Integrand size = 24, antiderivative size = 153 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^3} \, dx=-\frac {35495 \sqrt {1-2 x}}{1078 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^2}+\frac {429 \sqrt {1-2 x}}{98 (2+3 x) (3+5 x)^2}+\frac {1177080 \sqrt {1-2 x}}{5929 (3+5 x)}+\frac {134217}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {321825}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Output:
-35495/1078*(1-2*x)^(1/2)/(3+5*x)^2+3/14*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^2 +429/98*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^2+1177080*(1-2*x)^(1/2)/(17787+29645 *x)+134217/343*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))-321825/1331*55 ^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^3} \, dx=\frac {\sqrt {1-2 x} \left (26794499+127303347 x+201297915 x^2+105937200 x^3\right )}{11858 \left (6+19 x+15 x^2\right )^2}+\frac {134217}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {321825}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Input:
Integrate[1/(Sqrt[1 - 2*x]*(2 + 3*x)^3*(3 + 5*x)^3),x]
Output:
(Sqrt[1 - 2*x]*(26794499 + 127303347*x + 201297915*x^2 + 105937200*x^3))/( 11858*(6 + 19*x + 15*x^2)^2) + (134217*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - (321825*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121
Time = 0.29 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {114, 168, 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.
| \(\displaystyle \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{14} \int \frac {73-105 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^3}dx+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{7} \int \frac {7763-10725 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^3}dx+\frac {429 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{7} \left (-\frac {1}{22} \int \frac {6 (93053-106485 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {35495 \sqrt {1-2 x}}{11 (5 x+3)^2}\right )+\frac {429 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{7} \left (-\frac {3}{11} \int \frac {93053-106485 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {35495 \sqrt {1-2 x}}{11 (5 x+3)^2}\right )+\frac {429 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{7} \left (-\frac {3}{11} \left (-\frac {1}{11} \int \frac {3843979-2354160 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {784720 \sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {35495 \sqrt {1-2 x}}{11 (5 x+3)^2}\right )+\frac {429 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{7} \left (-\frac {3}{11} \left (\frac {1}{11} \left (16240257 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-26282375 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx\right )-\frac {784720 \sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {35495 \sqrt {1-2 x}}{11 (5 x+3)^2}\right )+\frac {429 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{7} \left (-\frac {3}{11} \left (\frac {1}{11} \left (26282375 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-16240257 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {784720 \sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {35495 \sqrt {1-2 x}}{11 (5 x+3)^2}\right )+\frac {429 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{7} \left (-\frac {3}{11} \left (\frac {1}{11} \left (10512950 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )-10826838 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )-\frac {784720 \sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {35495 \sqrt {1-2 x}}{11 (5 x+3)^2}\right )+\frac {429 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^2}\) |
Input:
Int[1/(Sqrt[1 - 2*x]*(2 + 3*x)^3*(3 + 5*x)^3),x]
Output:
(3*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2*(3 + 5*x)^2) + ((429*Sqrt[1 - 2*x])/(7*( 2 + 3*x)*(3 + 5*x)^2) + ((-35495*Sqrt[1 - 2*x])/(11*(3 + 5*x)^2) - (3*((-7 84720*Sqrt[1 - 2*x])/(11*(3 + 5*x)) + (-10826838*Sqrt[3/7]*ArcTanh[Sqrt[3/ 7]*Sqrt[1 - 2*x]] + 10512950*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]) /11))/11)/7)/14
