Integrand size = 24, antiderivative size = 181 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^2} \, dx=-\frac {8110915 \sqrt {1-2 x}}{1176 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 (3+5 x)}+\frac {83 \sqrt {1-2 x}}{18 (2+3 x)^3 (3+5 x)}+\frac {23173 \sqrt {1-2 x}}{504 (2+3 x)^2 (3+5 x)}+\frac {302668 \sqrt {1-2 x}}{441 (2+3 x) (3+5 x)}-\frac {55953383 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{196 \sqrt {21}}+8400 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
-55953383/4116*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+8400*arctanh(1 /11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-8110915/1176*(1-2*x)^(1/2)/(3+5*x)+7/ 12*(1-2*x)^(1/2)/(2+3*x)^4/(3+5*x)+83/18*(1-2*x)^(1/2)/(2+3*x)^3/(3+5*x)+2 3173/504*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)+302668/441*(1-2*x)^(1/2)/(2+3*x)/ (3+5*x)
Time = 0.39 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.55 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^2} \, dx=-\frac {\sqrt {1-2 x} \left (41029970+249642200 x+569295605 x^2+576721848 x^3+218994705 x^4\right )}{392 (2+3 x)^4 (3+5 x)}-\frac {55953383 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{196 \sqrt {21}}+8400 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
-1/392*(Sqrt[1 - 2*x]*(41029970 + 249642200*x + 569295605*x^2 + 576721848* x^3 + 218994705*x^4))/((2 + 3*x)^4*(3 + 5*x)) - (55953383*ArcTanh[Sqrt[3/7 ]*Sqrt[1 - 2*x]])/(196*Sqrt[21]) + 8400*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
Time = 0.28 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {109, 168, 27, 168, 168, 27, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2}}{(3 x+2)^5 (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{12} \int \frac {188-299 x}{\sqrt {1-2 x} (3 x+2)^4 (5 x+3)^2}dx+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{21} \int \frac {7 (3851-5810 x)}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^2}dx+\frac {166 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{3} \int \frac {3851-5810 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^2}dx+\frac {166 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{3} \left (\frac {1}{14} \int \frac {420898-579325 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^2}dx+\frac {23173 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {166 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{3} \left (\frac {1}{14} \left (\frac {1}{7} \int \frac {3 (10579981-12106720 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx+\frac {2421344 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {23173 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {166 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{3} \left (\frac {1}{14} \left (\frac {3}{7} \int \frac {10579981-12106720 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx+\frac {2421344 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {23173 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {166 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{3} \left (\frac {1}{14} \left (\frac {3}{7} \left (-\frac {1}{11} \int \frac {33 (13243851-8110915 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {8110915 \sqrt {1-2 x}}{5 x+3}\right )+\frac {2421344 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {23173 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {166 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{3} \left (\frac {1}{14} \left (\frac {3}{7} \left (-3 \int \frac {13243851-8110915 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {8110915 \sqrt {1-2 x}}{5 x+3}\right )+\frac {2421344 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {23173 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {166 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{3} \left (\frac {1}{14} \left (\frac {3}{7} \left (-3 \left (90552000 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-55953383 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {8110915 \sqrt {1-2 x}}{5 x+3}\right )+\frac {2421344 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {23173 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {166 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{3} \left (\frac {1}{14} \left (\frac {3}{7} \left (-3 \left (55953383 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-90552000 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {8110915 \sqrt {1-2 x}}{5 x+3}\right )+\frac {2421344 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {23173 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {166 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{3} \left (\frac {1}{14} \left (\frac {3}{7} \left (-3 \left (\frac {111906766 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-3292800 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {8110915 \sqrt {1-2 x}}{5 x+3}\right )+\frac {2421344 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {23173 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {166 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}\right )+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)}\) |
