Integrand size = 26, antiderivative size = 159 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}} \, dx=-\frac {8515}{7546 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {7090175 \sqrt {1-2 x}}{498036 (3+5 x)^{3/2}}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {765}{196 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac {707286025 \sqrt {1-2 x}}{5478396 \sqrt {3+5 x}}-\frac {1215945 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1372 \sqrt {7}} \] Output:
-8515/7546/(1-2*x)^(1/2)/(3+5*x)^(3/2)-7090175/498036*(1-2*x)^(1/2)/(3+5*x )^(3/2)+3/14/(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^(3/2)+765/196/(1-2*x)^(1/2)/( 2+3*x)/(3+5*x)^(3/2)+707286025/5478396*(1-2*x)^(1/2)/(3+5*x)^(1/2)-1215945 /9604*7^(1/2)*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))
Time = 0.20 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.53 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}} \, dx=\frac {8194676012+22311149965 x-16567908760 x^2-89836042575 x^3-63655742250 x^4}{5478396 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}-\frac {1215945 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1372 \sqrt {7}} \] Input:
Integrate[1/((1 - 2*x)^(3/2)*(2 + 3*x)^3*(3 + 5*x)^(5/2)),x]
Output:
(8194676012 + 22311149965*x - 16567908760*x^2 - 89836042575*x^3 - 63655742 250*x^4)/(5478396*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(3/2)) - (1215945*Ar cTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*Sqrt[7])
Time = 0.25 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {114, 27, 168, 27, 169, 27, 169, 27, 169, 27, 104, 217}
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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^3 (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{14} \int \frac {5 (19-48 x)}{2 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}}dx+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{28} \int \frac {19-48 x}{(1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}}dx+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {5}{28} \left (\frac {1}{7} \int \frac {2887-9180 x}{2 (1-2 x)^{3/2} (3 x+2) (5 x+3)^{5/2}}dx+\frac {153}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{28} \left (\frac {1}{14} \int \frac {2887-9180 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^{5/2}}dx+\frac {153}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{28} \left (\frac {1}{14} \left (-\frac {2}{77} \int -\frac {160991-204360 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}}dx-\frac {6812}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {153}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{28} \left (\frac {1}{14} \left (\frac {1}{77} \int \frac {160991-204360 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}}dx-\frac {6812}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {153}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{28} \left (\frac {1}{14} \left (\frac {1}{77} \left (-\frac {2}{33} \int \frac {18081589-17016420 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {2836070 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )-\frac {6812}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {153}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{28} \left (\frac {1}{14} \left (\frac {1}{77} \left (-\frac {1}{33} \int \frac {18081589-17016420 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {2836070 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )-\frac {6812}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {153}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{28} \left (\frac {1}{14} \left (\frac {1}{77} \left (\frac {1}{33} \left (\frac {2}{11} \int \frac {971053677}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {282914410 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {2836070 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )-\frac {6812}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {153}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{28} \left (\frac {1}{14} \left (\frac {1}{77} \left (\frac {1}{33} \left (88277607 \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {282914410 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {2836070 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )-\frac {6812}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {153}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {5}{28} \left (\frac {1}{14} \left (\frac {1}{77} \left (\frac {1}{33} \left (176555214 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {282914410 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {2836070 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )-\frac {6812}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {153}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5}{28} \left (\frac {1}{14} \left (\frac {1}{77} \left (\frac {1}{33} \left (\frac {282914410 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}-\frac {176555214 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}}\right )-\frac {2836070 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )-\frac {6812}{77 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {153}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}\) |
Input:
Int[1/((1 - 2*x)^(3/2)*(2 + 3*x)^3*(3 + 5*x)^(5/2)),x]
Output:
3/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (5*(153/(7*Sqrt[1 - 2*x ]*(2 + 3*x)*(3 + 5*x)^(3/2)) + (-6812/(77*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + ((-2836070*Sqrt[1 - 2*x])/(33*(3 + 5*x)^(3/2)) + ((282914410*Sqrt[1 - 2*x ])/(11*Sqrt[3 + 5*x]) - (176555214*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/Sqrt[7])/33)/77)/14))/28
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_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*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[((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] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs. \(2(120)=240\).
