Integrand size = 24, antiderivative size = 133 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)} \, dx=\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^3}+\frac {4313 \sqrt {1-2 x}}{72 (2+3 x)^2}+\frac {100145 \sqrt {1-2 x}}{168 (2+3 x)}+\frac {3454265 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{84 \sqrt {21}}-1210 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
7/12*(1-2*x)^(3/2)/(2+3*x)^4+3454265/1764*arctanh(1/7*21^(1/2)*(1-2*x)^(1/ 2))*21^(1/2)-1210*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+301/36*(1- 2*x)^(1/2)/(2+3*x)^3+4313/72*(1-2*x)^(1/2)/(2+3*x)^2+100145/168*(1-2*x)^(1 /2)/(2+3*x)
Time = 0.55 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.66 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)} \, dx=\frac {\sqrt {1-2 x} \left (844322+3730002 x+5498403 x^2+2703915 x^3\right )}{168 (2+3 x)^4}+\frac {3454265 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{84 \sqrt {21}}-1210 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(Sqrt[1 - 2*x]*(844322 + 3730002*x + 5498403*x^2 + 2703915*x^3))/(168*(2 + 3*x)^4) + (3454265*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(84*Sqrt[21]) - 1210 *Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
Time = 0.24 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {109, 27, 166, 25, 168, 27, 168, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^5 (5 x+3)} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{12} \int \frac {3 (65-53 x) \sqrt {1-2 x}}{(3 x+2)^4 (5 x+3)}dx+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int \frac {(65-53 x) \sqrt {1-2 x}}{(3 x+2)^4 (5 x+3)}dx+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{4} \left (\frac {301 \sqrt {1-2 x}}{9 (3 x+2)^3}-\frac {1}{9} \int -\frac {5259-7207 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)}dx\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{9} \int \frac {5259-7207 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)}dx+\frac {301 \sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{9} \left (\frac {1}{14} \int \frac {105 (3801-4313 x)}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {4313 \sqrt {1-2 x}}{2 (3 x+2)^2}\right )+\frac {301 \sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{9} \left (\frac {15}{2} \int \frac {3801-4313 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {4313 \sqrt {1-2 x}}{2 (3 x+2)^2}\right )+\frac {301 \sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{9} \left (\frac {15}{2} \left (\frac {1}{7} \int \frac {163521-100145 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {20029 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {4313 \sqrt {1-2 x}}{2 (3 x+2)^2}\right )+\frac {301 \sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{9} \left (\frac {15}{2} \left (\frac {1}{7} \left (1118040 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-690853 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {20029 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {4313 \sqrt {1-2 x}}{2 (3 x+2)^2}\right )+\frac {301 \sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{9} \left (\frac {15}{2} \left (\frac {1}{7} \left (690853 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-1118040 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {20029 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {4313 \sqrt {1-2 x}}{2 (3 x+2)^2}\right )+\frac {301 \sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{9} \left (\frac {15}{2} \left (\frac {1}{7} \left (\frac {1381706 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-40656 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {20029 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {4313 \sqrt {1-2 x}}{2 (3 x+2)^2}\right )+\frac {301 \sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4}\) |
(7*(1 - 2*x)^(3/2))/(12*(2 + 3*x)^4) + ((301*Sqrt[1 - 2*x])/(9*(2 + 3*x)^3 ) + ((4313*Sqrt[1 - 2*x])/(2*(2 + 3*x)^2) + (15*((20029*Sqrt[1 - 2*x])/(7* (2 + 3*x)) + ((1381706*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/Sqrt[21] - 40656* Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/7))/2)/9)/4
3.20.77.3.1 Defintions of rubi rules used
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 = 1.16 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.56
| method | result | size |
| risch | \(-\frac {5407830 x^{4}+8292891 x^{3}+1961601 x^{2}-2041358 x -844322}{168 \left (2+3 x \right )^{4} \sqrt {1-2 x}}+\frac {3454265 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1764}-1210 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}\) | \(74\) |
| derivativedivides | \(-1210 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {162 \left (\frac {100145 \left (1-2 x \right )^{\frac {7}{2}}}{504}-\frac {909931 \left (1-2 x \right )^{\frac {5}{2}}}{648}+\frac {2144065 \left (1-2 x \right )^{\frac {3}{2}}}{648}-\frac {5053615 \sqrt {1-2 x}}{1944}\right )}{\left (-4-6 x \right )^{4}}+\frac {3454265 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1764}\) | \(84\) |
| default | \(-1210 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {162 \left (\frac {100145 \left (1-2 x \right )^{\frac {7}{2}}}{504}-\frac {909931 \left (1-2 x \right )^{\frac {5}{2}}}{648}+\frac {2144065 \left (1-2 x \right )^{\frac {3}{2}}}{648}-\frac {5053615 \sqrt {1-2 x}}{1944}\right )}{\left (-4-6 x \right )^{4}}+\frac {3454265 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1764}\) | \(84\) |
