3.2957 \(\int \frac {\sqrt {2+3 x} (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=127 \[ -\frac {23 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{7 \sqrt {33}}+\frac {\sqrt {3 x+2} (5 x+3)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {23 \sqrt {3 x+2} \sqrt {5 x+3}}{7 \sqrt {1-2 x}}-\frac {139}{14} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

[Out]

-139/42*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-23/231*EllipticF(1/7*21^(1/2)*(1-2*x)^(
1/2),1/33*1155^(1/2))*33^(1/2)+1/3*(3+5*x)^(3/2)*(2+3*x)^(1/2)/(1-2*x)^(3/2)-23/7*(2+3*x)^(1/2)*(3+5*x)^(1/2)/
(1-2*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 = {97, 150, 158, 113, 119} \[ \frac {\sqrt {3 x+2} (5 x+3)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {23 \sqrt {3 x+2} \sqrt {5 x+3}}{7 \sqrt {1-2 x}}-\frac {23 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{7 \sqrt {33}}-\frac {139}{14} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-23*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]) + (Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/(3*(1 - 2*x)^(3/2)) - (1
39*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/14 - (23*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/(7*Sqrt[33])

Rule 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

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 150

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*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 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 {\sqrt {2+3 x} (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx &=\frac {\sqrt {2+3 x} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {\sqrt {3+5 x} \left (\frac {39}{2}+30 x\right )}{(1-2 x)^{3/2} \sqrt {2+3 x}} \, dx\\ &=-\frac {23 \sqrt {2+3 x} \sqrt {3+5 x}}{7 \sqrt {1-2 x}}+\frac {\sqrt {2+3 x} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {1}{21} \int \frac {-660-\frac {2085 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {23 \sqrt {2+3 x} \sqrt {3+5 x}}{7 \sqrt {1-2 x}}+\frac {\sqrt {2+3 x} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac {23}{14} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {139}{14} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {23 \sqrt {2+3 x} \sqrt {3+5 x}}{7 \sqrt {1-2 x}}+\frac {\sqrt {2+3 x} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {139}{14} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {23 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{7 \sqrt {33}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.16, size = 115, normalized size = 0.91 \[ -\frac {-70 \sqrt {2-4 x} (2 x-1) \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+2 \sqrt {3 x+2} \sqrt {5 x+3} (48-173 x)+139 \sqrt {2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{42 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/42*(2*(48 - 173*x)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] + 139*Sqrt[2 - 4*x]*(-1 + 2*x)*EllipticE[ArcSin[Sqrt[2/11]*S
qrt[3 + 5*x]], -33/2] - 70*Sqrt[2 - 4*x]*(-1 + 2*x)*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(1 - 2
*x)^(3/2)

________________________________________________________________________________________

fricas [F]  time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(5*x + 3)^(3/2)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(8*x^3 - 12*x^2 + 6*x - 1), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 0.02, size = 228, normalized size = 1.80 \[ \frac {\left (5190 x^{3}+5134 x^{2}-278 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+140 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+252 x +139 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-70 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-576\right ) \sqrt {-2 x +1}\, \sqrt {3 x +2}\, \sqrt {5 x +3}}{42 \left (2 x -1\right )^{2} \left (15 x^{2}+19 x +6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/42*(140*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)
-278*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)-70*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))+139*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))+5190*x^3+5134*x^2+2
52*x-576)*(-2*x+1)^(1/2)*(3*x+2)^(1/2)*(5*x+3)^(1/2)/(2*x-1)^2/(15*x^2+19*x+6)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

________________________________________________________________________________________