Integrand size = 24, antiderivative size = 131 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^2} \, dx=-\frac {6763 \sqrt {1-2 x}}{18 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)}+\frac {343 \sqrt {1-2 x}}{9 (2+3 x) (3+5 x)}-\frac {6665}{3} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+2288 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Output:
-6763*(1-2*x)^(1/2)/(54+90*x)+7/6*(1-2*x)^(3/2)/(2+3*x)^2/(3+5*x)+343/9*(1 -2*x)^(1/2)/(2+3*x)/(3+5*x)-6665/9*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^( 1/2))+2288/5*55^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^2} \, dx=-\frac {\sqrt {1-2 x} \left (8553+26380 x+20289 x^2\right )}{6 (2+3 x)^2 (3+5 x)}-\frac {6665}{3} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+2288 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Input:
Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^3*(3 + 5*x)^2),x]
Output:
-1/6*(Sqrt[1 - 2*x]*(8553 + 26380*x + 20289*x^2))/((2 + 3*x)^2*(3 + 5*x)) - (6665*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3 + 2288*Sqrt[11/5]*Ar cTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
Time = 0.23 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {109, 166, 25, 168, 27, 174, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^3 (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{6} \int \frac {(164-97 x) \sqrt {1-2 x}}{(3 x+2)^2 (5 x+3)^2}dx+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{6} \left (\frac {686 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)}-\frac {1}{3} \int -\frac {8821-10096 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx\right )+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \int \frac {8821-10096 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx+\frac {686 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (-\frac {1}{11} \int \frac {33 (11043-6763 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {6763 \sqrt {1-2 x}}{5 x+3}\right )+\frac {686 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (-3 \int \frac {11043-6763 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {6763 \sqrt {1-2 x}}{5 x+3}\right )+\frac {686 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (-3 \left (75504 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-46655 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {6763 \sqrt {1-2 x}}{5 x+3}\right )+\frac {686 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (-3 \left (46655 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-75504 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {6763 \sqrt {1-2 x}}{5 x+3}\right )+\frac {686 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (-3 \left (13330 \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-13728 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {6763 \sqrt {1-2 x}}{5 x+3}\right )+\frac {686 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)}\) |
Input:
Int[(1 - 2*x)^(5/2)/((2 + 3*x)^3*(3 + 5*x)^2),x]
Output:
(7*(1 - 2*x)^(3/2))/(6*(2 + 3*x)^2*(3 + 5*x)) + ((686*Sqrt[1 - 2*x])/(3*(2 + 3*x)*(3 + 5*x)) + ((-6763*Sqrt[1 - 2*x])/(3 + 5*x) - 3*(13330*Sqrt[7/3] *ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 13728*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sq rt[1 - 2*x]]))/3)/6
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
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 = 0.29 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.58
| method | result | size |
| risch | \(\frac {40578 x^{3}+32471 x^{2}-9274 x -8553}{6 \left (2+3 x \right )^{2} \sqrt {1-2 x}\, \left (3+5 x \right )}-\frac {6665 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{9}+\frac {2288 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{5}\) | \(76\) |
| derivativedivides | \(\frac {242 \sqrt {1-2 x}}{5 \left (-\frac {6}{5}-2 x \right )}+\frac {2288 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{5}+\frac {917 \left (1-2 x \right )^{\frac {3}{2}}-\frac {6517 \sqrt {1-2 x}}{3}}{\left (-4-6 x \right )^{2}}-\frac {6665 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{9}\) | \(82\) |
| default | \(\frac {242 \sqrt {1-2 x}}{5 \left (-\frac {6}{5}-2 x \right )}+\frac {2288 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{5}+\frac {917 \left (1-2 x \right )^{\frac {3}{2}}-\frac {6517 \sqrt {1-2 x}}{3}}{\left (-4-6 x \right )^{2}}-\frac {6665 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{9}\) | \(82\) |
| pseudoelliptic | \(\frac {-66650 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{2} \left (3+5 x \right ) \sqrt {21}+41184 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{2} \left (3+5 x \right ) \sqrt {55}-15 \sqrt {1-2 x}\, \left (20289 x^{2}+26380 x +8553\right )}{90 \left (2+3 x \right )^{2} \left (3+5 x \right )}\) | \(97\) |
| trager | \(-\frac {\left (20289 x^{2}+26380 x +8553\right ) \sqrt {1-2 x}}{6 \left (2+3 x \right )^{2} \left (3+5 x \right )}-\frac {6665 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{18}+\frac {1144 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{5}\) | \(123\) |
Input:
int((1-2*x)^(5/2)/(2+3*x)^3/(3+5*x)^2,x,method=_RETURNVERBOSE)
Output:
1/6*(40578*x^3+32471*x^2-9274*x-8553)/(2+3*x)^2/(1-2*x)^(1/2)/(3+5*x)-6665 /9*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))+2288/5*55^(1/2)*arctanh(1/ 11*55^(1/2)*(1-2*x)^(1/2))
Time = 0.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^2} \, dx=\frac {6864 \, \sqrt {\frac {11}{5}} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac {5 \, x - 5 \, \sqrt {\frac {11}{5}} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 6665 \, \sqrt {\frac {7}{3}} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac {3 \, x + 3 \, \sqrt {\frac {7}{3}} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - {\left (20289 \, x^{2} + 26380 \, x + 8553\right )} \sqrt {-2 \, x + 1}}{6 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \] Input:
integrate((1-2*x)^(5/2)/(2+3*x)^3/(3+5*x)^2,x, algorithm="fricas")
Output:
1/6*(6864*sqrt(11/5)*(45*x^3 + 87*x^2 + 56*x + 12)*log((5*x - 5*sqrt(11/5) *sqrt(-2*x + 1) - 8)/(5*x + 3)) + 6665*sqrt(7/3)*(45*x^3 + 87*x^2 + 56*x + 12)*log((3*x + 3*sqrt(7/3)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - (20289*x^2 + 26380*x + 8553)*sqrt(-2*x + 1))/(45*x^3 + 87*x^2 + 56*x + 12)
