\(\int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx\) [2784]

Optimal result

Integrand size = 28, antiderivative size = 280 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=-\frac {54281308 \sqrt {1-2 x} \sqrt {3+5 x}}{35756721 (2+3 x)^{5/2}}+\frac {1876198516 \sqrt {1-2 x} \sqrt {3+5 x}}{750891141 (2+3 x)^{3/2}}+\frac {129922578224 \sqrt {1-2 x} \sqrt {3+5 x}}{5256237987 \sqrt {2+3 x}}-\frac {2622980 \sqrt {1-2 x} (3+5 x)^{3/2}}{1702701 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac {60080 \sqrt {1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}-\frac {129922578224 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{477839817 \sqrt {33}}-\frac {3894280616 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{477839817 \sqrt {33}} \]

[Out]

-2/39*(1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(13/2)+370/1287*(1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(11/2)-129922578
224/15768713961*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-3894280616/15768713961*Elliptic
F(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2622980/1702701*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(7/
2)+60080/34749*(3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(9/2)-54281308/35756721*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)
^(5/2)[phone]/750891141*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)[phone]/5256237987*(1-2*x)^(1/2)*(3
+5*x)^(1/2)/(2+3*x)^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 157, 164, 114, 120} \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=-\frac {3894280616 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{477839817 \sqrt {33}}-\frac {129922578224 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{477839817 \sqrt {33}}+\frac {60080 \sqrt {1-2 x} (5 x+3)^{5/2}}{34749 (3 x+2)^{9/2}}+\frac {370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{1287 (3 x+2)^{11/2}}-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}-\frac {2622980 \sqrt {1-2 x} (5 x+3)^{3/2}}{1702701 (3 x+2)^{7/2}}+\frac {129922578224 \sqrt {1-2 x} \sqrt {5 x+3}}{5256237987 \sqrt {3 x+2}}+\frac {1876198516 \sqrt {1-2 x} \sqrt {5 x+3}}{750891141 (3 x+2)^{3/2}}-\frac {54281308 \sqrt {1-2 x} \sqrt {5 x+3}}{35756721 (3 x+2)^{5/2}} \]

[In]

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(15/2),x]

[Out]

(-54281308*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(35756721*(2 + 3*x)^(5/2)) + (1876198516*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/
(750891141*(2 + 3*x)^(3/2)) + (129922578224*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(5256237987*Sqrt[2 + 3*x]) - (2622980
*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(1702701*(2 + 3*x)^(7/2)) - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(39*(2 + 3*x)^
(13/2)) + (370*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(1287*(2 + 3*x)^(11/2)) + (60080*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)
)/(34749*(2 + 3*x)^(9/2)) - (129922578224*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(477839817*Sqrt[3
3]) - (3894280616*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(477839817*Sqrt[33])

Rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))), x] - Dist[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])

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> 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] &&  !LtQ[-(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])

Rule 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> 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] && Po
sQ[-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]))

Rule 155

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> 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] - Dist[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] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]

Rule 157

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> 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] + Dist[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]

Rule 164

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol]
:> Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(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] && SimplerQ[a + b*x, e + f*x] &&
SimplerQ[c + d*x, e + f*x]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {2}{39} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx \\ & = -\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}-\frac {4 \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2} \left (-1945+\frac {1675 x}{2}\right )}{(2+3 x)^{11/2}} \, dx}{1287} \\ & = -\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac {60080 \sqrt {1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}+\frac {8 \int \frac {\left (\frac {375445}{4}-\frac {210225 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx}{34749} \\ & = -\frac {2622980 \sqrt {1-2 x} (3+5 x)^{3/2}}{1702701 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac {60080 \sqrt {1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}+\frac {16 \int \frac {\left (\frac {38522835}{8}-5499150 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx}{5108103} \\ & = -\frac {54281308 \sqrt {1-2 x} \sqrt {3+5 x}}{35756721 (2+3 x)^{5/2}}-\frac {2622980 \sqrt {1-2 x} (3+5 x)^{3/2}}{1702701 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac {60080 \sqrt {1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}+\frac {32 \int \frac {\frac {1283806245}{16}-\frac {796081425 x}{8}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{536350815} \\ & = -\frac {54281308 \sqrt {1-2 x} \sqrt {3+5 x}}{35756721 (2+3 x)^{5/2}}+\frac {1876198516 \sqrt {1-2 x} \sqrt {3+5 x}}{750891141 (2+3 x)^{3/2}}-\frac {2622980 \sqrt {1-2 x} (3+5 x)^{3/2}}{1702701 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac {60080 \sqrt {1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}+\frac {64 \int \frac {\frac {14437282485}{4}-\frac {35178722175 x}{16}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{11263367115} \\ & = -\frac {54281308 \sqrt {1-2 x} \sqrt {3+5 x}}{35756721 (2+3 x)^{5/2}}+\frac {1876198516 \sqrt {1-2 x} \sqrt {3+5 x}}{750891141 (2+3 x)^{3/2}}+\frac {129922578224 \sqrt {1-2 x} \sqrt {3+5 x}}{5256237987 \sqrt {2+3 x}}-\frac {2622980 \sqrt {1-2 x} (3+5 x)^{3/2}}{1702701 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac {60080 \sqrt {1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}+\frac {128 \int \frac {\frac {1541948542725}{32}+\frac {609012085425 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{78843569805} \\ & = -\frac {54281308 \sqrt {1-2 x} \sqrt {3+5 x}}{35756721 (2+3 x)^{5/2}}+\frac {1876198516 \sqrt {1-2 x} \sqrt {3+5 x}}{750891141 (2+3 x)^{3/2}}+\frac {129922578224 \sqrt {1-2 x} \sqrt {3+5 x}}{5256237987 \sqrt {2+3 x}}-\frac {2622980 \sqrt {1-2 x} (3+5 x)^{3/2}}{1702701 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac {60080 \sqrt {1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}+\frac {1947140308 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{477839817}+\frac {129922578224 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{5256237987} \\ & = -\frac {54281308 \sqrt {1-2 x} \sqrt {3+5 x}}{35756721 (2+3 x)^{5/2}}+\frac {1876198516 \sqrt {1-2 x} \sqrt {3+5 x}}{750891141 (2+3 x)^{3/2}}+\frac {129922578224 \sqrt {1-2 x} \sqrt {3+5 x}}{5256237987 \sqrt {2+3 x}}-\frac {2622980 \sqrt {1-2 x} (3+5 x)^{3/2}}{1702701 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac {60080 \sqrt {1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}-\frac {129922578224 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{477839817 \sqrt {33}}-\frac {3894280616 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{477839817 \sqrt {33}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.94 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.41 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\frac {8 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (4382625184685+39086872650957 x+145238558453649 x^2+287874442427697 x^3+321056742490902 x^4+191022825888450 x^5+47356779762648 x^6\right )}{4 (2+3 x)^{13/2}}+i \sqrt {33} \left (16240322278 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-16727107355 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{15768713961} \]