(7*Sqrt[1 - 2*x])/(12*(2 + 3*x)^4*(3 + 5*x)) + ((166*Sqrt[1 - 2*x])/(3*(2 + 3*x)^3*(3 + 5*x)) + ((23173*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2*(3 + 5*x)) + ((2421344*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x)) + (3*((-8110915*Sqrt[1 - 2*x])/(3 + 5*x) - 3*((111906766*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/Sqrt[21] - 3292800*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])))/7)/14)/3)/12
3.20.18.3.1 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 + 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 = 1.12 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.48
| method | result | size |
| risch | \(\frac {437989410 x^{5}+934448991 x^{4}+561869362 x^{3}-70011205 x^{2}-167582260 x -41029970}{392 \left (2+3 x \right )^{4} \sqrt {1-2 x}\, \left (3+5 x \right )}-\frac {55953383 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4116}+8400 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}\) | \(86\) |
| derivativedivides | \(\frac {550 \sqrt {1-2 x}}{-\frac {6}{5}-2 x}+8400 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}+\frac {\frac {35067141 \left (1-2 x \right )^{\frac {7}{2}}}{196}-\frac {35319999 \left (1-2 x \right )^{\frac {5}{2}}}{28}+\frac {11859787 \left (1-2 x \right )^{\frac {3}{2}}}{4}-\frac {9293319 \sqrt {1-2 x}}{4}}{\left (-4-6 x \right )^{4}}-\frac {55953383 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4116}\) | \(100\) |
| default | \(\frac {550 \sqrt {1-2 x}}{-\frac {6}{5}-2 x}+8400 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}+\frac {\frac {35067141 \left (1-2 x \right )^{\frac {7}{2}}}{196}-\frac {35319999 \left (1-2 x \right )^{\frac {5}{2}}}{28}+\frac {11859787 \left (1-2 x \right )^{\frac {3}{2}}}{4}-\frac {9293319 \sqrt {1-2 x}}{4}}{\left (-4-6 x \right )^{4}}-\frac {55953383 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4116}\) | \(100\) |
| pseudoelliptic | \(\frac {-111906766 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{4} \left (3+5 x \right ) \sqrt {21}+69148800 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{4} \left (3+5 x \right ) \sqrt {55}-21 \sqrt {1-2 x}\, \left (218994705 x^{4}+576721848 x^{3}+569295605 x^{2}+249642200 x +41029970\right )}{8232 \left (2+3 x \right )^{4} \left (3+5 x \right )}\) | \(107\) |
| trager | \(-\frac {\left (218994705 x^{4}+576721848 x^{3}+569295605 x^{2}+249642200 x +41029970\right ) \sqrt {1-2 x}}{392 \left (2+3 x \right )^{4} \left (3+5 x \right )}+\frac {55953383 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{8232}+4200 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )\) | \(133\) |
1/392*(437989410*x^5+934448991*x^4+561869362*x^3-70011205*x^2-167582260*x- 41029970)/(2+3*x)^4/(1-2*x)^(1/2)/(3+5*x)-55953383/4116*arctanh(1/7*21^(1/ 2)*(1-2*x)^(1/2))*21^(1/2)+8400*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1 /2)
Time = 0.23 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.94 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^2} \, dx=\frac {34574400 \, \sqrt {55} {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (\frac {5 \, x - \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55953383 \, \sqrt {21} {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (218994705 \, x^{4} + 576721848 \, x^{3} + 569295605 \, x^{2} + 249642200 \, x + 41029970\right )} \sqrt {-2 \, x + 1}}{8232 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} \]
1/8232*(34574400*sqrt(55)*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368* x + 48)*log((5*x - sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 55953383*sqrt (21)*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)*log((3*x + sq rt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(218994705*x^4 + 576721848*x^3 + 569295605*x^2 + 249642200*x + 41029970)*sqrt(-2*x + 1))/(405*x^5 + 1323* x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)
Time = 141.52 (sec) , antiderivative size = 952, normalized size of antiderivative = 5.26 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^2} \, dx=\text {Too large to display} \]
46475*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt (21)/3))/7 - 4225*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5)) - 89760*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))) + 27192*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))) - 7392*Piecewise((sqrt(21)*(-5*log(sqrt(21)*sqrt( 1 - 2*x)/7 - 1)/32 + 5*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/32 - 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) - 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) < sq rt(21)/3))) + 1568*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/(192*(s qrt(21)*sqrt(1 - 2*x)/7 + 1)**3) + 1/(128*(sqrt(21)*sqrt(1 - 2*x)/7 + 1...
