Integrand size = 26, antiderivative size = 202 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{3/2}} \, dx=-\frac {4639661185 \sqrt {1-2 x}}{56448 \sqrt {3+5 x}}+\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 \sqrt {3+5 x}}+\frac {2513 \sqrt {1-2 x}}{360 (2+3 x)^4 \sqrt {3+5 x}}+\frac {12023 \sqrt {1-2 x}}{240 (2+3 x)^3 \sqrt {3+5 x}}+\frac {587477 \sqrt {1-2 x}}{1344 (2+3 x)^2 \sqrt {3+5 x}}+\frac {102293609 \sqrt {1-2 x}}{18816 (2+3 x) \sqrt {3+5 x}}+\frac {3538809681 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{6272 \sqrt {7}} \] Output:
-4639661185/56448*(1-2*x)^(1/2)/(3+5*x)^(1/2)+7/15*(1-2*x)^(3/2)/(2+3*x)^5 /(3+5*x)^(1/2)+2513/360*(1-2*x)^(1/2)/(2+3*x)^4/(3+5*x)^(1/2)+12023/240*(1 -2*x)^(1/2)/(2+3*x)^3/(3+5*x)^(1/2)+587477/1344*(1-2*x)^(1/2)/(2+3*x)^2/(3 +5*x)^(1/2)+102293609/18816*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^(1/2)+3538809681 /43904*7^(1/2)*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))
Time = 0.34 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.44 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{3/2}} \, dx=\frac {-\frac {7 \sqrt {1-2 x} \left (79638637088+601741553688 x+1818284414692 x^2+2746600901250 x^3+2074037896035 x^4+626354259975 x^5\right )}{(2+3 x)^5 \sqrt {3+5 x}}+17694048405 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{219520} \] Input:
Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^6*(3 + 5*x)^(3/2)),x]
Output:
((-7*Sqrt[1 - 2*x]*(79638637088 + 601741553688*x + 1818284414692*x^2 + 274 6600901250*x^3 + 2074037896035*x^4 + 626354259975*x^5))/((2 + 3*x)^5*Sqrt[ 3 + 5*x]) + 17694048405*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x ])])/219520
Time = 0.36 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.10, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {109, 27, 166, 27, 168, 27, 168, 27, 168, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^6 (5 x+3)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{15} \int \frac {(491-520 x) \sqrt {1-2 x}}{2 (3 x+2)^5 (5 x+3)^{3/2}}dx+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \int \frac {(491-520 x) \sqrt {1-2 x}}{(3 x+2)^5 (5 x+3)^{3/2}}dx+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{30} \left (\frac {2513 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}-\frac {1}{12} \int -\frac {11 (11273-17520 x)}{2 \sqrt {1-2 x} (3 x+2)^4 (5 x+3)^{3/2}}dx\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \left (\frac {11}{24} \int \frac {11273-17520 x}{\sqrt {1-2 x} (3 x+2)^4 (5 x+3)^{3/2}}dx+\frac {2513 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{30} \left (\frac {11}{24} \left (\frac {1}{21} \int \frac {105 (27175-39348 x)}{2 \sqrt {1-2 x} (3 x+2)^3 (5 x+3)^{3/2}}dx+\frac {3279 \sqrt {1-2 x}}{(3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {2513 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \left (\frac {11}{24} \left (\frac {5}{2} \int \frac {27175-39348 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^{3/2}}dx+\frac {3279 \sqrt {1-2 x}}{(3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {2513 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{30} \left (\frac {11}{24} \left (\frac {5}{2} \left (\frac {1}{14} \int \frac {5026859-6408840 x}{2 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}dx+\frac {160221 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {3279 \sqrt {1-2 x}}{(3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {2513 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \left (\frac {11}{24} \left (\frac {5}{2} \left (\frac {1}{28} \int \frac {5026859-6408840 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}dx+\frac {160221 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {3279 \sqrt {1-2 x}}{(3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {2513 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{30} \left (\frac {11}{24} \left (\frac {5}{2} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {593153153-557965140 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx+\frac {27898257 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {160221 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {3279 \sqrt {1-2 x}}{(3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {2513 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \left (\frac {11}{24} \left (\frac {5}{2} \left (\frac {1}{28} \left (\frac {1}{14} \int \frac {593153153-557965140 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx+\frac {27898257 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {160221 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {3279 \sqrt {1-2 x}}{(3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {2513 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{30} \left (\frac {11}{24} \left (\frac {5}{2} \left (\frac {1}{28} \left (\frac {1}{14} \left (-\frac {2}{11} \int \frac {31849287129}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {9279322370 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )+\frac {27898257 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {160221 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {3279 \sqrt {1-2 x}}{(3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {2513 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \left (\frac {11}{24} \left (\frac {5}{2} \left (\frac {1}{28} \left (\frac {1}{14} \left (-2895389739 \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {9279322370 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )+\frac {27898257 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {160221 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {3279 \sqrt {1-2 x}}{(3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {2513 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{30} \left (\frac {11}{24} \left (\frac {5}{2} \left (\frac {1}{28} \left (\frac {1}{14} \left (-5790779478 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {9279322370 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )+\frac {27898257 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {160221 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {3279 \sqrt {1-2 x}}{(3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {2513 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{30} \left (\frac {11}{24} \left (\frac {5}{2} \left (\frac {1}{28} \left (\frac {1}{14} \left (\frac {5790779478 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}}-\frac {9279322370 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )+\frac {27898257 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {160221 \sqrt {1-2 x}}{14 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {3279 \sqrt {1-2 x}}{(3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {2513 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 \sqrt {5 x+3}}\) |
Input:
Int[(1 - 2*x)^(5/2)/((2 + 3*x)^6*(3 + 5*x)^(3/2)),x]
Output:
(7*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^5*Sqrt[3 + 5*x]) + ((2513*Sqrt[1 - 2*x]) /(12*(2 + 3*x)^4*Sqrt[3 + 5*x]) + (11*((3279*Sqrt[1 - 2*x])/((2 + 3*x)^3*S qrt[3 + 5*x]) + (5*((160221*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2*Sqrt[3 + 5*x]) + ((27898257*Sqrt[1 - 2*x])/(7*(2 + 3*x)*Sqrt[3 + 5*x]) + ((-9279322370*Sq rt[1 - 2*x])/(11*Sqrt[3 + 5*x]) + (5790779478*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7 ]*Sqrt[3 + 5*x])])/Sqrt[7])/14)/28))/2))/24)/30
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*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[((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. \(345\) vs. \(2(157)=314\).
Time = 0.24 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.71
| method | result | size |
| default | \(-\frac {\left (21498268812075 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{6}+84559857327495 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+138544399011150 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+8768959639650 x^{5} \sqrt {-10 x^{2}-x +3}+121027291090200 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+29036530544490 x^{4} \sqrt {-10 x^{2}-x +3}+59452002640800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+38452412617500 x^{3} \sqrt {-10 x^{2}-x +3}+15570762596400 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +25455981805688 x^{2} \sqrt {-10 x^{2}-x +3}+1698628646880 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+8424381751632 x \sqrt {-10 x^{2}-x +3}+1114940919232 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{439040 \left (2+3 x \right )^{5} \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(346\) |
Input:
int((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/439040*(21498268812075*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x +3)^(1/2))*x^6+84559857327495*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x ^2-x+3)^(1/2))*x^5+138544399011150*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/( -10*x^2-x+3)^(1/2))*x^4+8768959639650*x^5*(-10*x^2-x+3)^(1/2)+121027291090 200*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+2903653 0544490*x^4*(-10*x^2-x+3)^(1/2)+59452002640800*7^(1/2)*arctan(1/14*(37*x+2 0)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+38452412617500*x^3*(-10*x^2-x+3)^(1/2) +15570762596400*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)) *x+25455981805688*x^2*(-10*x^2-x+3)^(1/2)+1698628646880*7^(1/2)*arctan(1/1 4*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+8424381751632*x*(-10*x^2-x+3)^(1/ 2)+1114940919232*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)^5/(-10*x^2-x+3 )^(1/2)/(3+5*x)^(1/2)
Time = 0.11 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{3/2}} \, dx=\frac {17694048405 \, \sqrt {7} {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (626354259975 \, x^{5} + 2074037896035 \, x^{4} + 2746600901250 \, x^{3} + 1818284414692 \, x^{2} + 601741553688 \, x + 79638637088\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{439040 \, {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} \] Input:
integrate((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^(3/2),x, algorithm="fricas")
Output:
1/439040*(17694048405*sqrt(7)*(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt (-2*x + 1)/(10*x^2 + x - 3)) - 14*(626354259975*x^5 + 2074037896035*x^4 + 2746600901250*x^3 + 1818284414692*x^2 + 601741553688*x + 79638637088)*sqrt (5*x + 3)*sqrt(-2*x + 1))/(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 336 0*x^2 + 880*x + 96)
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{3/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{6} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((1-2*x)**(5/2)/(2+3*x)**6/(3+5*x)**(3/2),x)
Output:
Integral((1 - 2*x)**(5/2)/((3*x + 2)**6*(5*x + 3)**(3/2)), x)
Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (157) = 314\).