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*(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] + Simp[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*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 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 = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.52
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \left (105937200 x^{3}+201297915 x^{2}+127303347 x +26794499\right )}{11858 \left (15 x^{2}+19 x +6\right )^{2} \sqrt {1-2 x}}+\frac {134217 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{343}-\frac {321825 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{1331}\) | \(79\) |
| derivativedivides | \(-\frac {972 \left (\frac {71 \left (1-2 x \right )^{\frac {3}{2}}}{196}-\frac {215 \sqrt {1-2 x}}{252}\right )}{\left (-4-6 x \right )^{2}}+\frac {134217 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{343}+\frac {-\frac {121875 \left (1-2 x \right )^{\frac {3}{2}}}{121}+\frac {24125 \sqrt {1-2 x}}{11}}{\left (-6-10 x \right )^{2}}-\frac {321825 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{1331}\) | \(94\) |
| default | \(-\frac {972 \left (\frac {71 \left (1-2 x \right )^{\frac {3}{2}}}{196}-\frac {215 \sqrt {1-2 x}}{252}\right )}{\left (-4-6 x \right )^{2}}+\frac {134217 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{343}+\frac {-\frac {121875 \left (1-2 x \right )^{\frac {3}{2}}}{121}+\frac {24125 \sqrt {1-2 x}}{11}}{\left (-6-10 x \right )^{2}}-\frac {321825 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{1331}\) | \(94\) |
| pseudoelliptic | \(\frac {357285654 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (15 x^{2}+19 x +6\right )^{2} \sqrt {21}-220771950 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (15 x^{2}+19 x +6\right )^{2} \sqrt {55}+77 \sqrt {1-2 x}\, \left (105937200 x^{3}+201297915 x^{2}+127303347 x +26794499\right )}{913066 \left (2+3 x \right )^{2} \left (3+5 x \right )^{2}}\) | \(102\) |
| trager | \(\frac {\left (105937200 x^{3}+201297915 x^{2}+127303347 x +26794499\right ) \sqrt {1-2 x}}{11858 \left (15 x^{2}+19 x +6\right )^{2}}+\frac {81 \operatorname {RootOf}\left (\textit {\_Z}^{2}-57658629\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-57658629\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-57658629\right )-34797 \sqrt {1-2 x}}{2+3 x}\right )}{686}+\frac {525 \operatorname {RootOf}\left (\textit {\_Z}^{2}-20667295\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-20667295\right ) x -8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-20667295\right )+33715 \sqrt {1-2 x}}{3+5 x}\right )}{2662}\) | \(127\) |
Input:
int(1/(1-2*x)^(1/2)/(2+3*x)^3/(3+5*x)^3,x,method=_RETURNVERBOSE)
Output:
-1/11858*(-1+2*x)*(105937200*x^3+201297915*x^2+127303347*x+26794499)/(15*x ^2+19*x+6)^2/(1-2*x)^(1/2)+134217/343*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x )^(1/2))-321825/1331*55^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
Time = 0.08 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^3} \, dx=\frac {15769425 \, \sqrt {\frac {5}{11}} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac {5 \, x + 11 \, \sqrt {\frac {5}{11}} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 16240257 \, \sqrt {\frac {3}{7}} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac {3 \, x - 7 \, \sqrt {\frac {3}{7}} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + {\left (105937200 \, x^{3} + 201297915 \, x^{2} + 127303347 \, x + 26794499\right )} \sqrt {-2 \, x + 1}}{11858 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \] Input:
integrate(1/(1-2*x)^(1/2)/(2+3*x)^3/(3+5*x)^3,x, algorithm="fricas")
Output:
1/11858*(15769425*sqrt(5/11)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*lo g((5*x + 11*sqrt(5/11)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 16240257*sqrt(3/7) *(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log((3*x - 7*sqrt(3/7)*sqrt(-2 *x + 1) - 5)/(3*x + 2)) + (105937200*x^3 + 201297915*x^2 + 127303347*x + 2 6794499)*sqrt(-2*x + 1))/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)
Result contains complex when optimal does not.
