3.3000 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=280 \[ -\frac{12417792656 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{111850585 \sqrt{33}}+\frac{412810345784 \sqrt{1-2 x} \sqrt{3 x+2}}{738213861 \sqrt{5 x+3}}-\frac{6208896328 \sqrt{1-2 x} \sqrt{3 x+2}}{67110351 (5 x+3)^{3/2}}+\frac{140700876 \sqrt{1-2 x}}{10168235 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{649224 \sqrt{1-2 x}}{1452605 (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac{3606 \sqrt{1-2 x}}{207515 (3 x+2)^{5/2} (5 x+3)^{3/2}}+\frac{632}{5929 \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}+\frac{4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{412810345784 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{111850585 \sqrt{33}} \]

[Out]

4/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + 632/(5929*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3
/2)) - (3606*Sqrt[1 - 2*x])/(207515*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (649224*Sqrt[1 - 2*x])/(1452605*(2 + 3*
x)^(3/2)*(3 + 5*x)^(3/2)) + (140700876*Sqrt[1 - 2*x])/(10168235*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (6208896328*S
qrt[1 - 2*x]*Sqrt[2 + 3*x])/(67110351*(3 + 5*x)^(3/2)) + (412810345784*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(738213861
*Sqrt[3 + 5*x]) - (412810345784*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(111850585*Sqrt[33]) - (124
17792656*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(111850585*Sqrt[33])

________________________________________________________________________________________

Rubi [A]  time = 0.124031, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {104, 152, 158, 113, 119} \[ \frac{412810345784 \sqrt{1-2 x} \sqrt{3 x+2}}{738213861 \sqrt{5 x+3}}-\frac{6208896328 \sqrt{1-2 x} \sqrt{3 x+2}}{67110351 (5 x+3)^{3/2}}+\frac{140700876 \sqrt{1-2 x}}{10168235 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{649224 \sqrt{1-2 x}}{1452605 (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac{3606 \sqrt{1-2 x}}{207515 (3 x+2)^{5/2} (5 x+3)^{3/2}}+\frac{632}{5929 \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}+\frac{4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{12417792656 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{111850585 \sqrt{33}}-\frac{412810345784 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{111850585 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

4/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + 632/(5929*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3
/2)) - (3606*Sqrt[1 - 2*x])/(207515*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (649224*Sqrt[1 - 2*x])/(1452605*(2 + 3*
x)^(3/2)*(3 + 5*x)^(3/2)) + (140700876*Sqrt[1 - 2*x])/(10168235*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (6208896328*S
qrt[1 - 2*x]*Sqrt[2 + 3*x])/(67110351*(3 + 5*x)^(3/2)) + (412810345784*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(738213861
*Sqrt[3 + 5*x]) - (412810345784*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(111850585*Sqrt[33]) - (124
17792656*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(111850585*Sqrt[33])

Rule 104

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 152

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 158

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]

Rule 113

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*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))])/b, 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 119

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d
), 2]*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))])/(
b*Sqrt[(b*e - a*f)/b]), 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)]))

