Integrand size = 22, antiderivative size = 76 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^2} \, dx=\frac {74}{27} \sqrt {1-2 x}+\frac {10}{27} (1-2 x)^{3/2}+\frac {7 \sqrt {1-2 x}}{27 (2+3 x)}-\frac {76}{27} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \] Output:
74/27*(1-2*x)^(1/2)+10/27*(1-2*x)^(3/2)+7*(1-2*x)^(1/2)/(54+81*x)-76/81*21 ^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))
Time = 0.10 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.76 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^2} \, dx=\frac {1}{81} \left (\frac {3 \sqrt {1-2 x} \left (175+212 x-60 x^2\right )}{2+3 x}-76 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right ) \] Input:
Integrate[((1 - 2*x)^(3/2)*(3 + 5*x))/(2 + 3*x)^2,x]
Output:
((3*Sqrt[1 - 2*x]*(175 + 212*x - 60*x^2))/(2 + 3*x) - 76*Sqrt[21]*ArcTanh[ Sqrt[3/7]*Sqrt[1 - 2*x]])/81
Time = 0.18 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {87, 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.
\(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)}{(3 x+2)^2} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {38}{21} \int \frac {(1-2 x)^{3/2}}{3 x+2}dx+\frac {(1-2 x)^{5/2}}{21 (3 x+2)}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {38}{21} \left (\frac {7}{3} \int \frac {\sqrt {1-2 x}}{3 x+2}dx+\frac {2}{9} (1-2 x)^{3/2}\right )+\frac {(1-2 x)^{5/2}}{21 (3 x+2)}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {38}{21} \left (\frac {7}{3} \left (\frac {7}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {2}{3} \sqrt {1-2 x}\right )+\frac {2}{9} (1-2 x)^{3/2}\right )+\frac {(1-2 x)^{5/2}}{21 (3 x+2)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {38}{21} \left (\frac {7}{3} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {7}{3} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {2}{9} (1-2 x)^{3/2}\right )+\frac {(1-2 x)^{5/2}}{21 (3 x+2)}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {38}{21} \left (\frac {7}{3} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {2}{3} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )+\frac {2}{9} (1-2 x)^{3/2}\right )+\frac {(1-2 x)^{5/2}}{21 (3 x+2)}\) |
Input:
Int[((1 - 2*x)^(3/2)*(3 + 5*x))/(2 + 3*x)^2,x]
Output:
(1 - 2*x)^(5/2)/(21*(2 + 3*x)) + (38*((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))/21
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 = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.67
method | result | size |
risch | \(\frac {120 x^{3}-484 x^{2}-138 x +175}{27 \left (2+3 x \right ) \sqrt {1-2 x}}-\frac {76 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{81}\) | \(51\) |
pseudoelliptic | \(\frac {-76 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right ) \sqrt {21}-3 \sqrt {1-2 x}\, \left (60 x^{2}-212 x -175\right )}{162+243 x}\) | \(52\) |
derivativedivides | \(\frac {10 \left (1-2 x \right )^{\frac {3}{2}}}{27}+\frac {74 \sqrt {1-2 x}}{27}-\frac {14 \sqrt {1-2 x}}{81 \left (-\frac {4}{3}-2 x \right )}-\frac {76 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{81}\) | \(54\) |
default | \(\frac {10 \left (1-2 x \right )^{\frac {3}{2}}}{27}+\frac {74 \sqrt {1-2 x}}{27}-\frac {14 \sqrt {1-2 x}}{81 \left (-\frac {4}{3}-2 x \right )}-\frac {76 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{81}\) | \(54\) |
trager | \(-\frac {\left (60 x^{2}-212 x -175\right ) \sqrt {1-2 x}}{27 \left (2+3 x \right )}+\frac {38 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{81}\) | \(72\) |
Input:
int((1-2*x)^(3/2)*(3+5*x)/(2+3*x)^2,x,method=_RETURNVERBOSE)
Output:
1/27*(120*x^3-484*x^2-138*x+175)/(2+3*x)/(1-2*x)^(1/2)-76/81*21^(1/2)*arct anh(1/7*21^(1/2)*(1-2*x)^(1/2))
Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.86 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^2} \, dx=\frac {38 \, \sqrt {\frac {7}{3}} {\left (3 \, x + 2\right )} \log \left (\frac {3 \, x + 3 \, \sqrt {\frac {7}{3}} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - {\left (60 \, x^{2} - 212 \, x - 175\right )} \sqrt {-2 \, x + 1}}{27 \, {\left (3 \, x + 2\right )}} \] Input:
integrate((1-2*x)^(3/2)*(3+5*x)/(2+3*x)^2,x, algorithm="fricas")
Output:
1/27*(38*sqrt(7/3)*(3*x + 2)*log((3*x + 3*sqrt(7/3)*sqrt(-2*x + 1) - 5)/(3 *x + 2)) - (60*x^2 - 212*x - 175)*sqrt(-2*x + 1))/(3*x + 2)
Time = 33.21 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.43 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^2} \, dx=\frac {10 \left (1 - 2 x\right )^{\frac {3}{2}}}{27} + \frac {74 \sqrt {1 - 2 x}}{27} + \frac {13 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{27} + \frac {196 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{27} \] Input:
integrate((1-2*x)**(3/2)*(3+5*x)/(2+3*x)**2,x)
Output:
10*(1 - 2*x)**(3/2)/27 + 74*sqrt(1 - 2*x)/27 + 13*sqrt(21)*(log(sqrt(1 - 2 *x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(21)/3))/27 + 196*Piecewise((s qrt(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)))/27
Time = 0.12 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^2} \, dx=\frac {10}{27} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {38}{81} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {74}{27} \, \sqrt {-2 \, x + 1} + \frac {7 \, \sqrt {-2 \, x + 1}}{27 \, {\left (3 \, x + 2\right )}} \] Input:
integrate((1-2*x)^(3/2)*(3+5*x)/(2+3*x)^2,x, algorithm="maxima")
Output:
10/27*(-2*x + 1)^(3/2) + 38/81*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1)) /(sqrt(21) + 3*sqrt(-2*x + 1))) + 74/27*sqrt(-2*x + 1) + 7/27*sqrt(-2*x + 1)/(3*x + 2)
Time = 0.13 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.97 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^2} \, dx=\frac {10}{27} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {38}{81} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {74}{27} \, \sqrt {-2 \, x + 1} + \frac {7 \, \sqrt {-2 \, x + 1}}{27 \, {\left (3 \, x + 2\right )}} \] Input:
integrate((1-2*x)^(3/2)*(3+5*x)/(2+3*x)^2,x, algorithm="giac")
Output:
10/27*(-2*x + 1)^(3/2) + 38/81*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(- 2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 74/27*sqrt(-2*x + 1) + 7/27*sqr t(-2*x + 1)/(3*x + 2)
Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^2} \, dx=\frac {14\,\sqrt {1-2\,x}}{81\,\left (2\,x+\frac {4}{3}\right )}+\frac {74\,\sqrt {1-2\,x}}{27}+\frac {10\,{\left (1-2\,x\right )}^{3/2}}{27}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,76{}\mathrm {i}}{81} \] Input:
int(((1 - 2*x)^(3/2)*(5*x + 3))/(3*x + 2)^2,x)
Output:
(21^(1/2)*atan((21^(1/2)*(1 - 2*x)^(1/2)*1i)/7)*76i)/81 + (14*(1 - 2*x)^(1 /2))/(81*(2*x + 4/3)) + (74*(1 - 2*x)^(1/2))/27 + (10*(1 - 2*x)^(3/2))/27
Time = 0.15 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.41 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^2} \, dx=\frac {-180 \sqrt {-2 x +1}\, x^{2}+636 \sqrt {-2 x +1}\, x +525 \sqrt {-2 x +1}+114 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x +76 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )-114 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x -76 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{243 x +162} \] Input:
int((1-2*x)^(3/2)*(3+5*x)/(2+3*x)^2,x)
Output:
( - 180*sqrt( - 2*x + 1)*x**2 + 636*sqrt( - 2*x + 1)*x + 525*sqrt( - 2*x + 1) + 114*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x + 76*sqrt(21)*log( 3*sqrt( - 2*x + 1) - sqrt(21)) - 114*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqr t(21))*x - 76*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21)))/(81*(3*x + 2))