Time = 0.14 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.97 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{3/2}} \, dx=-\frac {3538809681}{87808} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {4639661185 \, x}{28224 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {4844248403}{56448 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {343}{135 \, {\left (243 \, \sqrt {-10 \, x^{2} - x + 3} x^{5} + 810 \, \sqrt {-10 \, x^{2} - x + 3} x^{4} + 1080 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 720 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 240 \, \sqrt {-10 \, x^{2} - x + 3} x + 32 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {5341}{360 \, {\left (81 \, \sqrt {-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt {-10 \, x^{2} - x + 3} x + 16 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {242879}{2160 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {315689}{320 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {33314567}{2688 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \] Input:
integrate((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^(3/2),x, algorithm="maxima")
Output:
-3538809681/87808*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2) ) + 4639661185/28224*x/sqrt(-10*x^2 - x + 3) - 4844248403/56448/sqrt(-10*x ^2 - x + 3) + 343/135/(243*sqrt(-10*x^2 - x + 3)*x^5 + 810*sqrt(-10*x^2 - x + 3)*x^4 + 1080*sqrt(-10*x^2 - x + 3)*x^3 + 720*sqrt(-10*x^2 - x + 3)*x^ 2 + 240*sqrt(-10*x^2 - x + 3)*x + 32*sqrt(-10*x^2 - x + 3)) + 5341/360/(81 *sqrt(-10*x^2 - x + 3)*x^4 + 216*sqrt(-10*x^2 - x + 3)*x^3 + 216*sqrt(-10* x^2 - x + 3)*x^2 + 96*sqrt(-10*x^2 - x + 3)*x + 16*sqrt(-10*x^2 - x + 3)) + 242879/2160/(27*sqrt(-10*x^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqrt(-10*x^2 - x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) + 315689/320/(9*s qrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) + 33314567/2688/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3 ))
Leaf count of result is larger than twice the leaf count of optimal. 485 vs. \(2 (157) = 314\).
Time = 0.45 (sec) , antiderivative size = 485, normalized size of antiderivative = 2.40 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{3/2}} \, dx=-\frac {3538809681}{878080} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {3025}{2} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {121 \, \sqrt {10} {\left (34728039 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 30879615760 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 10961021460480 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 1791349451136000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {112299870108160000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {449199480432640000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{3136 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \] Input:
integrate((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^(3/2),x, algorithm="giac")
Output:
-3538809681/878080*sqrt(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)*sq rt(-10*x + 5) - sqrt(22)))) - 3025/2*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))) - 121/3136*sqrt(10)*(34728039*((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)))^9 + 30879615760*((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)))^7 + 10961021460480*((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)))^5 + 1791349451136000*((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 + 112299870108160000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 449199480432640000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(2 2)))/(((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)^5
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^6\,{\left (5\,x+3\right )}^{3/2}} \,d x \] Input:
int((1 - 2*x)^(5/2)/((3*x + 2)^6*(5*x + 3)^(3/2)),x)
Output:
int((1 - 2*x)^(5/2)/((3*x + 2)^6*(5*x + 3)^(3/2)), x)
Time = 0.31 (sec) , antiderivative size = 599, normalized size of antiderivative = 2.97 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{3/2}} \, dx =\text {Too large to display} \] Input:
int((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^(3/2),x)
Output:
( - 4299653762415*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin ((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**5 - 14332179208050*s qrt(5*x + 3)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)* sqrt(5))/sqrt(11))/2))/sqrt(2))*x**4 - 19109572277400*sqrt(5*x + 3)*sqrt(7 )*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/ 2))/sqrt(2))*x**3 - 12739714851600*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 - 4246571617200*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((s qrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x - 566209548960*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5)) /sqrt(11))/2))/sqrt(2)) + 4299653762415*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(3 3) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x **5 + 14332179208050*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(a sin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**4 + 1910957227740 0*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**3 + 12739714851600*sqrt(5*x + 3)*sqr t(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11 ))/2))/sqrt(2))*x**2 + 4246571617200*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 566209548960*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((...