3.2934 \(\int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=222 \[ \frac {1400888 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{132055}-\frac {46585232 \sqrt {1-2 x} \sqrt {3 x+2}}{290521 \sqrt {5 x+3}}+\frac {2101332 \sqrt {1-2 x}}{132055 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {14928 \sqrt {1-2 x}}{18865 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {138 \sqrt {1-2 x}}{2695 (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {46585232 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{132055} \]

[Out]

46585232/1452605*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+1400888/1452605*EllipticF(1/7*
21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+4/77/(2+3*x)^(5/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)+138/2695*(1-2*
x)^(1/2)/(2+3*x)^(5/2)/(3+5*x)^(1/2)+14928/18865*(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(1/2)+2101332/132055*(1-2
*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2)-46585232/290521*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)

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Rubi [A]  time = 0.08, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ -\frac {46585232 \sqrt {1-2 x} \sqrt {3 x+2}}{290521 \sqrt {5 x+3}}+\frac {2101332 \sqrt {1-2 x}}{132055 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {14928 \sqrt {1-2 x}}{18865 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {138 \sqrt {1-2 x}}{2695 (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {1400888 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{132055}+\frac {46585232 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{132055} \]

Antiderivative was successfully verified.

[In]

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

[Out]

4/(77*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (138*Sqrt[1 - 2*x])/(2695*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])
+ (14928*Sqrt[1 - 2*x])/(18865*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (2101332*Sqrt[1 - 2*x])/(132055*Sqrt[2 + 3*x]*
Sqrt[3 + 5*x]) - (46585232*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(290521*Sqrt[3 + 5*x]) + (46585232*Sqrt[3/11]*Elliptic
E[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/132055 + (1400888*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*
x]], 35/33])/132055

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)^{3/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2}{77} \int \frac {-\frac {163}{2}-105 x}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {4 \int \frac {-1291+\frac {1725 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx}{2695}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {14928 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} \sqrt {3+5 x}}-\frac {8 \int \frac {-\frac {301413}{4}+83970 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{56595}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {14928 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {2101332 \sqrt {1-2 x}}{132055 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {16 \int \frac {-\frac {12741465}{4}+\frac {7879995 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{396165}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {14928 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {2101332 \sqrt {1-2 x}}{132055 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {46585232 \sqrt {1-2 x} \sqrt {2+3 x}}{290521 \sqrt {3+5 x}}+\frac {32 \int \frac {-\frac {331786305}{8}-\frac {131020965 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4357815}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {14928 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {2101332 \sqrt {1-2 x}}{132055 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {46585232 \sqrt {1-2 x} \sqrt {2+3 x}}{290521 \sqrt {3+5 x}}-\frac {2101332 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{132055}-\frac {139755696 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1452605}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {14928 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {2101332 \sqrt {1-2 x}}{132055 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {46585232 \sqrt {1-2 x} \sqrt {2+3 x}}{290521 \sqrt {3+5 x}}+\frac {46585232 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{132055}+\frac {1400888 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{132055}\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 109, normalized size = 0.49 \[ \frac {2 \left (\frac {6289006320 x^4+9225477612 x^3+1919527182 x^2-2283681406 x-884250959}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}-2 \sqrt {2} \left (11646308 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-5867645 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )\right )}{1452605} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((-884250959 - 2283681406*x + 1919527182*x^2 + 9225477612*x^3 + 6289006320*x^4)/(Sqrt[1 - 2*x]*(2 + 3*x)^(5
/2)*Sqrt[3 + 5*x]) - 2*Sqrt[2]*(11646308*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5867645*Elliptic
F[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/1452605

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fricas [F]  time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{8100 \, x^{8} + 23220 \, x^{7} + 21141 \, x^{6} + 690 \, x^{5} - 9791 \, x^{4} - 4696 \, x^{3} + 424 \, x^{2} + 768 \, x + 144}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(8100*x^8 + 23220*x^7 + 21141*x^6 + 690*x^5 - 9791*x^4 - 4
696*x^3 + 424*x^2 + 768*x + 144), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [C]  time = 0.03, size = 314, normalized size = 1.41 \[ \frac {2 \sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-6289006320 x^{4}-9225477612 x^{3}[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-105617610 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1919527182 x^{2}[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-140823480 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] x [phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-46941160 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone]\right )}{1452605 \left (3 x +2\right )^{\frac {5}{2}} \left (10 x^{2}+x -3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1452605*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(209633544*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)-105617610*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)[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)-140823480*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)*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)*El
lipticE(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))-46941160*2^(1/2)*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)*Elli
pticF(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))-6289006320*x^4-9225477612*x^3-1919527182*x^2+2283681406*x+88425095
9)/(3*x+2)^(5/2)/(10*x^2+x-3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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