-\frac {98 \sqrt {1-2 x}}{243 \left (-\frac {4}{3}-2 x \right )}-\frac {560 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(63\) |
default | \(\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{9}+\frac {74 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {182 \sqrt {1-2 x}}{27}-\frac {98 \sqrt {1-2 x}}{243 \left (-\frac {4}{3}-2 x \right )}-\frac {560 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(63\) |
trager | \(\frac {\sqrt {1-2 x}\, \left (216 x^{3}-516 x^{2}+1474 x +1325\right )}{162+243 x}+\frac {280 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{243}\) | \(77\) |
-1/81*(432*x^4-1248*x^3+3464*x^2+1176*x-1325)/(2+3*x)/(1-2*x)^(1/2)-560/24 3*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^2} \, dx=\frac {280 \, \sqrt {7} \sqrt {3} {\left (3 \, x + 2\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) + 3 \, {\left (216 \, x^{3} - 516 \, x^{2} + 1474 \, x + 1325\right )} \sqrt {-2 \, x + 1}}{243 \, {\left (3 \, x + 2\right )}} \]
1/243*(280*sqrt(7)*sqrt(3)*(3*x + 2)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) + 3*(216*x^3 - 516*x^2 + 1474*x + 1325)*sqrt(-2*x + 1 ))/(3*x + 2)
Time = 28.24 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.21 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^2} \, dx=\frac {2 \left (1 - 2 x\right )^{\frac {5}{2}}}{9} + \frac {74 \left (1 - 2 x\right )^{\frac {3}{2}}}{81} + \frac {182 \sqrt {1 - 2 x}}{27} + \frac {287 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{243} + \frac {1372 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{81} \]
2*(1 - 2*x)**(5/2)/9 + 74*(1 - 2*x)**(3/2)/81 + 182*sqrt(1 - 2*x)/27 + 287 *sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(21)/ 3))/243 + 1372*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/4 + log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 + 1) ) - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (sqrt(1 - 2*x) > -sqrt(21)/ 3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/81
Time = 0.32 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^2} \, dx=\frac {2}{9} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {74}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {280}{243} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {182}{27} \, \sqrt {-2 \, x + 1} + \frac {49 \, \sqrt {-2 \, x + 1}}{81 \, {\left (3 \, x + 2\right )}} \]
2/9*(-2*x + 1)^(5/2) + 74/81*(-2*x + 1)^(3/2) + 280/243*sqrt(21)*log(-(sqr t(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 182/27*sqrt(-2* x + 1) + 49/81*sqrt(-2*x + 1)/(3*x + 2)
Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.01 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^2} \, dx=\frac {2}{9} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {74}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {280}{243} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {182}{27} \, \sqrt {-2 \, x + 1} + \frac {49 \, \sqrt {-2 \, x + 1}}{81 \, {\left (3 \, x + 2\right )}} \]
2/9*(2*x - 1)^2*sqrt(-2*x + 1) + 74/81*(-2*x + 1)^(3/2) + 280/243*sqrt(21) *log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)) ) + 182/27*sqrt(-2*x + 1) + 49/81*sqrt(-2*x + 1)/(3*x + 2)
Time = 0.06 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^2} \, dx=\frac {98\,\sqrt {1-2\,x}}{243\,\left (2\,x+\frac {4}{3}\right )}+\frac {182\,\sqrt {1-2\,x}}{27}+\frac {74\,{\left (1-2\,x\right )}^{3/2}}{81}+\frac {2\,{\left (1-2\,x\right )}^{5/2}}{9}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,560{}\mathrm {i}}{243} \]