Integrand size = 24, antiderivative size = 147 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^3} \, dx=-\frac {1015 \sqrt {1-2 x}}{6 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^2}+\frac {45 \sqrt {1-2 x}}{2 (2+3 x) (3+5 x)^2}+\frac {1020 \sqrt {1-2 x}}{3+5 x}+14073 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-13665 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Output:
-1015/6*(1-2*x)^(1/2)/(3+5*x)^2+7/6*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^2+45/2 *(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^2+1020*(1-2*x)^(1/2)/(3+5*x)+14073/7*21^(1/ 2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))-13665/11*55^(1/2)*arctanh(1/11*55^( 1/2)*(1-2*x)^(1/2))
Time = 0.30 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.65 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^3} \, dx=\frac {\sqrt {1-2 x} \left (23219+110315 x+174435 x^2+91800 x^3\right )}{2 \left (6+19 x+15 x^2\right )^2}+14073 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-13665 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Input:
Integrate[(1 - 2*x)^(3/2)/((2 + 3*x)^3*(3 + 5*x)^3),x]
Output:
(Sqrt[1 - 2*x]*(23219 + 110315*x + 174435*x^2 + 91800*x^3))/(2*(6 + 19*x + 15*x^2)^2) + 14073*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 13665*Sqr t[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
Time = 0.24 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {109, 168, 27, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2}}{(3 x+2)^3 (5 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{6} \int \frac {157-237 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^3}dx+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{7} \int \frac {7 (2441-3375 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)^3}dx+\frac {135 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\int \frac {2441-3375 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^3}dx+\frac {135 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{6} \left (-\frac {1}{22} \int \frac {198 (887-1015 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {1015 \sqrt {1-2 x}}{(5 x+3)^2}+\frac {135 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (-9 \int \frac {887-1015 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {1015 \sqrt {1-2 x}}{(5 x+3)^2}+\frac {135 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{6} \left (-9 \left (-\frac {1}{11} \int \frac {11 (3331-2040 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {680 \sqrt {1-2 x}}{5 x+3}\right )-\frac {1015 \sqrt {1-2 x}}{(5 x+3)^2}+\frac {135 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (-9 \left (-\int \frac {3331-2040 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {680 \sqrt {1-2 x}}{5 x+3}\right )-\frac {1015 \sqrt {1-2 x}}{(5 x+3)^2}+\frac {135 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{6} \left (-9 \left (14073 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-22775 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-\frac {680 \sqrt {1-2 x}}{5 x+3}\right )-\frac {1015 \sqrt {1-2 x}}{(5 x+3)^2}+\frac {135 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{6} \left (-9 \left (22775 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-14073 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {680 \sqrt {1-2 x}}{5 x+3}\right )-\frac {1015 \sqrt {1-2 x}}{(5 x+3)^2}+\frac {135 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{6} \left (-9 \left (-9382 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+9110 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )-\frac {680 \sqrt {1-2 x}}{5 x+3}\right )-\frac {1015 \sqrt {1-2 x}}{(5 x+3)^2}+\frac {135 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^2}\) |
Input:
Int[(1 - 2*x)^(3/2)/((2 + 3*x)^3*(3 + 5*x)^3),x]
Output:
(7*Sqrt[1 - 2*x])/(6*(2 + 3*x)^2*(3 + 5*x)^2) + ((-1015*Sqrt[1 - 2*x])/(3 + 5*x)^2 + (135*Sqrt[1 - 2*x])/((2 + 3*x)*(3 + 5*x)^2) - 9*((-680*Sqrt[1 - 2*x])/(3 + 5*x) - 9382*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + 9110* Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]))/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 + 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 = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.54
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \left (91800 x^{3}+174435 x^{2}+110315 x +23219\right )}{2 \left (15 x^{2}+19 x +6\right )^{2} \sqrt {1-2 x}}+\frac {14073 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{7}-\frac {13665 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{11}\) | \(79\) |
| derivativedivides | \(\frac {-5075 \left (1-2 x \right )^{\frac {3}{2}}+11055 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-\frac {13665 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{11}-\frac {324 \left (\frac {205 \left (1-2 x \right )^{\frac {3}{2}}}{36}-\frac {161 \sqrt {1-2 x}}{12}\right )}{\left (-4-6 x \right )^{2}}+\frac {14073 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{7}\) | \(94\) |