| pseudoelliptic | \(\frac {6908530 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{4} \sqrt {21}-4268880 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{4} \sqrt {55}+21 \sqrt {1-2 x}\, \left (2703915 x^{3}+5498403 x^{2}+3730002 x +844322\right )}{3528 \left (2+3 x \right )^{4}}\) | \(85\) |
| trager | \(\frac {\left (2703915 x^{3}+5498403 x^{2}+3730002 x +844322\right ) \sqrt {1-2 x}}{168 \left (2+3 x \right )^{4}}+605 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )+\frac {3454265 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{3528}\) | \(121\) |
-1/168*(5407830*x^4+8292891*x^3+1961601*x^2-2041358*x-844322)/(2+3*x)^4/(1 -2*x)^(1/2)+3454265/1764*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-1210 *arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.23 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.13 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)} \, dx=\frac {2134440 \, \sqrt {55} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 3454265 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (2703915 \, x^{3} + 5498403 \, x^{2} + 3730002 \, x + 844322\right )} \sqrt {-2 \, x + 1}}{3528 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
1/3528*(2134440*sqrt(55)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 3454265*sqrt(21)*(81*x^4 + 21 6*x^3 + 216*x^2 + 96*x + 16)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(2703915*x^3 + 5498403*x^2 + 3730002*x + 844322)*sqrt(-2*x + 1) )/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
Time = 125.25 (sec) , antiderivative size = 838, normalized size of antiderivative = 6.30 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)} \, dx=\text {Too large to display} \]
-6655*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt (21)/3))/7 + 605*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5)) + 15972*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))) - 57512*Piecewise((sqrt(2 1)*(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(2 1)*sqrt(1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/9 + 22736*Piecewise((sqrt(21)*(-5*log(sqrt(21)*sqr t(1 - 2*x)/7 - 1)/32 + 5*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/32 - 5/(32*(sqr t(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) - 1 /(48*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**3) - 5/(32*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2) - 1/(48*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**3))/7203, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/9 - 10976*Piecewise((sqrt(21)*(35*log(sqrt(21)*sqrt(1 - 2*x) /7 - 1)/256 - 35*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/256 + 35/(256*(sqrt(21) *sqrt(1 - 2*x)/7 + 1)) + 15/(256*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) + 5/(1 92*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**3) + 1/(128*(sqrt(21)*sqrt(1 - 2*x)/...
Time = 0.28 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.10 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)} \, dx=605 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {3454265}{3528} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2703915 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 19108551 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 45025365 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 35375305 \, \sqrt {-2 \, x + 1}}{84 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]
605*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 3454265/3528*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/84*(2703915*(-2*x + 1)^(7/2) - 19108551*(-2*x + 1 )^(5/2) + 45025365*(-2*x + 1)^(3/2) - 35375305*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)
Time = 0.28 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.05 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)} \, dx=605 \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {3454265}{3528} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2703915 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 19108551 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 45025365 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 35375305 \, \sqrt {-2 \, x + 1}}{1344 \, {\left (3 \, x + 2\right )}^{4}} \]
605*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sq rt(-2*x + 1))) - 3454265/3528*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2 *x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/1344*(2703915*(2*x - 1)^3*sqrt (-2*x + 1) + 19108551*(2*x - 1)^2*sqrt(-2*x + 1) - 45025365*(-2*x + 1)^(3/ 2) + 35375305*sqrt(-2*x + 1))/(3*x + 2)^4
Time = 0.10 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.80 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)} \, dx=\frac {3454265\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1764}-1210\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {5053615\,\sqrt {1-2\,x}}{972}-\frac {2144065\,{\left (1-2\,x\right )}^{3/2}}{324}+\frac {909931\,{\left (1-2\,x\right )}^{5/2}}{324}-\frac {100145\,{\left (1-2\,x\right )}^{7/2}}{252}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \]
(3454265*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/1764 - 1210*55^(1/2 )*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11) + ((5053615*(1 - 2*x)^(1/2))/972 - (2144065*(1 - 2*x)^(3/2))/324 + (909931*(1 - 2*x)^(5/2))/324 - (100145*(1 - 2*x)^(7/2))/252)/((2744*x)/27 + (98*(2*x - 1)^2)/3 + (28*(2*x - 1)^3)/3 + (2*x - 1)^4 - 1715/81)