\(\int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx\) [2776]

Optimal result

Integrand size = 28, antiderivative size = 249 \[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=-\frac {486785077 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{547296750}-\frac {3872003 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{30405375}-\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {8120161139 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{124385625 \sqrt {33}}-\frac {486785077 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{248771250 \sqrt {33}} \]

[Out]

-8120161139/4104725625*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-486785077/8209451250*Ell
ipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+326/10725*(1-2*x)^(3/2)*(3+5*x)^(7/2)*(2+3*x)^(1/2
)+2/65*(1-2*x)^(5/2)*(3+5*x)^(7/2)*(2+3*x)^(1/2)-3872003/30405375*(3+5*x)^(3/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)-12
1031/30405375*(3+5*x)^(5/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)+2314/111375*(3+5*x)^(7/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)-
486785077/547296750*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {103, 159, 164, 114, 120} \[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=-\frac {486785077 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{248771250 \sqrt {33}}-\frac {8120161139 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{124385625 \sqrt {33}}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{7/2}+\frac {326 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{7/2}}{10725}+\frac {2314 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}}{111375}-\frac {121031 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{30405375}-\frac {3872003 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{30405375}-\frac {486785077 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{547296750} \]

[In]

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

[Out]

(-486785077*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/547296750 - (3872003*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5
*x)^(3/2))/30405375 - (121031*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/30405375 + (2314*Sqrt[1 - 2*x]*Sqrt
[2 + 3*x]*(3 + 5*x)^(7/2))/111375 + (326*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))/10725 + (2*(1 - 2*x)^(
5/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))/65 - (8120161139*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(12438
5625*Sqrt[33]) - (486785077*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(248771250*Sqrt[33])

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b
*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Dist[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] && (Integ
ersQ[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 159

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Dist[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] /; FreeQ[{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]

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}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {2}{65} \int \frac {\left (-\frac {111}{2}-\frac {163 x}{2}\right ) (1-2 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx \\ & = \frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {4 \int \frac {\left (-\frac {5653}{2}-\frac {15041 x}{4}\right ) \sqrt {1-2 x} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx}{10725} \\ & = \frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {8 \int \frac {\left (-\frac {518563}{8}-\frac {121031 x}{8}\right ) (3+5 x)^{5/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1447875} \\ & = -\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}+\frac {8 \int \frac {(3+5 x)^{3/2} \left (\frac {71027395}{16}+\frac {58080045 x}{8}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{30405375} \\ & = -\frac {3872003 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{30405375}-\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {8 \int \frac {\left (-\frac {2382196995}{8}-\frac {7301776155 x}{16}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{456080625} \\ & = -\frac {486785077 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{547296750}-\frac {3872003 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{30405375}-\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}+\frac {8 \int \frac {\frac {308389708545}{32}+\frac {121802417085 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4104725625} \\ & = -\frac {486785077 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{547296750}-\frac {3872003 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{30405375}-\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}+\frac {486785077 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{497542500}+\frac {8120161139 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1368241875} \\ & = -\frac {486785077 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{547296750}-\frac {3872003 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{30405375}-\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {8120161139 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{124385625 \sqrt {33}}-\frac {486785077 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{248771250 \sqrt {33}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.25 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.45 \[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=\frac {15 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (495379991+2923422930 x-1730459250 x^2-7942630500 x^3+2577015000 x^4+8419950000 x^5\right )[phone] i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-16727107355 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{8209451250} \]

[In]

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

[Out]

(15*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(495379991 + 2923422930*x - 1730459250*x^2 - 7942630500*x^3 + 25
77015000*x^4 + 8419950000*x^5) + (16240322278*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (16727
107355*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/8209451250

Maple [A] (verified)

Time = 1.36 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.66

method	result	size
default	\(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (-3788977500000 x^{8}-4064539500000 x^{7}[phone] \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-16240322278 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )[phone] x^{6}[phone] x^{5}-1320576729750 x^{4}-2128036873050 x^{3}-19688021745 x^{2}[phone] x [phone]\right )}{8209451250 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(165\)
risch	\(-\frac {\left (8419950000 x^{5}[phone] x^{4}-7942630500 x^{3}-1730459250 x^{2}[phone] x [phone]\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{547296750 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {20559313903 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{60202642500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {8120161139 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{15050660625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(267\)
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {32482477 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6081075}+\frac {495379991 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{547296750}+\frac {20559313903 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{57466158750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {16240322278 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{28733079375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {59161 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{18711}+\frac {2020 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{429}-\frac {168098 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{11583}+\frac {200 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{5}}{13}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(294\)

[In]

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

[Out]

-1/8209451250*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(-3788977500000*x^8-4064539500000*x^7+15771272649*5^(1
/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))-16240322278*5^
(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*EllipticE((10+15*x)^(1/2),1/35*70^(1/2))[phone]
0*x^6+4547296260000*x^5-1320576729750*x^4-2128036873050*x^3-19688021745*x^2+315122962755*x+44584199190)/(30*x^
3+23*x^2-7*x-6)

Fricas [C] (verification not implemented)

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

Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.30 \[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=\frac {1}{547296750} \, {\left (8419950000 \, x^{5} + 2577015000 \, x^{4} - 7942630500 \, x^{3} - 1730459250 \, x^{2} + 2923422930 \, x + 495379991\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {551641713241}{738850612500} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {8120161139}{4104725625} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]

[In]

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

[Out]

1/547296750*(8419950000*x^5 + 2577015000*x^4 - 7942630500*x^3 - 1730459250*x^2 + 2923422930*x + 495379991)*sqr
t(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 551641713241/738850612500*sqrt(-30)*weierstrassPInverse(1159/675, 38
998/91125, x + 23/90) + 8120161139/4104725625*sqrt(-30)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInv
erse(1159/675, 38998/91125, x + 23/90))

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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