| default | \(\frac {-5075 \left (1-2 x \right )^{\frac {3}{2}}+11055 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-\frac {13665 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{11}-\frac {324 \left (\frac {205 \left (1-2 x \right )^{\frac {3}{2}}}{36}-\frac {161 \sqrt {1-2 x}}{12}\right )}{\left (-4-6 x \right )^{2}}+\frac {14073 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{7}\) | \(94\) |
| pseudoelliptic | \(\frac {309606 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (15 x^{2}+19 x +6\right )^{2} \sqrt {21}-191310 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (15 x^{2}+19 x +6\right )^{2} \sqrt {55}+77 \sqrt {1-2 x}\, \left (91800 x^{3}+174435 x^{2}+110315 x +23219\right )}{154 \left (2+3 x \right )^{2} \left (3+5 x \right )^{2}}\) | \(102\) |
| trager | \(\frac {\left (91800 x^{3}+174435 x^{2}+110315 x +23219\right ) \sqrt {1-2 x}}{2 \left (15 x^{2}+19 x +6\right )^{2}}+\frac {14073 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{14}-\frac {15 \operatorname {RootOf}\left (\textit {\_Z}^{2}-45645655\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-45645655\right ) x +50105 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-45645655\right )}{3+5 x}\right )}{22}\) | \(126\) |
Input:
int((1-2*x)^(3/2)/(2+3*x)^3/(3+5*x)^3,x,method=_RETURNVERBOSE)
Output:
-1/2*(-1+2*x)*(91800*x^3+174435*x^2+110315*x+23219)/(15*x^2+19*x+6)^2/(1-2 *x)^(1/2)+14073/7*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))-13665/11*55 ^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
Time = 0.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^3} \, dx=\frac {13665 \, \sqrt {\frac {5}{11}} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac {5 \, x + 11 \, \sqrt {\frac {5}{11}} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 14073 \, \sqrt {\frac {3}{7}} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac {3 \, x - 7 \, \sqrt {\frac {3}{7}} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + {\left (91800 \, x^{3} + 174435 \, x^{2} + 110315 \, x + 23219\right )} \sqrt {-2 \, x + 1}}{2 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \] Input:
integrate((1-2*x)^(3/2)/(2+3*x)^3/(3+5*x)^3,x, algorithm="fricas")
Output:
1/2*(13665*sqrt(5/11)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log((5*x + 11*sqrt(5/11)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 14073*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)) + (91800*x^3 + 174435*x^2 + 110315*x + 23219)*sqrt(-2*x + 1) )/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)
Time = 85.87 (sec) , antiderivative size = 654, normalized size of antiderivative = 4.45 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^3} \, dx =\text {Too large to display} \] Input:
integrate((1-2*x)**(3/2)/(2+3*x)**3/(3+5*x)**3,x)
Output:
-6934*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt (21)/3))/7 + 6934*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5))/11 + 8484*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))) - 1176*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))) + 22660*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) *sqrt(1 - 2*x)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)))/605, (sqr t(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5))) + 4840*Piecewis e((sqrt(55)*(3*log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/16 - 3*log(sqrt(55)*sqrt (1 - 2*x)/11 + 1)/16 + 3/(16*(sqrt(55)*sqrt(1 - 2*x)/11 + 1)) + 1/(16*(sqr t(55)*sqrt(1 - 2*x)/11 + 1)**2) + 3/(16*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)) - 1/(16*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)**2))/6655, (sqrt(1 - 2*x) > -sqrt(5 5)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))
Time = 0.12 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.99 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^3} \, dx=\frac {13665}{22} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {14073}{14} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (45900 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 312135 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 707200 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 533841 \, \sqrt {-2 \, x + 1}\right )}}{225 \, {\left (2 \, x - 1\right )}^{4} + 2040 \, {\left (2 \, x - 1\right )}^{3} + 6934 \, {\left (2 \, x - 1\right )}^{2} + 20944 \, x - 4543} \] Input:
integrate((1-2*x)^(3/2)/(2+3*x)^3/(3+5*x)^3,x, algorithm="maxima")
Output:
13665/22*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2 *x + 1))) - 14073/14*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2*(45900*(-2*x + 1)^(7/2) - 312135*(-2*x + 1)^(5/2 ) + 707200*(-2*x + 1)^(3/2) - 533841*sqrt(-2*x + 1))/(225*(2*x - 1)^4 + 20 40*(2*x - 1)^3 + 6934*(2*x - 1)^2 + 20944*x - 4543)
