3.2860 \(\int \frac {(2+3 x)^{5/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\)

Optimal. Leaf size=127 \[ -\frac {857 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{625 \sqrt {33}}-\frac {3}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}-\frac {74}{125} \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}-\frac {5161 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1250} \]

[Out]

-5161/3750*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-857/20625*EllipticF(1/7*21^(1/2)*(1-
2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-3/25*(2+3*x)^(3/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)-74/125*(1-2*x)^(1/2)*(2+3*
x)^(1/2)*(3+5*x)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {102, 154, 158, 113, 119} \[ -\frac {3}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}-\frac {74}{125} \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}-\frac {857 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{625 \sqrt {33}}-\frac {5161 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1250} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-74*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/125 - (3*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/25 - (51
61*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1250 - (857*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1
- 2*x]], 35/33])/(625*Sqrt[33])

Rule 102

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))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && 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 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && 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 {(2+3 x)^{5/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx &=-\frac {3}{25} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {1}{25} \int \frac {\left (-\frac {275}{2}-222 x\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {74}{125} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {3}{25} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {1}{375} \int \frac {4902+\frac {15483 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {74}{125} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {3}{25} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {857 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1250}+\frac {5161 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1250}\\ &=-\frac {74}{125} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {3}{25} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {5161 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1250}-\frac {857 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{625 \sqrt {33}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.25, size = 95, normalized size = 0.75 \[ \frac {5161 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-5 \left (518 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+3 \sqrt {2-4 x} \sqrt {3 x+2} \sqrt {5 x+3} (45 x+104)\right )}{1875 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5161*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5*(3*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(104
+ 45*x) + 518*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/(1875*Sqrt[2])

________________________________________________________________________________________

fricas [F]  time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{10 \, x^{2} + x - 3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(9*x^2 + 12*x + 4)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(10*x^2 + x - 3), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 0.01, size = 145, normalized size = 1.14 \[ \frac {\sqrt {3 x +2}\, \sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-40500 x^{4}-124650 x^{3}-62310 x^{2}+29940 x -5161 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+2590 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+18720\right )}{112500 x^{3}+86250 x^{2}-26250 x -22500} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3750*(3*x+2)^(1/2)*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(2590*2^(1/2)*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)*Ell
ipticF(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))-5161*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))-40500*x^4-124650*x^3-62310*x^2+29940*x+18720)/(30*x^3+23*x^2-7*x-6)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (3\,x+2\right )}^{5/2}}{\sqrt {1-2\,x}\,\sqrt {5\,x+3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________