Integrand size = 28, antiderivative size = 216 \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx=-\frac {188443}{700} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {2024}{35} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {193}{14} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {8 (2+3 x)^{7/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {6547351 E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{300 \sqrt {35}}+\frac {188443 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{300 \sqrt {35}} \] Output:
-188443/700*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)-2024/35*(1-2*x)^(1/2 )*(2+3*x)^(3/2)*(3+5*x)^(1/2)-193/14*(1-2*x)^(1/2)*(2+3*x)^(5/2)*(3+5*x)^( 1/2)-8*(2+3*x)^(7/2)*(3+5*x)^(1/2)/(1-2*x)^(1/2)+1/3*(2+3*x)^(7/2)*(3+5*x) ^(3/2)/(1-2*x)^(3/2)-6547351/10500*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1 /35*1155^(1/2))*35^(1/2)+188443/10500*EllipticF(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 = 8.75 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.58 \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx=-\frac {55 \sqrt {2+3 x} \sqrt {3+5 x} \left (1041609-2751916 x+567906 x^2+198180 x^3+40500 x^4\right )+72020861 i \sqrt {33-66 x} (-1+2 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-74187260 i \sqrt {33-66 x} (-1+2 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{115500 (1-2 x)^{3/2}} \] Input:
Integrate[((2 + 3*x)^(7/2)*(3 + 5*x)^(3/2))/(1 - 2*x)^(5/2),x]
Output:
-1/115500*(55*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(1041609 - 2751916*x + 567906*x^ 2 + 198180*x^3 + 40500*x^4) + (72020861*I)*Sqrt[33 - 66*x]*(-1 + 2*x)*Elli pticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (74187260*I)*Sqrt[33 - 66*x]*(-1 + 2*x)*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/(1 - 2*x)^(3/2)
Time = 0.30 (sec) , antiderivative size = 245, 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 = {108, 27, 167, 27, 171, 27, 171, 25, 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.
| \(\displaystyle \int \frac {(3 x+2)^{7/2} (5 x+3)^{3/2}}{(1-2 x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {(3 x+2)^{7/2} (5 x+3)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {3 (3 x+2)^{5/2} \sqrt {5 x+3} (50 x+31)}{2 (1-2 x)^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+2)^{7/2} (5 x+3)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {1}{2} \int \frac {(3 x+2)^{5/2} \sqrt {5 x+3} (50 x+31)}{(1-2 x)^{3/2}}dx\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{2} \left (-\frac {1}{11} \int -\frac {5 (3 x+2)^{3/2} \sqrt {5 x+3} (1341 x+838)}{\sqrt {1-2 x}}dx-\frac {112 (5 x+3)^{3/2} (3 x+2)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{3/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {5}{11} \int \frac {(3 x+2)^{3/2} \sqrt {5 x+3} (1341 x+838)}{\sqrt {1-2 x}}dx-\frac {112 (3 x+2)^{5/2} (5 x+3)^{3/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{3/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{2} \left (\frac {5}{11} \left (-\frac {1}{35} \int -\frac {\sqrt {3 x+2} \sqrt {5 x+3} (280578 x+177665)}{2 \sqrt {1-2 x}}dx-\frac {1341}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )-\frac {112 (3 x+2)^{5/2} (5 x+3)^{3/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{3/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {5}{11} \left (\frac {1}{70} \int \frac {\sqrt {3 x+2} \sqrt {5 x+3} (280578 x+177665)}{\sqrt {1-2 x}}dx-\frac {1341}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )-\frac {112 (3 x+2)^{5/2} (5 x+3)^{3/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{3/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{2} \left (\frac {5}{11} \left (\frac {1}{70} \left (-\frac {1}{25} \int -\frac {\sqrt {5 x+3} (19497591 x+12671053)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {280578}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {1341}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )-\frac {112 (3 x+2)^{5/2} (5 x+3)^{3/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{3/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {5}{11} \left (\frac {1}{70} \left (\frac {1}{25} \int \frac {\sqrt {5 x+3} (19497591 x+12671053)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {280578}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {1341}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )-\frac {112 (3 x+2)^{5/2} (5 x+3)^{3/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{3/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{2} \left (\frac {5}{11} \left (\frac {1}{70} \left (\frac {1}{25} \left (-\frac {1}{9} \int -\frac {99 (13094702 x+8290101)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-2166399 