Integrand size = 26, antiderivative size = 180 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 \sqrt {3+5 x}} \, dx=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {2023 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}+\frac {67187 \sqrt {1-2 x} \sqrt {3+5 x}}{2160 (2+3 x)^3}+\frac {2347559 \sqrt {1-2 x} \sqrt {3+5 x}}{12096 (2+3 x)^2}+\frac {245529161 \sqrt {1-2 x} \sqrt {3+5 x}}{169344 (2+3 x)}-\frac {104040277 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{6272 \sqrt {7}} \] Output:
7/15*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^5+2023/360*(1-2*x)^(1/2)*(3+5*x)^ (1/2)/(2+3*x)^4+67187/2160*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+2347559/1 2096*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+245529161*(1-2*x)^(1/2)*(3+5*x) ^(1/2)/(338688+508032*x)-104040277/43904*7^(1/2)*arctan(1/7*(1-2*x)^(1/2)* 7^(1/2)/(3+5*x)^(1/2))
Time = 0.31 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.47 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 \sqrt {3+5 x}} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (2341358496+13788819736 x+30475811404 x^2+29956486710 x^3+11048812245 x^4\right )}{(2+3 x)^5}-1560604155 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{658560} \] Input:
Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^6*Sqrt[3 + 5*x]),x]
Output:
((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2341358496 + 13788819736*x + 30475811404* x^2 + 29956486710*x^3 + 11048812245*x^4))/(2 + 3*x)^5 - 1560604155*Sqrt[7] *ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/658560
Time = 0.33 (sec) , antiderivative size = 200, 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 = {109, 27, 166, 27, 168, 27, 168, 27, 168, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^6 \sqrt {5 x+3}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{15} \int \frac {(421-380 x) \sqrt {1-2 x}}{2 (3 x+2)^5 \sqrt {5 x+3}}dx+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \int \frac {(421-380 x) \sqrt {1-2 x}}{(3 x+2)^5 \sqrt {5 x+3}}dx+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{30} \left (\frac {2023 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}-\frac {1}{12} \int -\frac {79903-115300 x}{2 \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx\right )+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \left (\frac {1}{24} \int \frac {79903-115300 x}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx+\frac {2023 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{30} \left (\frac {1}{24} \left (\frac {1}{21} \int \frac {35 (424189-537496 x)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {67187 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {2023 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \left (\frac {1}{24} \left (\frac {5}{6} \int \frac {424189-537496 x}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {67187 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {2023 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{30} \left (\frac {1}{24} \left (\frac {5}{6} \left (\frac {1}{14} \int \frac {50542267-46951180 x}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {2347559 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {67187 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {2023 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \left (\frac {1}{24} \left (\frac {5}{6} \left (\frac {1}{28} \int \frac {50542267-46951180 x}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {2347559 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {67187 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {2023 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{30} \left (\frac {1}{24} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {2809087479}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {245529161 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {2347559 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {67187 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {2023 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \left (\frac {1}{24} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {2809087479}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {245529161 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {2347559 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {67187 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {2023 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{30} \left (\frac {1}{24} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {2809087479}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {245529161 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {2347559 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {67187 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {2023 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{30} \left (\frac {1}{24} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {245529161 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {2809087479 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )+\frac {2347559 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {67187 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {2023 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}\) |
Input:
Int[(1 - 2*x)^(5/2)/((2 + 3*x)^6*Sqrt[3 + 5*x]),x]
Output:
(7*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(15*(2 + 3*x)^5) + ((2023*Sqrt[1 - 2*x]* Sqrt[3 + 5*x])/(12*(2 + 3*x)^4) + ((67187*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3* (2 + 3*x)^3) + (5*((2347559*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + ((245529161*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (2809087479*Arc Tan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/28))/6)/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_)^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])
Time = 0.24 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.74
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (11048812245 x^{4}+29956486710 x^{3}+30475811404 x^{2}+13788819736 x +2341358496\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{94080 \left (2+3 x \right )^{5} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {104040277 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{87808 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(134\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (379226809665 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+1264089365550 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+1685452487400 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+154683371430 x^{4} \sqrt {-10 x^{2}-x +3}+1123634991600 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+419390813940 x^{3} \sqrt {-10 x^{2}-x +3}+374544997200 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +426661359656 x^{2} \sqrt {-10 x^{2}-x +3}+49939332960 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+193043476304 x \sqrt {-10 x^{2}-x +3}+32779018944 \sqrt {-10 x^{2}-x +3}\right )}{1317120 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{5}}\) | \(298\) |
