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

[Out]

4/231/(1-2*x)^(3/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2)-412810345784/3691069305*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1
/33*1155^(1/2))*33^(1/2)-12417792656/3691069305*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)
+632/5929/(2+3*x)^(5/2)/(3+5*x)^(3/2)/(1-2*x)^(1/2)-3606/207515*(1-2*x)^(1/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2)+6492
24/1452605*(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(3/2)[phone]/10168235*(1-2*x)^(1/2)/(3+5*x)^(3/2)/(2+3*x)^(1
/2)-6208896328/67110351*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(3/2)[phone]/738213861*(1-2*x)^(1/2)*(2+3*x)
^(1/2)/(3+5*x)^(1/2)

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Rubi [A]  time = 0.12, antiderivative size = 280, normalized size of antiderivative = 1.00, 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 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)]))

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]

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*}

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Mathematica [A]  time = 0.44, size = 119, normalized size = 0.42 \[ \frac {2 \left (4 \sqrt {2} \left (51601293223 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-25989595870 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {557293966808400 x^6+873229924799280 x^5+84649478011164 x^4-430611138612568 x^3-149619576926754 x^2+52875828155808 x+23506658680609}{(1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )}{3691069305} \]

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

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fricas [F]  time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{81000 \, x^{10} + 240300 \, x^{9} + 210330 \, x^{8} - 41619 \, x^{7} - 160643 \, x^{6} - 58821 \, x^{5} + 28917 \, x^{4} + 22192 \, x^{3} + 936 \, x^{2} - 2160 \, x - 432}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]

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)

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maple [C]  time = 0.03, size = 501, normalized size = 1.79 \[ \frac {2 \sqrt {-2 x +1}\, \left (557293966808400 x^{6}[phone] x^{5}-18576465560280 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] x^{4}-26626267303068 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-430611138612568 x^{3}-5160129322300 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-149619576926754 x^{2}[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-3326668271360 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] x [phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1247500601760 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone]\right )}{3691069305 \left (3 x +2\right )^{\frac {5}{2}} \left (5 x +3\right )^{\frac {3}{2}} \left (2 x -1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3691069305*(-2*x+1)^(1/2)*(9356254513200*2^(1/2)*EllipticF(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x^4*(5*x+3)
^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)-18576465560280*2^(1/2)*EllipticE(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x^4
*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)[phone]*2^(1/2)*EllipticF(1/11*(110*x+66)^(1/2),1/2*I*66^(1
/2))*x^3*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)-26626267303068*2^(1/2)*EllipticE(1/11*(110*x+66)^(1/2),1/2
*I*66^(1/2))*x^3*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)[phone]*2^(1/2)*EllipticF(1/11*(110*x+66)^(1
/2),1/2*I*66^(1/2))*x^2*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)-5160129322300*2^(1/2)*EllipticE(1/11*(110*x
+66)^(1/2),1/2*I*66^(1/2))*x^2*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)-3326668271360*2^(1/2)*EllipticF(1/11
*(110*x+66)^(1/2),1/2*I*66^(1/2))*x*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)[phone]*2^(1/2)*EllipticE
(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)[phone]*x^6-124750
0601760*2^(1/2)*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)*EllipticF(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))+247
6862074704*2^(1/2)*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)*EllipticE(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))+
873229924799280*x^5+84649478011164*x^4-430611138612568*x^3-149619576926754*x^2+52875828155808*x+23506658680609
)/(3*x+2)^(5/2)/(5*x+3)^(3/2)/(2*x-1)^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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