Rubi steps

\begin{align*} \int \frac{1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac{2}{231} \int \frac{-\frac{309}{2}-165 x}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx\\ &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{632}{5929 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{4 \int \frac{\frac{83517}{4}+31995 x}{\sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx}{17787}\\ &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{632}{5929 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac{3606 \sqrt{1-2 x}}{207515 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{8 \int \frac{153282+\frac{189315 x}{4}}{\sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx}{622545}\\ &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{632}{5929 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac{3606 \sqrt{1-2 x}}{207515 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{649224 \sqrt{1-2 x}}{1452605 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{16 \int \frac{\frac{56833857}{8}-\frac{18259425 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx}{13073445}\\ &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{632}{5929 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac{3606 \sqrt{1-2 x}}{207515 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{649224 \sqrt{1-2 x}}{1452605 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{140700876 \sqrt{1-2 x}}{10168235 \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{32 \int \frac{\frac{2067907815}{4}-\frac{4748654565 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx}{91514115}\\ &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{632}{5929 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac{3606 \sqrt{1-2 x}}{207515 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{649224 \sqrt{1-2 x}}{1452605 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{140700876 \sqrt{1-2 x}}{10168235 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{6208896328 \sqrt{1-2 x} \sqrt{2+3 x}}{67110351 (3+5 x)^{3/2}}-\frac{64 \int \frac{\frac{338681488365}{16}-\frac{104775125535 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{3019965795}\\ &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{632}{5929 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac{3606 \sqrt{1-2 x}}{207515 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{649224 \sqrt{1-2 x}}{1452605 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{140700876 \sqrt{1-2 x}}{10168235 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{6208896328 \sqrt{1-2 x} \sqrt{2+3 x}}{67110351 (3+5 x)^{3/2}}+\frac{412810345784 \sqrt{1-2 x} \sqrt{2+3 x}}{738213861 \sqrt{3+5 x}}+\frac{128 \int \frac{\frac{551276253405}{2}+\frac{6966174585105 x}{16}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{33219623745}\\ &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{632}{5929 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac{3606 \sqrt{1-2 x}}{207515 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{649224 \sqrt{1-2 x}}{1452605 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{140700876 \sqrt{1-2 x}}{10168235 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{6208896328 \sqrt{1-2 x} \sqrt{2+3 x}}{67110351 (3+5 x)^{3/2}}+\frac{412810345784 \sqrt{1-2 x} \sqrt{2+3 x}}{738213861 \sqrt{3+5 x}}+\frac{6208896328 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{111850585}+\frac{412810345784 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1230356435}\\ &=\frac{4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{632}{5929 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac{3606 \sqrt{1-2 x}}{207515 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{649224 \sqrt{1-2 x}}{1452605 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{140700876 \sqrt{1-2 x}}{10168235 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{6208896328 \sqrt{1-2 x} \sqrt{2+3 x}}{67110351 (3+5 x)^{3/2}}+\frac{412810345784 \sqrt{1-2 x} \sqrt{2+3 x}}{738213861 \sqrt{3+5 x}}-\frac{412810345784 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{111850585 \sqrt{33}}-\frac{12417792656 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{111850585 \sqrt{33}}\\ \end{align*}

Mathematica [A]  time = 0.324144, size = 119, normalized size = 0.42 \[ \frac{2 \left (4 \sqrt{2} \left (51601293223 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-25989595870 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{557293966808400 x^6+873229924799280 x^5+84649478011164 x^4-430611138612568 x^3-149619576926754 x^2+52875828155808 x+23506658680609}{(1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )}{3691069305} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((23506658680609 + 52875828155808*x - 149619576926754*x^2 - 430611138612568*x^3 + 84649478011164*x^4 + 8732
29924799280*x^5 + 557293966808400*x^6)/((1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + 4*Sqrt[2]*(51601293
223*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 25989595870*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]
], -33/2])))/3691069305

________________________________________________________________________________________

Maple [C]  time = 0.028, size = 501, normalized size = 1.8 \begin{align*}{\frac{2}{3691069305\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 9356254513200\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}-18576465560280\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-26626267303068\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-5160129322300\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-3326668271360\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,{x}^{6}-1247500601760\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) [phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}-430611138612568\,{x}^{3}-149619576926754\,{x}^{2}[phone]\,x+23506658680609 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3691069305*(1-2*x)^(1/2)*(9356254513200*2^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^4*(2+3*x)^
(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)-18576465560280*2^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^4*(
2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)[phone]*2^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2)
)*x^3*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-26626267303068*2^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*6
6^(1/2))*x^3*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*2^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1
/2*I*66^(1/2))*x^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-5160129322300*2^(1/2)*EllipticE(1/11*(66+110*x)^(
1/2),1/2*I*66^(1/2))*x^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-3326668271360*2^(1/2)*EllipticF(1/11*(66+11
0*x)^(1/2),1/2*I*66^(1/2))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*2^(1/2)*EllipticE(1/11*(6
6+110*x)^(1/2),1/2*I*66^(1/2))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*x^6-1247500601760*2
^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))[phone]
*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))[phone]
9280*x^5+84649478011164*x^4-430611138612568*x^3-149619576926754*x^2+52875828155808*x+23506658680609)/(2+3*x)^(
5/2)/(3+5*x)^(3/2)/(2*x-1)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{81000 \, x^{10} + 240300 \, x^{9} + 210330 \, x^{8} - 41619 \, x^{7} - 160643 \, x^{6} - 58821 \, x^{5} + 28917 \, x^{4} + 22192 \, x^{3} + 936 \, x^{2} - 2160 \, x - 432}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(81000*x^10 + 240300*x^9 + 210330*x^8 - 41619*x^7 - 16064
3*x^6 - 58821*x^5 + 28917*x^4 + 22192*x^3 + 936*x^2 - 2160*x - 432), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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