Input:
int((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/94080*(-1+2*x)*(3+5*x)^(1/2)*(11048812245*x^4+29956486710*x^3+304758114 04*x^2+13788819736*x+2341358496)/(2+3*x)^5/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2 *x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)+104040277/87808*7^(1/2)*arctan(9/14*(20/3 +37/3*x)*7^(1/2)/(-90*(2/3+x)^2+67+111*x)^(1/2))*((1-2*x)*(3+5*x))^(1/2)/( 1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 0.11 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.73 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 \sqrt {3+5 x}} \, dx=-\frac {1560604155 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (11048812245 \, x^{4} + 29956486710 \, x^{3} + 30475811404 \, x^{2} + 13788819736 \, x + 2341358496\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1317120 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \] Input:
integrate((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^(1/2),x, algorithm="fricas")
Output:
-1/1317120*(1560604155*sqrt(7)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 2 40*x + 32)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(1 0*x^2 + x - 3)) - 14*(11048812245*x^4 + 29956486710*x^3 + 30475811404*x^2 + 13788819736*x + 2341358496)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(243*x^5 + 810 *x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 \sqrt {3+5 x}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{6} \sqrt {5 x + 3}}\, dx \] Input:
integrate((1-2*x)**(5/2)/(2+3*x)**6/(3+5*x)**(1/2),x)
Output:
Integral((1 - 2*x)**(5/2)/((3*x + 2)**6*sqrt(5*x + 3)), x)
Time = 0.12 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 \sqrt {3+5 x}} \, dx=\frac {104040277}{87808} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {49 \, \sqrt {-10 \, x^{2} - x + 3}}{45 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {637 \, \sqrt {-10 \, x^{2} - x + 3}}{120 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {67187 \, \sqrt {-10 \, x^{2} - x + 3}}{2160 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {2347559 \, \sqrt {-10 \, x^{2} - x + 3}}{12096 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {245529161 \, \sqrt {-10 \, x^{2} - x + 3}}{169344 \, {\left (3 \, x + 2\right )}} \] Input:
integrate((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^(1/2),x, algorithm="maxima")
Output:
104040277/87808*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 49/45*sqrt(-10*x^2 - x + 3)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 24 0*x + 32) + 637/120*sqrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 + 96 *x + 16) + 67187/2160*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 2347559/12096*sqrt(-10*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 245529161/169344 *sqrt(-10*x^2 - x + 3)/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (141) = 282\).
Time = 0.37 (sec) , antiderivative size = 426, normalized size of antiderivative = 2.37 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 \sqrt {3+5 x}} \, dx=\frac {104040277}{878080} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1331 \, \sqrt {10} {\left (706299 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 493892560 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 156884295680 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 24022907776000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {1441374466560000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {5765497866240000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{9408 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \] Input:
integrate((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^(1/2),x, algorithm="giac")
Output:
104040277/878080*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)*sqrt (-10*x + 5) - sqrt(22)))) + 1331/9408*sqrt(10)*(706299*((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 + 493892560*((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 + 1568842 95680*((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 + 24022907776000*((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 + 1441374466560000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22 ))/sqrt(5*x + 3) - 5765497866240000*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*sqr t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^5
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 \sqrt {3+5 x}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^6\,\sqrt {5\,x+3}} \,d x \] Input:
int((1 - 2*x)^(5/2)/((3*x + 2)^6*(5*x + 3)^(1/2)),x)
Output:
int((1 - 2*x)^(5/2)/((3*x + 2)^6*(5*x + 3)^(1/2)), x)
Time = 0.27 (sec) , antiderivative size = 537, normalized size of antiderivative = 2.98 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 \sqrt {3+5 x}} \, dx =\text {Too large to display} \] Input:
int((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^(1/2),x)
Output:
(379226809665*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1) *sqrt(5))/sqrt(11))/2))/sqrt(2))*x**5 + 1264089365550*sqrt(7)*atan((sqrt(3 3) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x **4 + 1685452487400*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2* x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**3 + 1123634991600*sqrt(7)*atan(( sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt (2))*x**2 + 374544997200*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 49939332960*sqrt(7)*atan(( sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt (2)) - 379226809665*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2* x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**5 - 1264089365550*sqrt(7)*atan(( sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt (2))*x**4 - 1685452487400*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt ( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**3 - 1123634991600*sqrt(7)* atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2) )/sqrt(2))*x**2 - 374544997200*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin( (sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x - 49939332960*sqrt(7)* atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2) )/sqrt(2)) + 77341685715*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**4 + 20969540697 0*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 + 213330679828*sqrt(5*x + 3)*sqrt...