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {280578}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {1341}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )-\frac {112 (3 x+2)^{5/2} (5 x+3)^{3/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{3/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {5}{11} \left (\frac {1}{70} \left (\frac {1}{25} \left (\frac {11}{2} \int \frac {13094702 x+8290101}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-2166399 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {280578}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {1341}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )-\frac {112 (3 x+2)^{5/2} (5 x+3)^{3/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{3/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{2} \left (\frac {5}{11} \left (\frac {1}{70} \left (\frac {1}{25} \left (\frac {11}{2} \left (\frac {2166399}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {13094702}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-2166399 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {280578}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {1341}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )-\frac {112 (3 x+2)^{5/2} (5 x+3)^{3/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{3/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{2} \left (\frac {5}{11} \left (\frac {1}{70} \left (\frac {1}{25} \left (\frac {11}{2} \left (\frac {2166399}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {13094702}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-2166399 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {280578}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {1341}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )-\frac {112 (3 x+2)^{5/2} (5 x+3)^{3/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{3/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{2} \left (\frac {5}{11} \left (\frac {1}{70} \left (\frac {1}{25} \left (\frac {11}{2} \left (-\frac {1444266}{5} \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {13094702}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-2166399 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {280578}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {1341}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )-\frac {112 (3 x+2)^{5/2} (5 x+3)^{3/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{3/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
Input:
Int[((2 + 3*x)^(7/2)*(3 + 5*x)^(3/2))/(1 - 2*x)^(5/2),x]
Output:
((2 + 3*x)^(7/2)*(3 + 5*x)^(3/2))/(3*(1 - 2*x)^(3/2)) + ((-112*(2 + 3*x)^( 5/2)*(3 + 5*x)^(3/2))/(11*Sqrt[1 - 2*x]) + (5*((-1341*Sqrt[1 - 2*x]*(2 + 3 *x)^(3/2)*(3 + 5*x)^(3/2))/35 + ((-280578*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/25 + (-2166399*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] + (1 1*((-13094702*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33] )/5 - (1444266*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33 ])/5))/2)/25)/70))/11)/2
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 + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 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[(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] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
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 = 0.65 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.07
| method | result | size |
| default | \(-\frac {\left (12998394 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-26189404 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+6075000 x^{6}-6499197 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+13094702 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+37422000 x^{5}+125270100 x^{4}-292994460 x^{3}-332548330 x^{2}+32790750 x +62496540\right ) \sqrt {3+5 x}\, \sqrt {2+3 x}}{21000 \left (1-2 x \right )^{\frac {3}{2}} \left (15 x^{2}+19 x +6\right )}\) | \(232\) |
| elliptic | \(\frac {\sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {1989 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{70}-\frac {265487 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2800}+\frac {2763367 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{9800 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {6547351 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{14700 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {135 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{28}+\frac {3773 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{192 \left (x -\frac {1}{2}\right )^{2}}+\frac {-\frac {57575}{8} x^{2}-\frac {218785}{24} x -\frac {11515}{4}}{\sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\left (15 x^{2}+19 x +6\right ) \sqrt {1-2 x}}\) | \(298\) |
Input:
int((2+3*x)^(7/2)*(3+5*x)^(3/2)/(1-2*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/21000*(12998394*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x*( 2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)-26189404*2^(1/2)*EllipticE(1/7*( 28+42*x)^(1/2),1/2*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)+ 6075000*x^6-6499197*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*Ell ipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))+13094702*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))+374 22000*x^5+125270100*x^4-292994460*x^3-332548330*x^2+32790750*x+62496540)*( 3+5*x)^(1/2)*(2+3*x)^(1/2)/(1-2*x)^(3/2)/(15*x^2+19*x+6)
Time = 0.09 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.48 \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx=-\frac {225 \, {\left (40500 \, x^{4} + 198180 \, x^{3} + 567906 \, x^{2} - 2751916 \, x + 1041609\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 111232736 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 294630795 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{472500 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \] Input:
integrate((2+3*x)^(7/2)*(3+5*x)^(3/2)/(1-2*x)^(5/2),x, algorithm="fricas")
Output:
-1/472500*(225*(40500*x^4 + 198180*x^3 + 567906*x^2 - 2751916*x + 1041609) *sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 111232736*sqrt(-30)*(4*x^2 - 4*x + 1)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) - 29463079 5*sqrt(-30)*(4*x^2 - 4*x + 1)*weierstrassZeta(1159/675, 38998/91125, weier strassPInverse(1159/675, 38998/91125, x + 23/90)))/(4*x^2 - 4*x + 1)
Timed out. \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((2+3*x)**(7/2)*(3+5*x)**(3/2)/(1-2*x)**(5/2),x)
Output:
Timed out
\[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((2+3*x)^(7/2)*(3+5*x)^(3/2)/(1-2*x)^(5/2),x, algorithm="maxima")
Output:
integrate((5*x + 3)^(3/2)*(3*x + 2)^(7/2)/(-2*x + 1)^(5/2), x)
\[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((2+3*x)^(7/2)*(3+5*x)^(3/2)/(1-2*x)^(5/2),x, algorithm="giac")
Output:
integrate((5*x + 3)^(3/2)*(3*x + 2)^(7/2)/(-2*x + 1)^(5/2), x)
Timed out. \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{5/2}} \,d x \] Input:
int(((3*x + 2)^(7/2)*(5*x + 3)^(3/2))/(1 - 2*x)^(5/2),x)
Output:
int(((3*x + 2)^(7/2)*(5*x + 3)^(3/2))/(1 - 2*x)^(5/2), x)
\[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx=\frac {-10732500 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{4}-52517700 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}-150495090 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}-658780672 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x +477015288 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}-90824846700 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{120 x^{5}-28 x^{4}-90 x^{3}+27 x^{2}+17 x -6}d x \right ) x^{2}+90824846700 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{120 x^{5}-28 x^{4}-90 x^{3}+27 x^{2}+17 x -6}d x \right ) x -22706211675 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{120 x^{5}-28 x^{4}-90 x^{3}+27 x^{2}+17 x -6}d x \right )+36340401552 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{120 x^{5}-28 x^{4}-90 x^{3}+27 x^{2}+17 x -6}d x \right ) x^{2}-36340401552 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{120 x^{5}-28 x^{4}-90 x^{3}+27 x^{2}+17 x -6}d x \right ) x +9085100388 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{120 x^{5}-28 x^{4}-90 x^{3}+27 x^{2}+17 x -6}d x \right )}{2226000 x^{2}-2226000 x +556500} \] Input:
int((2+3*x)^(7/2)*(3+5*x)^(3/2)/(1-2*x)^(5/2),x)
Output:
( - 10732500*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**4 - 52517700* sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 - 150495090*sqrt(3*x + 2 )*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 - 658780672*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x + 477015288*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) - 90824846700*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 )/(120*x**5 - 28*x**4 - 90*x**3 + 27*x**2 + 17*x - 6),x)*x**2 + 9082484670 0*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(120*x**5 - 28*x **4 - 90*x**3 + 27*x**2 + 17*x - 6),x)*x - 22706211675*int((sqrt(3*x + 2)* sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(120*x**5 - 28*x**4 - 90*x**3 + 27*x* *2 + 17*x - 6),x) + 36340401552*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2 *x + 1))/(120*x**5 - 28*x**4 - 90*x**3 + 27*x**2 + 17*x - 6),x)*x**2 - 363 40401552*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(120*x**5 - 28 *x**4 - 90*x**3 + 27*x**2 + 17*x - 6),x)*x + 9085100388*int((sqrt(3*x + 2) *sqrt(5*x + 3)*sqrt( - 2*x + 1))/(120*x**5 - 28*x**4 - 90*x**3 + 27*x**2 + 17*x - 6),x))/(556500*(4*x**2 - 4*x + 1))