3.2891 \(\int \frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=186 \[ \frac {673523 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{8750 \sqrt {33}}+\frac {\sqrt {5 x+3} (3 x+2)^{7/2}}{\sqrt {1-2 x}}+\frac {12}{7} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}+\frac {2517}{350} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}+\frac {29293}{875} \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}+\frac {4071079 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{17500} \]

[Out]

4071079/52500*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+673523/288750*EllipticF(1/7*21^(1
/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+(2+3*x)^(7/2)*(3+5*x)^(1/2)/(1-2*x)^(1/2)+2517/350*(2+3*x)^(3/2)*(
1-2*x)^(1/2)*(3+5*x)^(1/2)+12/7*(2+3*x)^(5/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)+29293/875*(1-2*x)^(1/2)*(2+3*x)^(1/2
)*(3+5*x)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {97, 154, 158, 113, 119} \[ \frac {\sqrt {5 x+3} (3 x+2)^{7/2}}{\sqrt {1-2 x}}+\frac {12}{7} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}+\frac {2517}{350} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}+\frac {29293}{875} \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}+\frac {673523 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8750 \sqrt {33}}+\frac {4071079 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{17500} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(29293*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/875 + (2517*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/350
+ (12*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/7 + ((2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] + (40710
79*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/17500 + (673523*EllipticF[ArcSin[Sqrt[3/7]*Sq
rt[1 - 2*x]], 35/33])/(8750*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 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)^{7/2} \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx &=\frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}-\int \frac {(2+3 x)^{5/2} \left (\frac {73}{2}+60 x\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {12}{7} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {1}{35} \int \frac {\left (-3845-\frac {12585 x}{2}\right ) (2+3 x)^{3/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {2517}{350} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {12}{7} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}-\frac {1}{875} \int \frac {\sqrt {2+3 x} \left (\frac {1083625}{4}+439395 x\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {29293}{875} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2517}{350} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {12}{7} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {\int \frac {-9665070-\frac {61066185 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{13125}\\ &=\frac {29293}{875} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2517}{350} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {12}{7} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}-\frac {673523 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{17500}-\frac {4071079 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{17500}\\ &=\frac {29293}{875} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2517}{350} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {12}{7} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {4071079 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{17500}+\frac {673523 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8750 \sqrt {33}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.30, size = 115, normalized size = 0.62 \[ \frac {2050510 \sqrt {2-4 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )-30 \sqrt {3 x+2} \sqrt {5 x+3} \left (6750 x^3+26010 x^2+54757 x-109756\right )-4071079 \sqrt {2-4 x} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{52500 \sqrt {1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-30*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-109756 + 54757*x + 26010*x^2 + 6750*x^3) - 4071079*Sqrt[2 - 4*x]*EllipticE[
ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 2050510*Sqrt[2 - 4*x]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -
33/2])/(52500*Sqrt[1 - 2*x])

________________________________________________________________________________________

fricas [F]  time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{4 \, x^{2} - 4 \, x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((27*x^3 + 54*x^2 + 36*x + 8)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 0.04, size = 150, normalized size = 0.81 \[ -\frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (-3037500 x^{5}-15552000 x^{4}-40681350 x^{3}[phone] x^{2}[phone] x -4071079 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+2050510 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone]\right )}{52500 \left (30 x^{3}+23 x^{2}-7 x -6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/52500*(3*x+2)^(1/2)*(5*x+3)^(1/2)*(-2*x+1)^(1/2)*(2050510*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))-4071079*2^(1/2)*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)*E
llipticE(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))-3037500*x^5-15552000*x^4-40681350*x^3+13496910*x^2+52704660*x+1
9756080)/(30*x^3+23*x^2-7*x-6)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (3\,x+2\right )}^{7/2}\,\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

________________________________________________________________________________________