Integrand size = 24, antiderivative size = 108 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^5} \, dx=-\frac {(1-2 x)^{5/2}}{252 (2+3 x)^4}+\frac {277 (1-2 x)^{5/2}}{5292 (2+3 x)^3}-\frac {14423 (1-2 x)^{3/2}}{31752 (2+3 x)^2}+\frac {14423 \sqrt {1-2 x}}{31752 (2+3 x)}-\frac {14423 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{15876 \sqrt {21}} \]
-1/252*(1-2*x)^(5/2)/(2+3*x)^4+277/5292*(1-2*x)^(5/2)/(2+3*x)^3-14423/3175 2*(1-2*x)^(3/2)/(2+3*x)^2-14423/333396*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2)) *21^(1/2)+14423/31752*(1-2*x)^(1/2)/(2+3*x)
Time = 0.23 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.60 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^5} \, dx=\frac {\frac {21 \sqrt {1-2 x} \left (60890+453730 x+988035 x^2+668979 x^3\right )}{2 (2+3 x)^4}-14423 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{333396} \]
((21*Sqrt[1 - 2*x]*(60890 + 453730*x + 988035*x^2 + 668979*x^3))/(2*(2 + 3 *x)^4) - 14423*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/333396
Time = 0.19 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {100, 87, 51, 51, 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.
\(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)^2}{(3 x+2)^5} \, dx\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {1}{252} \int \frac {(1-2 x)^{3/2} (2100 x+1123)}{(3 x+2)^4}dx-\frac {(1-2 x)^{5/2}}{252 (3 x+2)^4}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {1}{252} \left (\frac {14423}{21} \int \frac {(1-2 x)^{3/2}}{(3 x+2)^3}dx+\frac {277 (1-2 x)^{5/2}}{21 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2}}{252 (3 x+2)^4}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{252} \left (\frac {14423}{21} \left (-\frac {1}{2} \int \frac {\sqrt {1-2 x}}{(3 x+2)^2}dx-\frac {(1-2 x)^{3/2}}{6 (3 x+2)^2}\right )+\frac {277 (1-2 x)^{5/2}}{21 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2}}{252 (3 x+2)^4}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{252} \left (\frac {14423}{21} \left (\frac {1}{2} \left (\frac {1}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {\sqrt {1-2 x}}{3 (3 x+2)}\right )-\frac {(1-2 x)^{3/2}}{6 (3 x+2)^2}\right )+\frac {277 (1-2 x)^{5/2}}{21 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2}}{252 (3 x+2)^4}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{252} \left (\frac {14423}{21} \left (\frac {1}{2} \left (\frac {\sqrt {1-2 x}}{3 (3 x+2)}-\frac {1}{3} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {(1-2 x)^{3/2}}{6 (3 x+2)^2}\right )+\frac {277 (1-2 x)^{5/2}}{21 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2}}{252 (3 x+2)^4}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{252} \left (\frac {14423}{21} \left (\frac {1}{2} \left (\frac {\sqrt {1-2 x}}{3 (3 x+2)}-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3 \sqrt {21}}\right )-\frac {(1-2 x)^{3/2}}{6 (3 x+2)^2}\right )+\frac {277 (1-2 x)^{5/2}}{21 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2}}{252 (3 x+2)^4}\) |
-1/252*(1 - 2*x)^(5/2)/(2 + 3*x)^4 + ((277*(1 - 2*x)^(5/2))/(21*(2 + 3*x)^ 3) + (14423*(-1/6*(1 - 2*x)^(3/2)/(2 + 3*x)^2 + (Sqrt[1 - 2*x]/(3*(2 + 3*x )) - (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(3*Sqrt[21]))/2))/21)/252
3.19.79.3.1 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] :> 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*((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 = 1.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.52
method | result | size |
risch | \(-\frac {1337958 x^{4}+1307091 x^{3}-80575 x^{2}-331950 x -60890}{31752 \left (2+3 x \right )^{4} \sqrt {1-2 x}}-\frac {14423 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{333396}\) | \(56\) |
pseudoelliptic | \(\frac {-28846 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{4} \sqrt {21}+21 \sqrt {1-2 x}\, \left (668979 x^{3}+988035 x^{2}+453730 x +60890\right )}{666792 \left (2+3 x \right )^{4}}\) | \(60\) |
derivativedivides | \(\frac {-\frac {8259 \left (1-2 x \right )^{\frac {7}{2}}}{196}+\frac {189667 \left (1-2 x \right )^{\frac {5}{2}}}{756}-\frac {158653 \left (1-2 x \right )^{\frac {3}{2}}}{324}+\frac {100961 \sqrt {1-2 x}}{324}}{\left (-4-6 x \right )^{4}}-\frac {14423 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{333396}\) | \(66\) |
default | \(\frac {-\frac {8259 \left (1-2 x \right )^{\frac {7}{2}}}{196}+\frac {189667 \left (1-2 x \right )^{\frac {5}{2}}}{756}-\frac {158653 \left (1-2 x \right )^{\frac {3}{2}}}{324}+\frac {100961 \sqrt {1-2 x}}{324}}{\left (-4-6 x \right )^{4}}-\frac {14423 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{333396}\) | \(66\) |
trager | \(\frac {\left (668979 x^{3}+988035 x^{2}+453730 x +60890\right ) \sqrt {1-2 x}}{31752 \left (2+3 x \right )^{4}}+\frac {14423 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{666792}\) | \(77\) |
-1/31752*(1337958*x^4+1307091*x^3-80575*x^2-331950*x-60890)/(2+3*x)^4/(1-2 *x)^(1/2)-14423/333396*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.92 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^5} \, dx=\frac {14423 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (668979 \, x^{3} + 988035 \, x^{2} + 453730 \, x + 60890\right )} \sqrt {-2 \, x + 1}}{666792 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
1/666792*(14423*sqrt(21)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(668979*x^3 + 988035*x^2 + 453730*x + 60890)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16 )
Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^5} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^5} \, dx=\frac {14423}{666792} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {668979 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 3983007 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 7773997 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 4947089 \, \sqrt {-2 \, x + 1}}{15876 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]
14423/666792*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqr t(-2*x + 1))) - 1/15876*(668979*(-2*x + 1)^(7/2) - 3983007*(-2*x + 1)^(5/2 ) + 7773997*(-2*x + 1)^(3/2) - 4947089*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 7 56*(2*x - 1)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)
Time = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^5} \, dx=\frac {14423}{666792} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {668979 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 3983007 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 7773997 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 4947089 \, \sqrt {-2 \, x + 1}}{254016 \, {\left (3 \, x + 2\right )}^{4}} \]
14423/666792*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21 ) + 3*sqrt(-2*x + 1))) + 1/254016*(668979*(2*x - 1)^3*sqrt(-2*x + 1) + 398 3007*(2*x - 1)^2*sqrt(-2*x + 1) - 7773997*(-2*x + 1)^(3/2) + 4947089*sqrt( -2*x + 1))/(3*x + 2)^4
Time = 1.65 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.82 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^5} \, dx=\frac {\frac {100961\,\sqrt {1-2\,x}}{26244}-\frac {158653\,{\left (1-2\,x\right )}^{3/2}}{26244}+\frac {189667\,{\left (1-2\,x\right )}^{5/2}}{61236}-\frac {2753\,{\left (1-2\,x\right )}^{7/2}}{5292}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}}-\frac {14423\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{333396} \]