Integrand size = 22, antiderivative size = 68 \[ \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^3} \, dx=\frac {\sqrt {1-2 x}}{18 (2+3 x)^2}-\frac {71 \sqrt {1-2 x}}{126 (2+3 x)}+\frac {23 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}} \] Output:
1/18*(1-2*x)^(1/2)/(2+3*x)^2-71*(1-2*x)^(1/2)/(252+378*x)+23/441*21^(1/2)* arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))
Time = 0.14 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^3} \, dx=-\frac {\sqrt {1-2 x} (45+71 x)}{42 (2+3 x)^2}+\frac {23 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}} \] Input:
Integrate[(Sqrt[1 - 2*x]*(3 + 5*x))/(2 + 3*x)^3,x]
Output:
-1/42*(Sqrt[1 - 2*x]*(45 + 71*x))/(2 + 3*x)^2 + (23*ArcTanh[Sqrt[3/7]*Sqrt [1 - 2*x]])/(21*Sqrt[21])
Time = 0.17 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {87, 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.
\(\displaystyle \int \frac {\sqrt {1-2 x} (5 x+3)}{(3 x+2)^3} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {23}{14} \int \frac {\sqrt {1-2 x}}{(3 x+2)^2}dx+\frac {(1-2 x)^{3/2}}{42 (3 x+2)^2}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {23}{14} \left (-\frac {1}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{3 (3 x+2)}\right )+\frac {(1-2 x)^{3/2}}{42 (3 x+2)^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {23}{14} \left (\frac {1}{3} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{3 (3 x+2)}\right )+\frac {(1-2 x)^{3/2}}{42 (3 x+2)^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {23}{14} \left (\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3 \sqrt {21}}-\frac {\sqrt {1-2 x}}{3 (3 x+2)}\right )+\frac {(1-2 x)^{3/2}}{42 (3 x+2)^2}\) |
Input:
Int[(Sqrt[1 - 2*x]*(3 + 5*x))/(2 + 3*x)^3,x]
Output:
(1 - 2*x)^(3/2)/(42*(2 + 3*x)^2) + (23*(-1/3*Sqrt[1 - 2*x]/(2 + 3*x) + (2* ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(3*Sqrt[21])))/14
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)^(-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 = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.68
method | result | size |
risch | \(\frac {142 x^{2}+19 x -45}{42 \left (2+3 x \right )^{2} \sqrt {1-2 x}}+\frac {23 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{441}\) | \(46\) |
derivativedivides | \(-\frac {36 \left (-\frac {71 \left (1-2 x \right )^{\frac {3}{2}}}{756}+\frac {23 \sqrt {1-2 x}}{108}\right )}{\left (-4-6 x \right )^{2}}+\frac {23 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{441}\) | \(48\) |
default | \(-\frac {36 \left (-\frac {71 \left (1-2 x \right )^{\frac {3}{2}}}{756}+\frac {23 \sqrt {1-2 x}}{108}\right )}{\left (-4-6 x \right )^{2}}+\frac {23 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{441}\) | \(48\) |
pseudoelliptic | \(\frac {46 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{2} \sqrt {21}-21 \sqrt {1-2 x}\, \left (71 x +45\right )}{882 \left (2+3 x \right )^{2}}\) | \(50\) |
trager | \(-\frac {\left (71 x +45\right ) \sqrt {1-2 x}}{42 \left (2+3 x \right )^{2}}-\frac {23 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{882}\) | \(67\) |
Input:
int((1-2*x)^(1/2)*(3+5*x)/(2+3*x)^3,x,method=_RETURNVERBOSE)
Output:
1/42*(142*x^2+19*x-45)/(2+3*x)^2/(1-2*x)^(1/2)+23/441*21^(1/2)*arctanh(1/7 *21^(1/2)*(1-2*x)^(1/2))
Time = 0.09 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^3} \, dx=\frac {23 \, \sqrt {21} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (71 \, x + 45\right )} \sqrt {-2 \, x + 1}}{882 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \] Input:
integrate((1-2*x)^(1/2)*(3+5*x)/(2+3*x)^3,x, algorithm="fricas")
Output:
1/882*(23*sqrt(21)*(9*x^2 + 12*x + 4)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(71*x + 45)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)
Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (56) = 112\).
Time = 97.16 (sec) , antiderivative size = 332, normalized size of antiderivative = 4.88 \[ \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^3} \, dx=- \frac {10 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{189} - \frac {148 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{9} - \frac {56 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{9} \] Input:
integrate((1-2*x)**(1/2)*(3+5*x)/(2+3*x)**3,x)
Output:
-10*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(2 1)/3))/189 - 148*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)))/9 - 56*Piecewise((sqrt(21)*(3*log(sq rt(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) < sq rt(21)/3)))/9
Time = 0.11 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^3} \, dx=-\frac {23}{882} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {71 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 161 \, \sqrt {-2 \, x + 1}}{21 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \] Input:
integrate((1-2*x)^(1/2)*(3+5*x)/(2+3*x)^3,x, algorithm="maxima")
Output:
-23/882*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2* x + 1))) + 1/21*(71*(-2*x + 1)^(3/2) - 161*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)
Time = 0.13 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^3} \, dx=-\frac {23}{882} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {71 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 161 \, \sqrt {-2 \, x + 1}}{84 \, {\left (3 \, x + 2\right )}^{2}} \] Input:
integrate((1-2*x)^(1/2)*(3+5*x)/(2+3*x)^3,x, algorithm="giac")
Output:
-23/882*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3 *sqrt(-2*x + 1))) + 1/84*(71*(-2*x + 1)^(3/2) - 161*sqrt(-2*x + 1))/(3*x + 2)^2
Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^3} \, dx=\frac {23\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{441}-\frac {\frac {23\,\sqrt {1-2\,x}}{27}-\frac {71\,{\left (1-2\,x\right )}^{3/2}}{189}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}} \] Input:
int(((1 - 2*x)^(1/2)*(5*x + 3))/(3*x + 2)^3,x)
Output:
(23*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/441 - ((23*(1 - 2*x)^(1/ 2))/27 - (71*(1 - 2*x)^(3/2))/189)/((28*x)/3 + (2*x - 1)^2 + 7/9)
Time = 0.15 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.07 \[ \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^3} \, dx=\frac {-1491 \sqrt {-2 x +1}\, x -945 \sqrt {-2 x +1}-207 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{2}-276 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x -92 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )+207 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{2}+276 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x +92 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{7938 x^{2}+10584 x +3528} \] Input:
int((1-2*x)^(1/2)*(3+5*x)/(2+3*x)^3,x)
Output:
( - 1491*sqrt( - 2*x + 1)*x - 945*sqrt( - 2*x + 1) - 207*sqrt(21)*log(3*sq rt( - 2*x + 1) - sqrt(21))*x**2 - 276*sqrt(21)*log(3*sqrt( - 2*x + 1) - sq rt(21))*x - 92*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21)) + 207*sqrt(21)* log(3*sqrt( - 2*x + 1) + sqrt(21))*x**2 + 276*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x + 92*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21)))/(882*( 9*x**2 + 12*x + 4))