3.2979 \(\int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx\)

Optimal. Leaf size=218 \[ -\frac {837304 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{924385 \sqrt {33}}+\frac {26062156 \sqrt {1-2 x} \sqrt {5 x+3}}{10168235 \sqrt {3 x+2}}+\frac {349904 \sqrt {1-2 x} \sqrt {5 x+3}}{1452605 (3 x+2)^{3/2}}-\frac {806 \sqrt {1-2 x} \sqrt {5 x+3}}{207515 (3 x+2)^{5/2}}+\frac {1336 \sqrt {5 x+3}}{17787 \sqrt {1-2 x} (3 x+2)^{5/2}}+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{5/2}}-\frac {26062156 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{924385 \sqrt {33}} \]

[Out]

-26062156/30504705*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-837304/30504705*EllipticF(1/
7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+4/231*(3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^(5/2)+1336/17787*
(3+5*x)^(1/2)/(2+3*x)^(5/2)/(1-2*x)^(1/2)-806/207515*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)+349904/1452605*
(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)[phone]/10168235*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)

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Rubi [A]  time = 0.09, antiderivative size = 218, 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} \[ \frac {26062156 \sqrt {1-2 x} \sqrt {5 x+3}}{10168235 \sqrt {3 x+2}}+\frac {349904 \sqrt {1-2 x} \sqrt {5 x+3}}{1452605 (3 x+2)^{3/2}}-\frac {806 \sqrt {1-2 x} \sqrt {5 x+3}}{207515 (3 x+2)^{5/2}}+\frac {1336 \sqrt {5 x+3}}{17787 \sqrt {1-2 x} (3 x+2)^{5/2}}+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{5/2}}-\frac {837304 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{924385 \sqrt {33}}-\frac {26062156 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{924385 \sqrt {33}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3 + 5*x])/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)) + (1336*Sqrt[3 + 5*x])/(17787*Sqrt[1 - 2*x]*(2 + 3*x)^
(5/2)) - (806*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(207515*(2 + 3*x)^(5/2)) + (349904*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14
52605*(2 + 3*x)^(3/2)) + (26062156*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(10168235*Sqrt[2 + 3*x]) - (26062156*EllipticE
[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(924385*Sqrt[33]) - (837304*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]
], 35/33])/(924385*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} \sqrt {3+5 x}} \, dx &=\frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{5/2}}-\frac {2}{231} \int \frac {-\frac {229}{2}-105 x}{(1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx\\ &=\frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac {1336 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{5/2}}+\frac {4 \int \frac {\frac {32997}{4}+12525 x}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{17787}\\ &=\frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac {1336 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {806 \sqrt {1-2 x} \sqrt {3+5 x}}{207515 (2+3 x)^{5/2}}+\frac {8 \int \frac {\frac {137259}{2}+\frac {18135 x}{4}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{622545}\\ &=\frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac {1336 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {806 \sqrt {1-2 x} \sqrt {3+5 x}}{207515 (2+3 x)^{5/2}}+\frac {349904 \sqrt {1-2 x} \sqrt {3+5 x}}{1452605 (2+3 x)^{3/2}}+\frac {16 \int \frac {\frac {14298057}{8}-984105 x}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{13073445}\\ &=\frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac {1336 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {806 \sqrt {1-2 x} \sqrt {3+5 x}}{207515 (2+3 x)^{5/2}}+\frac {349904 \sqrt {1-2 x} \sqrt {3+5 x}}{1452605 (2+3 x)^{3/2}}+\frac {26062156 \sqrt {1-2 x} \sqrt {3+5 x}}{10168235 \sqrt {2+3 x}}+\frac {32 \int \frac {\frac {93140595}{4}+\frac {293199255 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{91514115}\\ &=\frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac {1336 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {806 \sqrt {1-2 x} \sqrt {3+5 x}}{207515 (2+3 x)^{5/2}}+\frac {349904 \sqrt {1-2 x} \sqrt {3+5 x}}{1452605 (2+3 x)^{3/2}}+\frac {26062156 \sqrt {1-2 x} \sqrt {3+5 x}}{10168235 \sqrt {2+3 x}}+\frac {418652 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{924385}+\frac {26062156 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{10168235}\\ &=\frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac {1336 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {806 \sqrt {1-2 x} \sqrt {3+5 x}}{207515 (2+3 x)^{5/2}}+\frac {349904 \sqrt {1-2 x} \sqrt {3+5 x}}{1452605 (2+3 x)^{3/2}}+\frac {26062156 \sqrt {1-2 x} \sqrt {3+5 x}}{10168235 \sqrt {2+3 x}}-\frac {26062156 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{924385 \sqrt {33}}-\frac {837304 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{924385 \sqrt {33}}\\ \end {align*}

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Mathematica [A]  time = 0.38, size = 107, normalized size = 0.49 \[ \frac {-24493280 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {2 \sqrt {10 x+6} \left (1407356424 x^4+513206712 x^3-914077314 x^2-176797172 x+165071409\right )}{(1-2 x)^{3/2} (3 x+2)^{5/2}}[phone] E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{30504705 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*Sqrt[6 + 10*x]*(165071409 - 176797172*x - 914077314*x^2 + 513206712*x^3 + 1407356424*x^4))/((1 - 2*x)^(3/2
)*(2 + 3*x)^(5/2)) + 52124312*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 24493280*EllipticF[ArcSin[S
qrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(30504705*Sqrt[2])

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fricas [F]  time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{3240 \, x^{8} + 5724 \, x^{7} + 378 \, x^{6} - 4179 \, x^{5} - 1547 \, x^{4} + 1008 \, x^{3} + 504 \, x^{2} - 80 \, x - 48}, x\right ) \]

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)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(3240*x^8 + 5724*x^7 + 378*x^6 - 4179*x^5 - 1547*x^4 + 10
08*x^3 + 504*x^2 - 80*x - 48), x)

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

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)^(1/2),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.03, size = 406, normalized size = 1.86 \[ -\frac {2 \sqrt {-2 x +1}\, \left (-7036782120 x^{5}-6788102832 x^{4}[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-110219760 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] x^{3}[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-91849800 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] x^{2}-52124312 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-294965529 x -52124312 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-495214227\right )}{30504705 \left (3 x +2\right )^{\frac {5}{2}} \left (2 x -1\right )^{2} \sqrt {5 x +3}} \]

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)^(1/2),x)

[Out]

-2/30504705*(-2*x+1)^(1/2)*(234559404*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)-110219760*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)[phone]*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)-91849800*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)-52124312*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]*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)-52124312*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))[phone]*2^(1/2)*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)*EllipticF(1/1
1*(110*x+66)^(1/2),1/2*I*66^(1/2))-7036782120*x^5-6788102832*x^4+3030766434*x^3+3626217802*x^2-294965529*x-495
214227)/(3*x+2)^(5/2)/(2*x-1)^2/(5*x+3)^(1/2)

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

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)^(1/2),x, algorithm="maxima")

[Out]

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

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)^(1/2)),x)

[Out]

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

[Out]

Timed out

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