Time = 13.50 (sec) , antiderivative size = 6346, normalized size of antiderivative = 41.48 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(1-2*x)**(1/2)/(2+3*x)**3/(3+5*x)**3,x)
Output:
234926334720000*sqrt(2)*I*(x - 1/2)**(23/2)/(5916667680000*(x - 1/2)**12 + 53644453632000*(x - 1/2)**11 + 212763369772800*(x - 1/2)**10 + 4821444282 54720*(x - 1/2)**9 + 682784662823648*(x - 1/2)**8 + 618752016260224*(x - 1 /2)**7 + 350409449828592*(x - 1/2)**6 + 113381774768416*(x - 1/2)**5 + 160 48523266853*(x - 1/2)**4) + 1863787142448000*sqrt(2)*I*(x - 1/2)**(21/2)/( 5916667680000*(x - 1/2)**12 + 53644453632000*(x - 1/2)**11 + 2127633697728 00*(x - 1/2)**10 + 482144428254720*(x - 1/2)**9 + 682784662823648*(x - 1/2 )**8 + 618752016260224*(x - 1/2)**7 + 350409449828592*(x - 1/2)**6 + 11338 1774768416*(x - 1/2)**5 + 16048523266853*(x - 1/2)**4) + 6336048975379200* sqrt(2)*I*(x - 1/2)**(19/2)/(5916667680000*(x - 1/2)**12 + 53644453632000* (x - 1/2)**11 + 212763369772800*(x - 1/2)**10 + 482144428254720*(x - 1/2)* *9 + 682784662823648*(x - 1/2)**8 + 618752016260224*(x - 1/2)**7 + 3504094 49828592*(x - 1/2)**6 + 113381774768416*(x - 1/2)**5 + 16048523266853*(x - 1/2)**4) + 11964721362058080*sqrt(2)*I*(x - 1/2)**(17/2)/(5916667680000*( x - 1/2)**12 + 53644453632000*(x - 1/2)**11 + 212763369772800*(x - 1/2)**1 0 + 482144428254720*(x - 1/2)**9 + 682784662823648*(x - 1/2)**8 + 61875201 6260224*(x - 1/2)**7 + 350409449828592*(x - 1/2)**6 + 113381774768416*(x - 1/2)**5 + 16048523266853*(x - 1/2)**4) + 13554148250345472*sqrt(2)*I*(x - 1/2)**(15/2)/(5916667680000*(x - 1/2)**12 + 53644453632000*(x - 1/2)**11 + 212763369772800*(x - 1/2)**10 + 482144428254720*(x - 1/2)**9 + 682784...
Time = 0.11 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^3} \, dx=\frac {321825}{2662} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {134217}{686} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (52968600 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 360203715 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 816108324 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 616051205 \, \sqrt {-2 \, x + 1}\right )}}{5929 \, {\left (225 \, {\left (2 \, x - 1\right )}^{4} + 2040 \, {\left (2 \, x - 1\right )}^{3} + 6934 \, {\left (2 \, x - 1\right )}^{2} + 20944 \, x - 4543\right )}} \] Input:
integrate(1/(1-2*x)^(1/2)/(2+3*x)^3/(3+5*x)^3,x, algorithm="maxima")
Output:
321825/2662*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt (-2*x + 1))) - 134217/686*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqr t(21) + 3*sqrt(-2*x + 1))) - 2/5929*(52968600*(-2*x + 1)^(7/2) - 360203715 *(-2*x + 1)^(5/2) + 816108324*(-2*x + 1)^(3/2) - 616051205*sqrt(-2*x + 1)) /(225*(2*x - 1)^4 + 2040*(2*x - 1)^3 + 6934*(2*x - 1)^2 + 20944*x - 4543)
Time = 0.12 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^3} \, dx=\frac {321825}{2662} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {134217}{686} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2 \, {\left (52968600 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 360203715 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 816108324 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 616051205 \, \sqrt {-2 \, x + 1}\right )}}{5929 \, {\left (15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \] Input:
integrate(1/(1-2*x)^(1/2)/(2+3*x)^3/(3+5*x)^3,x, algorithm="giac")
Output:
321825/2662*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55 ) + 5*sqrt(-2*x + 1))) - 134217/686*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*s qrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/5929*(52968600*(2*x - 1) ^3*sqrt(-2*x + 1) + 360203715*(2*x - 1)^2*sqrt(-2*x + 1) - 816108324*(-2*x + 1)^(3/2) + 616051205*sqrt(-2*x + 1))/(15*(2*x - 1)^2 + 136*x + 9)^2
Time = 1.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^3} \, dx=\frac {134217\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {321825\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331}+\frac {\frac {3200266\,\sqrt {1-2\,x}}{3465}-\frac {544072216\,{\left (1-2\,x\right )}^{3/2}}{444675}+\frac {16009054\,{\left (1-2\,x\right )}^{5/2}}{29645}-\frac {470832\,{\left (1-2\,x\right )}^{7/2}}{5929}}{\frac {20944\,x}{225}+\frac {6934\,{\left (2\,x-1\right )}^2}{225}+\frac {136\,{\left (2\,x-1\right )}^3}{15}+{\left (2\,x-1\right )}^4-\frac {4543}{225}} \] Input:
int(1/((1 - 2*x)^(1/2)*(3*x + 2)^3*(5*x + 3)^3),x)
Output:
(134217*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/343 - (321825*55^(1/ 2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/1331 + ((3200266*(1 - 2*x)^(1/2)) /3465 - (544072216*(1 - 2*x)^(3/2))/444675 + (16009054*(1 - 2*x)^(5/2))/29 645 - (470832*(1 - 2*x)^(7/2))/5929)/((20944*x)/225 + (6934*(2*x - 1)^2)/2 25 + (136*(2*x - 1)^3)/15 + (2*x - 1)^4 - 4543/225)
Time = 0.15 (sec) , antiderivative size = 443, normalized size of antiderivative = 2.90 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^3} \, dx=\frac {8157164400 \sqrt {-2 x +1}\, x^{3}+15499939455 \sqrt {-2 x +1}\, x^{2}+9802357719 \sqrt {-2 x +1}\, x +2063176423 \sqrt {-2 x +1}+24836844375 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{4}+62920005750 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{3}+59718812475 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{2}+25168002300 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x +3973895100 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right )-24836844375 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{4}-62920005750 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{3}-59718812475 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{2}-25168002300 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x -3973895100 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right )-40194636075 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{4}-101826411390 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{3}-96645769407 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{2}-40730564556 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x -6431141772 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )+40194636075 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{4}+101826411390 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{3}+96645769407 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{2}+40730564556 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x +6431141772 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{205439850 x^{4}+520447620 x^{3}+493968706 x^{2}+208179048 x +32870376} \] Input:
int(1/(1-2*x)^(1/2)/(2+3*x)^3/(3+5*x)^3,x)
Output:
(8157164400*sqrt( - 2*x + 1)*x**3 + 15499939455*sqrt( - 2*x + 1)*x**2 + 98 02357719*sqrt( - 2*x + 1)*x + 2063176423*sqrt( - 2*x + 1) + 24836844375*sq rt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x**4 + 62920005750*sqrt(55)*log( 5*sqrt( - 2*x + 1) - sqrt(55))*x**3 + 59718812475*sqrt(55)*log(5*sqrt( - 2 *x + 1) - sqrt(55))*x**2 + 25168002300*sqrt(55)*log(5*sqrt( - 2*x + 1) - s qrt(55))*x + 3973895100*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55)) - 2483 6844375*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x**4 - 62920005750*sqr t(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x**3 - 59718812475*sqrt(55)*log(5 *sqrt( - 2*x + 1) + sqrt(55))*x**2 - 25168002300*sqrt(55)*log(5*sqrt( - 2* x + 1) + sqrt(55))*x - 3973895100*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(5 5)) - 40194636075*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**4 - 10182 6411390*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**3 - 96645769407*sqr t(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**2 - 40730564556*sqrt(21)*log(3 *sqrt( - 2*x + 1) - sqrt(21))*x - 6431141772*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21)) + 40194636075*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x **4 + 101826411390*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x**3 + 9664 5769407*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x**2 + 40730564556*sqr t(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x + 6431141772*sqrt(21)*log(3*sqr t( - 2*x + 1) + sqrt(21)))/(913066*(225*x**4 + 570*x**3 + 541*x**2 + 228*x + 36))