Time = 0.25 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.87
| method | result | size |
| default | \(-\frac {2184870773250 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+4442570572275 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+2485897413120 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+891180391500 x^{4} \sqrt {-10 x^{2}-x +3}-412697812725 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+1257704596050 x^{3} \sqrt {-10 x^{2}-x +3}-757421868060 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +231950722640 x^{2} \sqrt {-10 x^{2}-x +3}-174789661860 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-312356099510 x \sqrt {-10 x^{2}-x +3}-114725464168 \sqrt {-10 x^{2}-x +3}}{76697544 \left (2+3 x \right )^{2} \sqrt {1-2 x}\, \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(298\) |
Input:
int(1/(1-2*x)^(3/2)/(2+3*x)^3/(3+5*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/76697544*(2184870773250*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2- x+3)^(1/2))*x^5+4442570572275*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x ^2-x+3)^(1/2))*x^4+2485897413120*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-1 0*x^2-x+3)^(1/2))*x^3+891180391500*x^4*(-10*x^2-x+3)^(1/2)-412697812725*7^ (1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+1257704596050 *x^3*(-10*x^2-x+3)^(1/2)-757421868060*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2 )/(-10*x^2-x+3)^(1/2))*x+231950722640*x^2*(-10*x^2-x+3)^(1/2)-174789661860 *7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-312356099510*x *(-10*x^2-x+3)^(1/2)-114725464168*(-10*x^2-x+3)^(1/2))/(2+3*x)^2/(1-2*x)^( 1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)
Time = 0.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.82 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}} \, dx=-\frac {4855268385 \, \sqrt {7} {\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (63655742250 \, x^{4} + 89836042575 \, x^{3} + 16567908760 \, x^{2} - 22311149965 \, x - 8194676012\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{76697544 \, {\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )}} \] Input:
integrate(1/(1-2*x)^(3/2)/(2+3*x)^3/(3+5*x)^(5/2),x, algorithm="fricas")
Output:
-1/76697544*(4855268385*sqrt(7)*(450*x^5 + 915*x^4 + 512*x^3 - 85*x^2 - 15 6*x - 36)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10 *x^2 + x - 3)) - 14*(63655742250*x^4 + 89836042575*x^3 + 16567908760*x^2 - 22311149965*x - 8194676012)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(450*x^5 + 915* x^4 + 512*x^3 - 85*x^2 - 156*x - 36)
\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}} \, dx=\int \frac {1}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/(1-2*x)**(3/2)/(2+3*x)**3/(3+5*x)**(5/2),x)
Output:
Integral(1/((1 - 2*x)**(3/2)*(3*x + 2)**3*(5*x + 3)**(5/2)), x)
\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{3} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(1-2*x)^(3/2)/(2+3*x)^3/(3+5*x)^(5/2),x, algorithm="maxima")
Output:
integrate(1/((5*x + 3)^(5/2)*(3*x + 2)^3*(-2*x + 1)^(3/2)), x)
Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (120) = 240\).
Time = 0.36 (sec) , antiderivative size = 398, normalized size of antiderivative = 2.50 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}} \, dx=-\frac {125}{63888} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {243189}{38416} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {11875}{2662} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {64 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{2282665 \, {\left (2 \, x - 1\right )}} + \frac {891 \, \sqrt {10} {\left (67 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {16120 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {64480 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{98 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \] Input:
integrate(1/(1-2*x)^(3/2)/(2+3*x)^3/(3+5*x)^(5/2),x, algorithm="giac")
Output:
-125/63888*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 243189/38416*sqr t(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt (-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(2 2)))) + 11875/2662*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) - 64/2282665 *sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) + 891/98*sqrt(10)*(67*((s qrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2 )*sqrt(-10*x + 5) - sqrt(22)))^3 + 16120*(sqrt(2)*sqrt(-10*x + 5) - sqrt(2 2))/sqrt(5*x + 3) - 64480*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22 )))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3) /(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^2
Timed out. \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{5/2}} \,d x \] Input:
int(1/((1 - 2*x)^(3/2)*(3*x + 2)^3*(5*x + 3)^(5/2)),x)
Output:
int(1/((1 - 2*x)^(3/2)*(3*x + 2)^3*(5*x + 3)^(5/2)), x)
Time = 0.26 (sec) , antiderivative size = 433, normalized size of antiderivative = 2.72 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}} \, dx=\frac {218487077325 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x^{3}+422408349495 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x^{2}+271895029560 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x +58263220620 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )-218487077325 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x^{3}-422408349495 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x^{2}-271895029560 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x -58263220620 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )-445590195750 x^{4}-628852298025 x^{3}-115975361320 x^{2}+156178049755 x +57362732084}{38348772 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (45 x^{3}+87 x^{2}+56 x +12\right )} \] Input:
int(1/(1-2*x)^(3/2)/(2+3*x)^3/(3+5*x)^(5/2),x)
Output:
(218487077325*sqrt(5*x + 3)*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) - sqrt (35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**3 + 422 408349495*sqrt(5*x + 3)*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) - sqrt(35) *tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 + 2718950 29560*sqrt(5*x + 3)*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan (asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 58263220620*sq rt(5*x + 3)*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((s qrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) - 218487077325*sqrt(5*x + 3)*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2* x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**3 - 422408349495*sqrt(5*x + 3)*s qrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 - 271895029560*sqrt(5*x + 3)*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*s qrt(5))/sqrt(11))/2))/sqrt(2))*x - 58263220620*sqrt(5*x + 3)*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/ sqrt(11))/2))/sqrt(2)) - 445590195750*x**4 - 628852298025*x**3 - 115975361 320*x**2 + 156178049755*x + 57362732084)/(38348772*sqrt(5*x + 3)*sqrt( - 2 *x + 1)*(45*x**3 + 87*x**2 + 56*x + 12))