Integrand size = 28, antiderivative size = 218 \[ \int \frac {(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=-\frac {41741369 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{623700}-\frac {896659 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{62370}-\frac {45005 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{12474}-\frac {170}{297} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac {1}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {725140729 E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{133650 \sqrt {35}}+\frac {41741369 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{267300 \sqrt {35}} \] Output:
-41741369/623700*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)-896659/62370*(1 -2*x)^(1/2)*(2+3*x)^(3/2)*(3+5*x)^(1/2)-45005/12474*(1-2*x)^(1/2)*(2+3*x)^ (5/2)*(3+5*x)^(1/2)-170/297*(1-2*x)^(1/2)*(2+3*x)^(5/2)*(3+5*x)^(3/2)-1/11 *(1-2*x)^(1/2)*(2+3*x)^(5/2)*(3+5*x)^(5/2)-725140729/4677750*EllipticE(1/1 1*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)+41741369/9355500*Ellipt icF(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)
Result contains complex when optimal does not.
Time = 7.82 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.51 \[ \int \frac {(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\frac {1450281458 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-5 \left (3 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (75000749+86822370 x+81985950 x^2+48384000 x^3+12757500 x^4\right )+298781231 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{9355500} \] Input:
Integrate[((2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/Sqrt[1 - 2*x],x]
Output:
((1450281458*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 5*( 3*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(75000749 + 86822370*x + 81985 950*x^2 + 48384000*x^3 + 12757500*x^4) + (298781231*I)*Sqrt[33]*EllipticF[ I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/9355500
Time = 0.32 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {112, 27, 171, 25, 171, 27, 171, 27, 171, 27, 176, 123, 129}
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.
| \(\displaystyle \int \frac {(3 x+2)^{5/2} (5 x+3)^{5/2}}{\sqrt {1-2 x}} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {1}{11} \int \frac {5 (3 x+2)^{3/2} (5 x+3)^{3/2} (68 x+43)}{2 \sqrt {1-2 x}}dx-\frac {1}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{22} \int \frac {(3 x+2)^{3/2} (5 x+3)^{3/2} (68 x+43)}{\sqrt {1-2 x}}dx-\frac {1}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {5}{22} \left (-\frac {1}{45} \int -\frac {\sqrt {3 x+2} (5 x+3)^{3/2} (9477 x+6080)}{\sqrt {1-2 x}}dx-\frac {68}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {1}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5}{22} \left (\frac {1}{45} \int \frac {\sqrt {3 x+2} (5 x+3)^{3/2} (9477 x+6080)}{\sqrt {1-2 x}}dx-\frac {68}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {1}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {5}{22} \left (\frac {1}{45} \left (-\frac {1}{35} \int -\frac {(5 x+3)^{3/2} (1978098 x+1296619)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {9477}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {68}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {1}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{22} \left (\frac {1}{45} \left (\frac {1}{70} \int \frac {(5 x+3)^{3/2} (1978098 x+1296619)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {9477}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {68}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {1}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {5}{22} \left (\frac {1}{45} \left (\frac {1}{70} \left (-\frac {1}{15} \int -\frac {3 \sqrt {5 x+3} (43624697 x+28350726)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {659366}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {9477}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {68}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {1}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{22} \left (\frac {1}{45} \left (\frac {1}{70} \left (\frac {1}{5} \int \frac {\sqrt {5 x+3} (43624697 x+28350726)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {659366}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {9477}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {68}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {1}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {5}{22} \left (\frac {1}{45} \left (\frac {1}{70} \left (\frac {1}{5} \left (-\frac {1}{9} \int -\frac {2900562916 x+1836312083}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {43624697}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {659366}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {9477}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {68}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {1}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{22} \left (\frac {1}{45} \left (\frac {1}{70} \left (\frac {1}{5} \left (\frac {1}{18} \int \frac {2900562916 x+1836312083}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {43624697}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {659366}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {9477}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {68}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {1}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {5}{22} \left (\frac {1}{45} \left (\frac {1}{70} \left (\frac {1}{5} \left (\frac {1}{18} \left (\frac {479871667}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {2900562916}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {43624697}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {659366}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {9477}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {68}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {1}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {5}{22} \left (\frac {1}{45} \left (\frac {1}{70} \left (\frac {1}{5} \left (\frac {1}{18} \left (\frac {479871667}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2900562916}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {43624697}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {659366}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {9477}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {68}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {1}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {5}{22} \left (\frac {1}{45} \left (\frac {1}{70} \left (\frac {1}{5} \left (\frac {1}{18} \left (-\frac {87249394}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {2900562916}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {43624697}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {659366}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {9477}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {68}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {1}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\) |
Input:
Int[((2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/Sqrt[1 - 2*x],x]
Output:
