\(\int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx\) [2745]

Optimal result

Integrand size = 28, antiderivative size = 222 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx=\frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {176 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {12276 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1706144 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {10312712 \sqrt {1-2 x} \sqrt {2+3 x}}{1029 \sqrt {3+5 x}}+\frac {10312712 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}+\frac {310208 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1715} \]

[Out]

10312712/5145*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+310208/5145*EllipticF(1/7*21^(1/2
)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+2/3*(1-2*x)^(1/2)/(2+3*x)^(7/2)/(3+5*x)^(1/2)+176/35*(1-2*x)^(1/2)/(
2+3*x)^(5/2)/(3+5*x)^(1/2)+12276/245*(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(1/2)+1706144/1715*(1-2*x)^(1/2)/(2+3
*x)^(1/2)/(3+5*x)^(1/2)-10312712/1029*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {100, 157, 164, 114, 120} \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx=\frac {310208 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1715}+\frac {10312712 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}-\frac {10312712 \sqrt {1-2 x} \sqrt {3 x+2}}{1029 \sqrt {5 x+3}}+\frac {1706144 \sqrt {1-2 x}}{1715 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {12276 \sqrt {1-2 x}}{245 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {176 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}} \]

[In]

Int[(1 - 2*x)^(3/2)/((2 + 3*x)^(9/2)*(3 + 5*x)^(3/2)),x]

[Out]

(2*Sqrt[1 - 2*x])/(3*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x]) + (176*Sqrt[1 - 2*x])/(35*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) +
(12276*Sqrt[1 - 2*x])/(245*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (1706144*Sqrt[1 - 2*x])/(1715*Sqrt[2 + 3*x]*Sqrt[
3 + 5*x]) - (10312712*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1029*Sqrt[3 + 5*x]) + (10312712*Sqrt[11/3]*EllipticE[ArcSi
n[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1715 + (310208*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/1715

Rule 100

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

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 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 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {2}{21} \int \frac {154-231 x}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx \\ & = \frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {176 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {4}{735} \int \frac {\frac {33649}{2}-23100 x}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx \\ & = \frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {176 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {12276 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {8 \int \frac {1272579-\frac {2900205 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{15435} \\ & = \frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {176 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {12276 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1706144 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {16 \int \frac {\frac {217164255}{4}-33589710 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{108045} \\ & = \frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {176 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {12276 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1706144 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {10312712 \sqrt {1-2 x} \sqrt {2+3 x}}{1029 \sqrt {3+5 x}}-\frac {32 \int \frac {\frac {2827810755}{4}+\frac {4466693385 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1188495} \\ & = \frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {176 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {12276 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1706144 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {10312712 \sqrt {1-2 x} \sqrt {2+3 x}}{1029 \sqrt {3+5 x}}-\frac {1706144 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1715}-\frac {10312712 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1715} \\ & = \frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {176 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {12276 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1706144 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {10312712 \sqrt {1-2 x} \sqrt {2+3 x}}{1029 \sqrt {3+5 x}}+\frac {10312712 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}+\frac {310208 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.33 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.47 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx=\frac {2 \left (-\frac {3 \sqrt {1-2 x} \left (130497191+793777840 x+1809835578 x^2+1833255216 x^3+696108060 x^4\right )}{(2+3 x)^{7/2} \sqrt {3+5 x}}-4 i \sqrt {33} \left (1289089 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1327865 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{5145} \]

[In]

Integrate[(1 - 2*x)^(3/2)/((2 + 3*x)^(9/2)*(3 + 5*x)^(3/2)),x]

[Out]

(2*((-3*Sqrt[1 - 2*x]*(130497191 + 793777840*x + 1809835578*x^2 + 1833255216*x^3 + 696108060*x^4))/((2 + 3*x)^
(7/2)*Sqrt[3 + 5*x]) - (4*I)*Sqrt[33]*(1289089*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 1327865*EllipticF
[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/5145

Maple [A] (verified)

Time = 1.30 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.33

method	result	size
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{81 \left (\frac {2}{3}+x \right )^{4}}-\frac {878 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{945 \left (\frac {2}{3}+x \right )^{3}}-\frac {67558 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2205 \left (\frac {2}{3}+x \right )^{2}}-\frac {7482962 \left (-30 x^{2}-3 x +9\right )}{5145 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {13057712 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{36015 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {20625424 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{36015 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {550 \left (-30 x^{2}-5 x +10\right )}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(295\)
default	\(\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (135214596 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-139221612 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}[phone] \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-278443224 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}[phone] \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-185628816 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}[phone] \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-41250848 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-4176648360 x^{5}-8911207116 x^{4}-5359247820 x^{3}[phone] x^{2}[phone] x [phone]\right )}{5145 \left (2+3 x \right )^{\frac {7}{2}} \left (10 x^{2}+x -3\right )}\)	\(409\)

[In]

int((1-2*x)^(3/2)/(2+3*x)^(9/2)/(3+5*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

(-(-1+2*x)*(3+5*x)*(2+3*x))^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2)*(-2/81*(-30*x^3-23*x^2+7*x+6)^(1/2
)/(2/3+x)^4-878/945*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3-67558/2205*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^2-7
482962/5145*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/2)-13057712/36015*(10+15*x)^(1/2)*(21-42*x)^(1/2)*(-1
5*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))-20625424/36015*(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)))-550*(-30*x^2-5*x+10)/((x+3/5)*(-30*x^2-5*x+10))^(1/2))

Fricas [C] (verification not implemented)

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

Time = 0.08 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx=-\frac {2 \, {\left (135 \, {\left (696108060 \, x^{4} + 1833255216 \, x^{3} + 1809835578 \, x^{2} + 793777840 \, x + 130497191\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 87601166 \, \sqrt {-30} {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 232036020 \, \sqrt {-30} {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{231525 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} \]

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^(9/2)/(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

-2/231525*(135*(696108060*x^4 + 1833255216*x^3 + 1809835578*x^2 + 793777840*x + 130497191)*sqrt(5*x + 3)*sqrt(
3*x + 2)*sqrt(-2*x + 1) - 87601166*sqrt(-30)*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)*weierstra
ssPInverse(1159/675, 38998/91125, x + 23/90) + 232036020*sqrt(-30)*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 +
368*x + 48)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(4
05*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)

Sympy [F(-1)]

Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate((1-2*x)**(3/2)/(2+3*x)**(9/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {9}{2}}} \,d x } \]

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^(9/2)/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

integrate((-2*x + 1)^(3/2)/((5*x + 3)^(3/2)*(3*x + 2)^(9/2)), x)

Giac [F]

\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {9}{2}}} \,d x } \]

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^(9/2)/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

integrate((-2*x + 1)^(3/2)/((5*x + 3)^(3/2)*(3*x + 2)^(9/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^{9/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]

[In]

int((1 - 2*x)^(3/2)/((3*x + 2)^(9/2)*(5*x + 3)^(3/2)),x)

[Out]

int((1 - 2*x)^(3/2)/((3*x + 2)^(9/2)*(5*x + 3)^(3/2)), x)