Integrand size = 22, antiderivative size = 81 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^3} \, dx=-\frac {71}{63} \sqrt {1-2 x}+\frac {(1-2 x)^{5/2}}{42 (2+3 x)^2}-\frac {71 (1-2 x)^{3/2}}{126 (2+3 x)}+\frac {71 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9 \sqrt {21}} \]
1/42*(1-2*x)^(5/2)/(2+3*x)^2-71/126*(1-2*x)^(3/2)/(2+3*x)+71/189*arctanh(1 /7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-71/63*(1-2*x)^(1/2)
Time = 0.15 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^3} \, dx=-\frac {\sqrt {1-2 x} \left (101+235 x+120 x^2\right )}{18 (2+3 x)^2}+\frac {71 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9 \sqrt {21}} \]
-1/18*(Sqrt[1 - 2*x]*(101 + 235*x + 120*x^2))/(2 + 3*x)^2 + (71*ArcTanh[Sq rt[3/7]*Sqrt[1 - 2*x]])/(9*Sqrt[21])
Time = 0.18 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {87, 51, 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.
\(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)}{(3 x+2)^3} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {71}{42} \int \frac {(1-2 x)^{3/2}}{(3 x+2)^2}dx+\frac {(1-2 x)^{5/2}}{42 (3 x+2)^2}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {71}{42} \left (-\int \frac {\sqrt {1-2 x}}{3 x+2}dx-\frac {(1-2 x)^{3/2}}{3 (3 x+2)}\right )+\frac {(1-2 x)^{5/2}}{42 (3 x+2)^2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {71}{42} \left (-\frac {7}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {(1-2 x)^{3/2}}{3 (3 x+2)}-\frac {2}{3} \sqrt {1-2 x}\right )+\frac {(1-2 x)^{5/2}}{42 (3 x+2)^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {71}{42} \left (\frac {7}{3} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {(1-2 x)^{3/2}}{3 (3 x+2)}-\frac {2}{3} \sqrt {1-2 x}\right )+\frac {(1-2 x)^{5/2}}{42 (3 x+2)^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {71}{42} \left (\frac {2}{3} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {(1-2 x)^{3/2}}{3 (3 x+2)}-\frac {2}{3} \sqrt {1-2 x}\right )+\frac {(1-2 x)^{5/2}}{42 (3 x+2)^2}\) |
(1 - 2*x)^(5/2)/(42*(2 + 3*x)^2) + (71*((-2*Sqrt[1 - 2*x])/3 - (1 - 2*x)^( 3/2)/(3*(2 + 3*x)) + (2*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3))/42
3.19.66.3.1 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 + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
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 = 0.98 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.63
method | result | size |
risch | \(\frac {240 x^{3}+350 x^{2}-33 x -101}{18 \left (2+3 x \right )^{2} \sqrt {1-2 x}}+\frac {71 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{189}\) | \(51\) |
pseudoelliptic | \(\frac {142 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{2} \sqrt {21}-21 \sqrt {1-2 x}\, \left (120 x^{2}+235 x +101\right )}{378 \left (2+3 x \right )^{2}}\) | \(55\) |
derivativedivides | \(-\frac {20 \sqrt {1-2 x}}{27}-\frac {4 \left (-\frac {25 \left (1-2 x \right )^{\frac {3}{2}}}{4}+\frac {511 \sqrt {1-2 x}}{36}\right )}{3 \left (-4-6 x \right )^{2}}+\frac {71 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{189}\) | \(57\) |
default | \(-\frac {20 \sqrt {1-2 x}}{27}-\frac {4 \left (-\frac {25 \left (1-2 x \right )^{\frac {3}{2}}}{4}+\frac {511 \sqrt {1-2 x}}{36}\right )}{3 \left (-4-6 x \right )^{2}}+\frac {71 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{189}\) | \(57\) |
trager | \(-\frac {\left (120 x^{2}+235 x +101\right ) \sqrt {1-2 x}}{18 \left (2+3 x \right )^{2}}+\frac {71 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{378}\) | \(72\) |
1/18*(240*x^3+350*x^2-33*x-101)/(2+3*x)^2/(1-2*x)^(1/2)+71/189*arctanh(1/7 *21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^3} \, dx=\frac {71 \, \sqrt {21} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (120 \, x^{2} + 235 \, x + 101\right )} \sqrt {-2 \, x + 1}}{378 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
1/378*(71*sqrt(21)*(9*x^2 + 12*x + 4)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(120*x^2 + 235*x + 101)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)
Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (68) = 136\).
Time = 75.94 (sec) , antiderivative size = 343, normalized size of antiderivative = 4.23 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^3} \, dx=- \frac {20 \sqrt {1 - 2 x}}{27} - \frac {16 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{63} - \frac {364 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{9} - \frac {392 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{27} \]
-20*sqrt(1 - 2*x)/27 - 16*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log( sqrt(1 - 2*x) + sqrt(21)/3))/63 - 364*Piecewise((sqrt(21)*(-log(sqrt(21)*s qrt(1 - 2*x)/7 - 1)/4 + log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/4 - 1/(4*(sqrt(2 1)*sqrt(1 - 2*x)/7 + 1)) - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (sqr t(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/9 - 392*Piecewi se((sqrt(21)*(3*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/16 - 3*log(sqrt(21)*sqrt (1 - 2*x)/7 + 1)/16 + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) + 1/(16*(sqrt( 21)*sqrt(1 - 2*x)/7 + 1)**2) + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) - 1/( 16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/27
Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^3} \, dx=-\frac {71}{378} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {20}{27} \, \sqrt {-2 \, x + 1} + \frac {225 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 511 \, \sqrt {-2 \, x + 1}}{27 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]
-71/378*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2* x + 1))) - 20/27*sqrt(-2*x + 1) + 1/27*(225*(-2*x + 1)^(3/2) - 511*sqrt(-2 *x + 1))/(9*(2*x - 1)^2 + 84*x + 7)
Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.95 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^3} \, dx=-\frac {71}{378} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {20}{27} \, \sqrt {-2 \, x + 1} + \frac {225 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 511 \, \sqrt {-2 \, x + 1}}{108 \, {\left (3 \, x + 2\right )}^{2}} \]
-71/378*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3 *sqrt(-2*x + 1))) - 20/27*sqrt(-2*x + 1) + 1/108*(225*(-2*x + 1)^(3/2) - 5 11*sqrt(-2*x + 1))/(3*x + 2)^2
Time = 1.38 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.78 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^3} \, dx=\frac {71\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{189}-\frac {20\,\sqrt {1-2\,x}}{27}-\frac {\frac {511\,\sqrt {1-2\,x}}{243}-\frac {25\,{\left (1-2\,x\right )}^{3/2}}{27}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}} \]