-1/11*(Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2)) + (5*((-68*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/45 + ((-9477*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/35 + ((-659366*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x) ^(3/2))/5 + ((-43624697*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ((- 2900562916*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (87249394*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33]) /5)/18)/5)/70)/45))/22
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_) )^(p_), x_] :> Simp[(a + b*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (IntegersQ[2*m, 2*n, 2*p ] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] /Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !L tQ[-(b*c - a*d)/d, 0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d ), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)/b, 0])
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x _)]), x_] :> Simp[2*(Rt[-b/d, 2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[ Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ [(b*e - a*f)/b, 0] && PosQ[-b/d] && !(SimplerQ[c + d*x, a + b*x] && GtQ[(d *e - c*f)/d, 0] && GtQ[-d/b, 0]) && !(SimplerQ[c + d*x, a + b*x] && GtQ[(( -b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) && !(SimplerQ[e + f*x, a + b*x] && GtQ [((-d)*e + c*f)/f, 0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f /b]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f Int[Sqrt[e + f*x]/(Sqrt[a + b*x ]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
Time = 0.74 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.72
| method | result | size |
| default | \(-\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (11481750000 x^{7}+52348275000 x^{6}+1439615001 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-2900562916 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+104493240000 x^{5}+122253448500 x^{4}+101481939900 x^{3}+18760348110 x^{2}-31378183890 x -13500134820\right )}{18711000 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(158\) |
| risch | \(\frac {\left (12757500 x^{4}+48384000 x^{3}+81985950 x^{2}+86822370 x +75000749\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{623700 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {1836312083 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \operatorname {EllipticF}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{68607000 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {725140729 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \operatorname {EllipticF}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{17151750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(261\) |
| elliptic | \(\frac {\sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {964693 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6930}-\frac {75000749 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{623700}+\frac {1836312083 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{26195400 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {725140729 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{6548850 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {182191 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1386}-\frac {2560 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{33}-\frac {225 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{11}\right )}{\left (15 x^{2}+19 x +6\right ) \sqrt {1-2 x}}\) | \(290\) |
Input:
int((2+3*x)^(5/2)*(3+5*x)^(5/2)/(1-2*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/18711000*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(11481750000*x^7+523 48275000*x^6+1439615001*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2) *EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-2900562916*2^(1/2)*(2+3*x)^(1 /2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2 ))+104493240000*x^5+122253448500*x^4+101481939900*x^3+18760348110*x^2-3137 8183890*x-13500134820)/(30*x^3+23*x^2-7*x-6)
Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.32 \[ \int \frac {(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=-\frac {1}{623700} \, {\left (12757500 \, x^{4} + 48384000 \, x^{3} + 81985950 \, x^{2} + 86822370 \, x + 75000749\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {49277570201}{841995000} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {725140729}{4677750} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \] Input:
integrate((2+3*x)^(5/2)*(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="fricas")
Output:
-1/623700*(12757500*x^4 + 48384000*x^3 + 81985950*x^2 + 86822370*x + 75000 749)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 49277570201/841995000*sq rt(-30)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 725140729/ 4677750*sqrt(-30)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInver se(1159/675, 38998/91125, x + 23/90))
Timed out. \[ \int \frac {(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\text {Timed out} \] Input:
integrate((2+3*x)**(5/2)*(3+5*x)**(5/2)/(1-2*x)**(1/2),x)
Output:
Timed out
\[ \int \frac {(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}{\sqrt {-2 \, x + 1}} \,d x } \] Input:
integrate((2+3*x)^(5/2)*(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="maxima")
Output:
integrate((5*x + 3)^(5/2)*(3*x + 2)^(5/2)/sqrt(-2*x + 1), x)
\[ \int \frac {(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}{\sqrt {-2 \, x + 1}} \,d x } \] Input:
integrate((2+3*x)^(5/2)*(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="giac")
Output:
integrate((5*x + 3)^(5/2)*(3*x + 2)^(5/2)/sqrt(-2*x + 1), x)
Timed out. \[ \int \frac {(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\int \frac {{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}} \,d x \] Input:
int(((3*x + 2)^(5/2)*(5*x + 3)^(5/2))/(1 - 2*x)^(1/2),x)
Output:
int(((3*x + 2)^(5/2)*(5*x + 3)^(5/2))/(1 - 2*x)^(1/2), x)
\[ \int \frac {(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=-\frac {225 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{4}}{11}-\frac {2560 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}}{33}-\frac {182191 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{1386}-\frac {964693 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x}{6930}-\frac {23520731 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{106260}+\frac {725140729 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{30 x^{3}+23 x^{2}-7 x -6}d x \right )}{159390}-\frac {129350983 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{30 x^{3}+23 x^{2}-7 x -6}d x \right )}{70840} \] Input:
int((2+3*x)^(5/2)*(3+5*x)^(5/2)/(1-2*x)^(1/2),x)
Output:
( - 13041000*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**4 - 49459200* sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 - 83807860*sqrt(3*x + 2) *sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 - 88751756*sqrt(3*x + 2)*sqrt(5*x + 3 )*sqrt( - 2*x + 1)*x - 141124386*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) + 2900562916*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/( 30*x**3 + 23*x**2 - 7*x - 6),x) - 1164158847*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(30*x**3 + 23*x**2 - 7*x - 6),x))/637560