[In]

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(15/2),x]

[Out]

(8*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(4382625184685 + 39086872650957*x + 145238558453649*x^2 + 287874442427697*x
^3 + 321056742490902*x^4 + 191022825888450*x^5 + 47356779762648*x^6))/(4*(2 + 3*x)^(13/2)) + I*Sqrt[33]*(16240
322278*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 16727107355*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))
)/15768713961

Maple [A] (verified)

Time = 1.31 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.25

method	result	size
elliptic	\(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6908733 \left (\frac {2}{3}+x \right )^{7}}+\frac {518 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{938223 \left (\frac {2}{3}+x \right )^{6}}-\frac {478462 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{75996063 \left (\frac {2}{3}+x \right )^{5}}+\frac {17427370 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1241269029 \left (\frac {2}{3}+x \right )^{4}}+\frac {13028276 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2896294401 \left (\frac {2}{3}+x \right )^{3}}+\frac {1876198516 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6758020269 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {1299225782240}{5256237987} x^{2}-\frac {129922578224}{5256237987} x +\frac {129922578224}{1752079329}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {164474511224 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{110380997727 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {259845156448 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{110380997727 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\)	\(350\)
default	\(\frac {2 \left (47356779762648 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{6} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-45989031044484 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{6} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}[phone] \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-183956124177936 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}[phone] \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-306593540296560 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}[phone] \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-272527591374720 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}[phone] \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-136263795687360 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+1420703392879440 x^{8}[phone] \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-36337012183296 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+5872755115941444 x^{7}[phone] \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-4037445798144 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+9778559734528578 x^{6}+7880198067307566 x^{5}+2331269398474443 x^{4}-982548126959616 x^{3}-1058407652589420 x^{2}-338633978304558 x -39443626662165\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{15768713961 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {13}{2}}}\)	\(694\)

[In]

int((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(15/2),x,method=_RETURNVERBOSE)

[Out]

-1/(10*x^2+x-3)/(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-(-1+2*x)*(3+5*x)*(2+3*x))^(1/2)*(-98/6908733*(-30*
x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^7+518/938223*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^6-478462/75996063*(-30*x^3-2
3*x^2+7*x+6)^(1/2)/(2/3+x)^5+17427370/1241269029*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^4+13028276/2896294401*(-
30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3+1876198516/6758020269*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^2+129922578224
/15768713961*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/2)[phone]/110380997727*(10+15*x)^(1/2)*(21-42*
x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))[phone]/11
0380997727*(10+15*x)^(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*(-7/6*EllipticE((10+15
*x)^(1/2),1/35*70^(1/2))+1/2*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.07 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\frac {2 \, {\left (135 \, {\left (47356779762648 \, x^{6} + 191022825888450 \, x^{5} + 321056742490902 \, x^{4} + 287874442427697 \, x^{3} + 145238558453649 \, x^{2} + 39086872650957 \, x + 4382625184685\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 1103283426482 \, \sqrt {-30} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 2923258010040 \, \sqrt {-30} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{709592128245 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(15/2),x, algorithm="fricas")

[Out]

2/709592128245*(135*(47356779762648*x^6 + 191022825888450*x^5 + 321056742490902*x^4 + 287874442427697*x^3 + 14
5238558453649*x^2 + 39086872650957*x + 4382625184685)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 11032834264
82*sqrt(-30)*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*weierstrassP
Inverse(1159/675, 38998/91125, x + 23/90) + 2923258010040*sqrt(-30)*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*
x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675
, 38998/91125, x + 23/90)))/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 12
8)

Sympy [F(-1)]

Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\text {Timed out} \]

[In]

integrate((1-2*x)**(5/2)*(3+5*x)**(5/2)/(2+3*x)**(15/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {15}{2}}} \,d x } \]

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(15/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(15/2), x)

Giac [F]

\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {15}{2}}} \,d x } \]

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(15/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(15/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{15/2}} \,d x \]

[In]

int(((1 - 2*x)^(5/2)*(5*x + 3)^(5/2))/(3*x + 2)^(15/2),x)

[Out]

int(((1 - 2*x)^(5/2)*(5*x + 3)^(5/2))/(3*x + 2)^(15/2), x)