Time = 54.77 (sec) , antiderivative size = 488, normalized size of antiderivative = 3.73 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^2} \, dx=363 \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 ) - 231 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right ) - \frac {12544 \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 )}{3} + \frac {2744 \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 )}{3} - 5324 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right ) \] Input:
integrate((1-2*x)**(5/2)/(2+3*x)**3/(3+5*x)**2,x)
Output:
363*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(2 1)/3)) - 231*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5)) - 12544*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)))/3 + 2744*Piecewise((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)))/3 - 5324*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/4 + log(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/4 - 1/(4*(sqrt(55)*sqr t(1 - 2*x)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)))/605, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))
Time = 0.12 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^2} \, dx=-\frac {1144}{5} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {6665}{18} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {20289 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 93338 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 107261 \, \sqrt {-2 \, x + 1}}{3 \, {\left (45 \, {\left (2 \, x - 1\right )}^{3} + 309 \, {\left (2 \, x - 1\right )}^{2} + 1414 \, x - 168\right )}} \] Input:
integrate((1-2*x)^(5/2)/(2+3*x)^3/(3+5*x)^2,x, algorithm="maxima")
Output:
-1144/5*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2* x + 1))) + 6665/18*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/3*(20289*(-2*x + 1)^(5/2) - 93338*(-2*x + 1)^(3/2) + 107261*sqrt(-2*x + 1))/(45*(2*x - 1)^3 + 309*(2*x - 1)^2 + 1414*x - 168 )
Time = 0.13 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.94 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^2} \, dx=-\frac {1144}{5} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {6665}{18} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {121 \, \sqrt {-2 \, x + 1}}{5 \, x + 3} + \frac {7 \, {\left (393 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 931 \, \sqrt {-2 \, x + 1}\right )}}{12 \, {\left (3 \, x + 2\right )}^{2}} \] Input:
integrate((1-2*x)^(5/2)/(2+3*x)^3/(3+5*x)^2,x, algorithm="giac")
Output:
-1144/5*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 6665/18*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2* x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 121*sqrt(-2*x + 1)/(5*x + 3) + 7/ 12*(393*(-2*x + 1)^(3/2) - 931*sqrt(-2*x + 1))/(3*x + 2)^2
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^2} \, dx=\frac {2288\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{5}-\frac {6665\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{9}-\frac {\frac {107261\,\sqrt {1-2\,x}}{135}-\frac {93338\,{\left (1-2\,x\right )}^{3/2}}{135}+\frac {6763\,{\left (1-2\,x\right )}^{5/2}}{45}}{\frac {1414\,x}{45}+\frac {103\,{\left (2\,x-1\right )}^2}{15}+{\left (2\,x-1\right )}^3-\frac {56}{15}} \] Input:
int((1 - 2*x)^(5/2)/((3*x + 2)^3*(5*x + 3)^2),x)
Output:
(2288*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/5 - (6665*21^(1/2)*at anh((21^(1/2)*(1 - 2*x)^(1/2))/7))/9 - ((107261*(1 - 2*x)^(1/2))/135 - (93 338*(1 - 2*x)^(3/2))/135 + (6763*(1 - 2*x)^(5/2))/45)/((1414*x)/45 + (103* (2*x - 1)^2)/15 + (2*x - 1)^3 - 56/15)
Time = 0.16 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.65 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^2} \, dx=\frac {-304335 \sqrt {-2 x +1}\, x^{2}-395700 \sqrt {-2 x +1}\, x -128295 \sqrt {-2 x +1}-926640 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{3}-1791504 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{2}-1153152 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x -247104 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right )+926640 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{3}+1791504 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{2}+1153152 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x +247104 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right )+1499625 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{3}+2899275 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{2}+1866200 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x +399900 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )-1499625 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{3}-2899275 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{2}-1866200 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x -399900 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{4050 x^{3}+7830 x^{2}+5040 x +1080} \] Input:
int((1-2*x)^(5/2)/(2+3*x)^3/(3+5*x)^2,x)
Output:
( - 304335*sqrt( - 2*x + 1)*x**2 - 395700*sqrt( - 2*x + 1)*x - 128295*sqrt ( - 2*x + 1) - 926640*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x**3 - 1 791504*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x**2 - 1153152*sqrt(55) *log(5*sqrt( - 2*x + 1) - sqrt(55))*x - 247104*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55)) + 926640*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x**3 + 1791504*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x**2 + 1153152*sqrt (55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x + 247104*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55)) + 1499625*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21)) *x**3 + 2899275*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**2 + 1866200 *sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x + 399900*sqrt(21)*log(3*sqr t( - 2*x + 1) - sqrt(21)) - 1499625*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt (21))*x**3 - 2899275*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x**2 - 18 66200*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x - 399900*sqrt(21)*log( 3*sqrt( - 2*x + 1) + sqrt(21)))/(90*(45*x**3 + 87*x**2 + 56*x + 12))