Time = 0.29 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.91 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^2} \, dx=-4200 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {55953383}{8232} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {218994705 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 2029422516 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 7051481738 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 10887812348 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 6303237941 \, \sqrt {-2 \, x + 1}}{196 \, {\left (405 \, {\left (2 \, x - 1\right )}^{5} + 4671 \, {\left (2 \, x - 1\right )}^{4} + 21546 \, {\left (2 \, x - 1\right )}^{3} + 49686 \, {\left (2 \, x - 1\right )}^{2} + 114562 \, x - 30870\right )}} \]
-4200*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 55953383/8232*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(2 1) + 3*sqrt(-2*x + 1))) - 1/196*(218994705*(-2*x + 1)^(9/2) - 2029422516*( -2*x + 1)^(7/2) + 7051481738*(-2*x + 1)^(5/2) - 10887812348*(-2*x + 1)^(3/ 2) + 6303237941*sqrt(-2*x + 1))/(405*(2*x - 1)^5 + 4671*(2*x - 1)^4 + 2154 6*(2*x - 1)^3 + 49686*(2*x - 1)^2 + 114562*x - 30870)
Time = 0.28 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.86 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^2} \, dx=-4200 \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {55953383}{8232} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {1375 \, \sqrt {-2 \, x + 1}}{5 \, x + 3} - \frac {35067141 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 247239993 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 581129563 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 455372631 \, \sqrt {-2 \, x + 1}}{3136 \, {\left (3 \, x + 2\right )}^{4}} \]
-4200*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5* sqrt(-2*x + 1))) + 55953383/8232*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt (-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1375*sqrt(-2*x + 1)/(5*x + 3) - 1/3136*(35067141*(2*x - 1)^3*sqrt(-2*x + 1) + 247239993*(2*x - 1)^2*sqr t(-2*x + 1) - 581129563*(-2*x + 1)^(3/2) + 455372631*sqrt(-2*x + 1))/(3*x + 2)^4
Time = 1.60 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.70 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^2} \, dx=8400\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )-\frac {55953383\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{4116}-\frac {\frac {128637509\,\sqrt {1-2\,x}}{1620}-\frac {55550063\,{\left (1-2\,x\right )}^{3/2}}{405}+\frac {503677267\,{\left (1-2\,x\right )}^{5/2}}{5670}-\frac {169118543\,{\left (1-2\,x\right )}^{7/2}}{6615}+\frac {1622183\,{\left (1-2\,x\right )}^{9/2}}{588}}{\frac {114562\,x}{405}+\frac {16562\,{\left (2\,x-1\right )}^2}{135}+\frac {266\,{\left (2\,x-1\right )}^3}{5}+\frac {173\,{\left (2\,x-1\right )}^4}{15}+{\left (2\,x-1\right )}^5-\frac {686}{9}} \]
8400*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11) - (55953383*21^(1/2)*at anh((21^(1/2)*(1 - 2*x)^(1/2))/7))/4116 - ((128637509*(1 - 2*x)^(1/2))/162 0 - (55550063*(1 - 2*x)^(3/2))/405 + (503677267*(1 - 2*x)^(5/2))/5670 - (1 69118543*(1 - 2*x)^(7/2))/6615 + (1622183*(1 - 2*x)^(9/2))/588)/((114562*x )/405 + (16562*(2*x - 1)^2)/135 + (266*(2*x - 1)^3)/5 + (173*(2*x - 1)^4)/ 15 + (2*x - 1)^5 - 686/9)