Integrand size = 24, antiderivative size = 89 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^2} \, dx=-\frac {146}{81} \sqrt {1-2 x}-\frac {146}{567} (1-2 x)^{3/2}-\frac {5}{9} (1-2 x)^{5/2}-\frac {(1-2 x)^{5/2}}{63 (2+3 x)}+\frac {146}{81} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
-146/567*(1-2*x)^(3/2)-5/9*(1-2*x)^(5/2)-1/63*(1-2*x)^(5/2)/(2+3*x)+146/24 3*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-146/81*(1-2*x)^(1/2)
Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^2} \, dx=\frac {1}{243} \left (-\frac {3 \sqrt {1-2 x} \left (425+187 x-300 x^2+540 x^3\right )}{2+3 x}+146 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right ) \]
((-3*Sqrt[1 - 2*x]*(425 + 187*x - 300*x^2 + 540*x^3))/(2 + 3*x) + 146*Sqrt [21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/243
Time = 0.19 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {100, 90, 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.
\(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)^2}{(3 x+2)^2} \, dx\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {1}{63} \int \frac {(1-2 x)^{3/2} (525 x+277)}{3 x+2}dx-\frac {(1-2 x)^{5/2}}{63 (3 x+2)}\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {1}{63} \left (-73 \int \frac {(1-2 x)^{3/2}}{3 x+2}dx-35 (1-2 x)^{5/2}\right )-\frac {(1-2 x)^{5/2}}{63 (3 x+2)}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{63} \left (-73 \left (\frac {7}{3} \int \frac {\sqrt {1-2 x}}{3 x+2}dx+\frac {2}{9} (1-2 x)^{3/2}\right )-35 (1-2 x)^{5/2}\right )-\frac {(1-2 x)^{5/2}}{63 (3 x+2)}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{63} \left (-73 \left (\frac {7}{3} \left (\frac {7}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {2}{3} \sqrt {1-2 x}\right )+\frac {2}{9} (1-2 x)^{3/2}\right )-35 (1-2 x)^{5/2}\right )-\frac {(1-2 x)^{5/2}}{63 (3 x+2)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{63} \left (-73 \left (\frac {7}{3} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {7}{3} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {2}{9} (1-2 x)^{3/2}\right )-35 (1-2 x)^{5/2}\right )-\frac {(1-2 x)^{5/2}}{63 (3 x+2)}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{63} \left (-73 \left (\frac {7}{3} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {2}{3} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )+\frac {2}{9} (1-2 x)^{3/2}\right )-35 (1-2 x)^{5/2}\right )-\frac {(1-2 x)^{5/2}}{63 (3 x+2)}\) |
-1/63*(1 - 2*x)^(5/2)/(2 + 3*x) + (-35*(1 - 2*x)^(5/2) - 73*((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))/63
3.19.76.3.1 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*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
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 = 56, normalized size of antiderivative = 0.63
method | result | size |
risch | \(\frac {1080 x^{4}-1140 x^{3}+674 x^{2}+663 x -425}{81 \left (2+3 x \right ) \sqrt {1-2 x}}+\frac {146 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(56\) |
pseudoelliptic | \(\frac {146 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right ) \sqrt {21}-3 \sqrt {1-2 x}\, \left (540 x^{3}-300 x^{2}+187 x +425\right )}{486+729 x}\) | \(57\) |
derivativedivides | \(-\frac {5 \left (1-2 x \right )^{\frac {5}{2}}}{9}-\frac {20 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {16 \sqrt {1-2 x}}{9}+\frac {14 \sqrt {1-2 x}}{243 \left (-\frac {4}{3}-2 x \right )}+\frac {146 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(63\) |
default | \(-\frac {5 \left (1-2 x \right )^{\frac {5}{2}}}{9}-\frac {20 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {16 \sqrt {1-2 x}}{9}+\frac {14 \sqrt {1-2 x}}{243 \left (-\frac {4}{3}-2 x \right )}+\frac {146 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(63\) |
trager | \(-\frac {\left (540 x^{3}-300 x^{2}+187 x +425\right ) \sqrt {1-2 x}}{81 \left (2+3 x \right )}-\frac {73 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{243}\) | \(77\) |
1/81*(1080*x^4-1140*x^3+674*x^2+663*x-425)/(2+3*x)/(1-2*x)^(1/2)+146/243*a rctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^2} \, dx=\frac {73 \, \sqrt {7} \sqrt {3} {\left (3 \, x + 2\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) - 3 \, {\left (540 \, x^{3} - 300 \, x^{2} + 187 \, x + 425\right )} \sqrt {-2 \, x + 1}}{243 \, {\left (3 \, x + 2\right )}} \]
1/243*(73*sqrt(7)*sqrt(3)*(3*x + 2)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) - 3*(540*x^3 - 300*x^2 + 187*x + 425)*sqrt(-2*x + 1)) /(3*x + 2)
Time = 31.73 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.24 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^2} \, dx=- \frac {5 \left (1 - 2 x\right )^{\frac {5}{2}}}{9} - \frac {20 \left (1 - 2 x\right )^{\frac {3}{2}}}{81} - \frac {16 \sqrt {1 - 2 x}}{9} - \frac {74 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{243} - \frac {196 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{81} \]
-5*(1 - 2*x)**(5/2)/9 - 20*(1 - 2*x)**(3/2)/81 - 16*sqrt(1 - 2*x)/9 - 74*s qrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(21)/3) )/243 - 196*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/4 + lo g(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 = 0.28 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^2} \, dx=-\frac {5}{9} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {20}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {73}{243} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {16}{9} \, \sqrt {-2 \, x + 1} - \frac {7 \, \sqrt {-2 \, x + 1}}{81 \, {\left (3 \, x + 2\right )}} \]
-5/9*(-2*x + 1)^(5/2) - 20/81*(-2*x + 1)^(3/2) - 73/243*sqrt(21)*log(-(sqr t(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 16/9*sqrt(-2*x + 1) - 7/81*sqrt(-2*x + 1)/(3*x + 2)
Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.01 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^2} \, dx=-\frac {5}{9} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {20}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {73}{243} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {16}{9} \, \sqrt {-2 \, x + 1} - \frac {7 \, \sqrt {-2 \, x + 1}}{81 \, {\left (3 \, x + 2\right )}} \]
-5/9*(2*x - 1)^2*sqrt(-2*x + 1) - 20/81*(-2*x + 1)^(3/2) - 73/243*sqrt(21) *log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)) ) - 16/9*sqrt(-2*x + 1) - 7/81*sqrt(-2*x + 1)/(3*x + 2)
Time = 1.75 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^2} \, dx=-\frac {14\,\sqrt {1-2\,x}}{243\,\left (2\,x+\frac {4}{3}\right )}-\frac {16\,\sqrt {1-2\,x}}{9}-\frac {20\,{\left (1-2\,x\right )}^{3/2}}{81}-\frac {5\,{\left (1-2\,x\right )}^{5/2}}{9}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,146{}\mathrm {i}}{243} \]