Time = 0.14 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.01 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^3} \, dx=\frac {13665}{22} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {14073}{14} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2 \, {\left (45900 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 312135 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 707200 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 533841 \, \sqrt {-2 \, x + 1}\right )}}{{\left (15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \] Input:
integrate((1-2*x)^(3/2)/(2+3*x)^3/(3+5*x)^3,x, algorithm="giac")
Output:
13665/22*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 14073/14*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(- 2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2*(45900*(2*x - 1)^3*sqrt(-2*x + 1) + 312135*(2*x - 1)^2*sqrt(-2*x + 1) - 707200*(-2*x + 1)^(3/2) + 53384 1*sqrt(-2*x + 1))/(15*(2*x - 1)^2 + 136*x + 9)^2
Time = 1.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.73 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^3} \, dx=\frac {14073\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{7}-\frac {13665\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{11}+\frac {\frac {355894\,\sqrt {1-2\,x}}{75}-\frac {56576\,{\left (1-2\,x\right )}^{3/2}}{9}+\frac {41618\,{\left (1-2\,x\right )}^{5/2}}{15}-408\,{\left (1-2\,x\right )}^{7/2}}{\frac {20944\,x}{225}+\frac {6934\,{\left (2\,x-1\right )}^2}{225}+\frac {136\,{\left (2\,x-1\right )}^3}{15}+{\left (2\,x-1\right )}^4-\frac {4543}{225}} \] Input:
int((1 - 2*x)^(3/2)/((3*x + 2)^3*(5*x + 3)^3),x)
Output:
(14073*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/7 - (13665*55^(1/2)*a tanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/11 + ((355894*(1 - 2*x)^(1/2))/75 - ( 56576*(1 - 2*x)^(3/2))/9 + (41618*(1 - 2*x)^(5/2))/15 - 408*(1 - 2*x)^(7/2 ))/((20944*x)/225 + (6934*(2*x - 1)^2)/225 + (136*(2*x - 1)^3)/15 + (2*x - 1)^4 - 4543/225)
Time = 0.16 (sec) , antiderivative size = 443, normalized size of antiderivative = 3.01 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^3} \, dx=\frac {7068600 \sqrt {-2 x +1}\, x^{3}+13431495 \sqrt {-2 x +1}\, x^{2}+8494255 \sqrt {-2 x +1}\, x +1787863 \sqrt {-2 x +1}+21522375 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{4}+54523350 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{3}+51749355 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{2}+21809340 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x +3443580 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right )-21522375 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{4}-54523350 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{3}-51749355 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{2}-21809340 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x -3443580 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right )-34830675 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{4}-88237710 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{3}-83748423 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{2}-35295084 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x -5572908 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )+34830675 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{4}+88237710 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{3}+83748423 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{2}+35295084 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x +5572908 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{34650 x^{4}+87780 x^{3}+83314 x^{2}+35112 x +5544} \] Input:
int((1-2*x)^(3/2)/(2+3*x)^3/(3+5*x)^3,x)
Output:
(7068600*sqrt( - 2*x + 1)*x**3 + 13431495*sqrt( - 2*x + 1)*x**2 + 8494255* sqrt( - 2*x + 1)*x + 1787863*sqrt( - 2*x + 1) + 21522375*sqrt(55)*log(5*sq rt( - 2*x + 1) - sqrt(55))*x**4 + 54523350*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x**3 + 51749355*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x **2 + 21809340*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x + 3443580*sqr t(55)*log(5*sqrt( - 2*x + 1) - sqrt(55)) - 21522375*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x**4 - 54523350*sqrt(55)*log(5*sqrt( - 2*x + 1) + sq rt(55))*x**3 - 51749355*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x**2 - 21809340*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x - 3443580*sqrt(55) *log(5*sqrt( - 2*x + 1) + sqrt(55)) - 34830675*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**4 - 88237710*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21 ))*x**3 - 83748423*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**2 - 3529 5084*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x - 5572908*sqrt(21)*log( 3*sqrt( - 2*x + 1) - sqrt(21)) + 34830675*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x**4 + 88237710*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x* *3 + 83748423*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x**2 + 35295084* sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x + 5572908*sqrt(21)*log(3*sqr t( - 2*x + 1) + sqrt(21)))/(154*(225*x**4 + 570*x**3 + 541*x**2 + 228*x + 36))