\(\int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\) [2989]

Optimal result

Integrand size = 28, antiderivative size = 249 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {4839325048 \sqrt {1-2 x} \sqrt {2+3 x}}{67110351 \sqrt {3+5 x}}+\frac {4839325048 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{10168235 \sqrt {33}}+\frac {145418632 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{10168235 \sqrt {33}} \]

[Out]

4839325048/335551755*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)[phone]/335551755*Ellipt
icF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+4/231/(1-2*x)^(3/2)/(2+3*x)^(5/2)/(3+5*x)^(1/2)+1616/
17787/(2+3*x)^(5/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)-2206/207515*(1-2*x)^(1/2)/(2+3*x)^(5/2)/(3+5*x)^(1/2)+499564/1
452605*(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(1/2)[phone]/10168235*(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2)-4
839325048/67110351*(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 = {106, 157, 164, 114, 120} \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\frac {145418632 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{10168235 \sqrt {33}}+\frac {4839325048 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{10168235 \sqrt {33}}-\frac {4839325048 \sqrt {1-2 x} \sqrt {3 x+2}}{67110351 \sqrt {5 x+3}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {499564 \sqrt {1-2 x}}{1452605 (3 x+2)^{3/2} \sqrt {5 x+3}}-\frac {2206 \sqrt {1-2 x}}{207515 (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {1616}{17787 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt {5 x+3}} \]

[In]

Int[1/((1 - 2*x)^(5/2)*(2 + 3*x)^(7/2)*(3 + 5*x)^(3/2)),x]

[Out]

4/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + 1616/(17787*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x
]) - (2206*Sqrt[1 - 2*x])/(207515*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (499564*Sqrt[1 - 2*x])/(1452605*(2 + 3*x)^(
3/2)*Sqrt[3 + 5*x]) + (72709316*Sqrt[1 - 2*x])/(10168235*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (4839325048*Sqrt[1 - 2
*x]*Sqrt[2 + 3*x])/(67110351*Sqrt[3 + 5*x]) + (4839325048*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(
10168235*Sqrt[33]) + (145418632*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(10168235*Sqrt[33])

Rule 106

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegersQ[2*m, 2*n, 2*p]

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 {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2}{231} \int \frac {-\frac {269}{2}-135 x}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx \\ & = \frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {4 \int \frac {\frac {55457}{4}+21210 x}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx}{17787} \\ & = \frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {8 \int \frac {\frac {429823}{4}+\frac {82725 x}{4}}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx}{622545} \\ & = \frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {16 \int \frac {\frac {32051607}{8}-\frac {16860285 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{13073445} \\ & = \frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {32 \int \frac {\frac {661979505}{4}-\frac {817979805 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{91514115} \\ & = \frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {4839325048 \sqrt {1-2 x} \sqrt {2+3 x}}{67110351 \sqrt {3+5 x}}-\frac {64 \int \frac {\frac {34464999645}{16}+\frac {27221203395 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1006655265} \\ & = \frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {4839325048 \sqrt {1-2 x} \sqrt {2+3 x}}{67110351 \sqrt {3+5 x}}-\frac {72709316 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{10168235}-\frac {4839325048 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{111850585} \\ & = \frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {4839325048 \sqrt {1-2 x} \sqrt {2+3 x}}{67110351 \sqrt {3+5 x}}+\frac {4839325048 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{10168235 \sqrt {33}}+\frac {145418632 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{10168235 \sqrt {33}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.64 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.44 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\frac {2 \left (-\frac {91855922241+53503915182 x-673871013766 x^2-559512908172 x^3+1263428429256 x^4+1306617762960 x^5}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}-4 i \sqrt {33} \left (604915631 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-623092960 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{335551755} \]

[In]

Integrate[1/((1 - 2*x)^(5/2)*(2 + 3*x)^(7/2)*(3 + 5*x)^(3/2)),x]

[Out]

(2*(-((91855922241 + 53503915182*x - 673871013766*x^2 - 559512908172*x^3 + 1263428429256*x^4 + 1306617762960*x
^5)/((1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])) - (4*I)*Sqrt[33]*(604915631*EllipticE[I*ArcSinh[Sqrt[9 +
15*x]], -2/33] - 623092960*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/335551755

Maple [A] (verified)

Time = 4.72 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.30

method	result	size
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {6 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1715 \left (\frac {2}{3}+x \right )^{3}}-\frac {6 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{35 \left (\frac {2}{3}+x \right )^{2}}-\frac {817326 \left (-30 x^{2}-3 x +9\right )}{84035 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {6127111048 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2348862285 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {9678650096 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{2348862285 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {16 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{871563 \left (x -\frac {1}{2}\right )^{2}}-\frac {17216 \left (-30 x^{2}-38 x -12\right )}{67110351 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}-\frac {6250 \left (-30 x^{2}-5 x +10\right )}{1331 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(323\)
default	\(-\frac {2 \sqrt {1-2 x}\, \left (43553925432 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-42299110656 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}[phone] \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-35249258880 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-9678650096 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}[phone] \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-9678650096 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )[phone] \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )[phone] x^{5}[phone] x^{4}-559512908172 x^{3}-673871013766 x^{2}[phone] x [phone]\right )}{335551755 \left (2+3 x \right )^{\frac {5}{2}} \left (-1+2 x \right )^{2} \sqrt {3+5 x}}\)	\(406\)

[In]

int(1/(1-2*x)^(5/2)/(2+3*x)^(7/2)/(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)*(-6/1715*(-30*x^3-23*x^2+7*x+6)^(1
/2)/(2/3+x)^3-6/35*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^2-817326/84035*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9
))^(1/2)-6127111048/2348862285*(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)*El
lipticF((10+15*x)^(1/2),1/35*70^(1/2))-9678650096/2348862285*(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)))+16/871563*(-30*x^3-23*x^2+7*x+6)^(1/2)/(x-1/2)^2-17216/67110351*(-30*x^2-38*x-12)/((x-1/2)*(-30*x^2
-38*x-12))^(1/2)-6250/1331*(-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 = 0.07 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=-\frac {2 \, {\left (45 \, {\left (1306617762960 \, x^{5} + 1263428429256 \, x^{4} - 559512908172 \, x^{3} - 673871013766 \, x^{2} + 53503915182 \, x + 91855922241\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 41103880264 \, \sqrt {-30} {\left (540 \, x^{6} + 864 \, x^{5} + 99 \, x^{4} - 425 \, x^{3} - 154 \, x^{2} + 52 \, x + 24\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 108884813580 \, \sqrt {-30} {\left (540 \, x^{6} + 864 \, x^{5} + 99 \, x^{4} - 425 \, x^{3} - 154 \, x^{2} + 52 \, x + 24\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{15099828975 \, {\left (540 \, x^{6} + 864 \, x^{5} + 99 \, x^{4} - 425 \, x^{3} - 154 \, x^{2} + 52 \, x + 24\right )}} \]

[In]

integrate(1/(1-2*x)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

-2/15099828975*(45*(1306617762960*x^5 + 1263428429256*x^4 - 559512908172*x^3 - 673871013766*x^2 + 53503915182*
x + 91855922241)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 41103880264*sqrt(-30)*(540*x^6 + 864*x^5 + 99*x^
4 - 425*x^3 - 154*x^2 + 52*x + 24)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 108884813580*sqrt(-
30)*(540*x^6 + 864*x^5 + 99*x^4 - 425*x^3 - 154*x^2 + 52*x + 24)*weierstrassZeta(1159/675, 38998/91125, weiers
trassPInverse(1159/675, 38998/91125, x + 23/90)))/(540*x^6 + 864*x^5 + 99*x^4 - 425*x^3 - 154*x^2 + 52*x + 24)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(1-2*x)**(5/2)/(2+3*x)**(7/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/(1-2*x)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^(7/2)*(-2*x + 1)^(5/2)), x)

Giac [F]

\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/(1-2*x)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^(7/2)*(-2*x + 1)^(5/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]

[In]

int(1/((1 - 2*x)^(5/2)*(3*x + 2)^(7/2)*(5*x + 3)^(3/2)),x)

[Out]

int(1/((1 - 2*x)^(5/2)*(3*x + 2)^(7/2)*(5*x + 3)^(3/2)), x)