3.29.92 \(\int \frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx\) [2892]

Optimal. Leaf size=155 \[ \frac {419}{50} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {9}{5} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {7279}{125} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {4817 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{250 \sqrt {33}} \]

[Out]

7279/375*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+4817/8250*EllipticF(1/7*21^(1/2)*(1-2*
x)^(1/2),1/33*1155^(1/2))*33^(1/2)+(2+3*x)^(5/2)*(3+5*x)^(1/2)/(1-2*x)^(1/2)+9/5*(2+3*x)^(3/2)*(1-2*x)^(1/2)*(
3+5*x)^(1/2)+419/50*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {99, 159, 164, 114, 120} \begin {gather*} \frac {4817 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{250 \sqrt {33}}+\frac {7279}{125} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{5/2}}{\sqrt {1-2 x}}+\frac {9}{5} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}+\frac {419}{50} \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(419*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/50 + (9*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/5 + ((2 +
3*x)^(5/2)*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] + (7279*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])
/125 + (4817*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(250*Sqrt[33])

Rule 99

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 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*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)))], 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 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(Rt[-b/d,
2]/(b*Sqrt[(b*e - a*f)/b]))*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)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] && Po
sQ[-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 159

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 164

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 {3+5 x}}{(1-2 x)^{3/2}} \, dx &=\frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}-\int \frac {(2+3 x)^{3/2} \left (\frac {55}{2}+45 x\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {9}{5} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {1}{25} \int \frac {\left (-\frac {3875}{2}-\frac {6285 x}{2}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {419}{50} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {9}{5} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}-\frac {1}{375} \int \frac {\frac {276495}{4}+109185 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {419}{50} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {9}{5} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}-\frac {4817}{500} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {7279}{125} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {419}{50} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {9}{5} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {7279}{125} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {4817 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{250 \sqrt {33}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 5.98, size = 110, normalized size = 0.71 \begin {gather*} \frac {-30 \sqrt {2+3 x} \sqrt {3+5 x} \left (-799+328 x+90 x^2\right )-29116 \sqrt {2-4 x} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+14665 \sqrt {2-4 x} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{1500 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-30*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-799 + 328*x + 90*x^2) - 29116*Sqrt[2 - 4*x]*EllipticE[ArcSin[Sqrt[2/11]*Sqr
t[3 + 5*x]], -33/2] + 14665*Sqrt[2 - 4*x]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(1500*Sqrt[1 - 2
*x])

________________________________________________________________________________________

Maple [A]
time = 0.10, size = 143, normalized size = 0.92

method	result	size
default	\(\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (14451 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-29116 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+40500 x^{4}+198900 x^{3}-156390 x^{2}-396390 x -143820\right )}{45000 x^{3}+34500 x^{2}-10500 x -9000}\)	\(143\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {9 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{10}+\frac {373 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{100}-\frac {18433 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{2100 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {7279 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{525 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {49 \left (-30 x^{2}-38 x -12\right )}{8 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(240\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1500*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(14451*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*Ell
ipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-29116*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/
7*(28+42*x)^(1/2),1/2*70^(1/2))+40500*x^4+198900*x^3-156390*x^2-396390*x-143820)/(30*x^3+23*x^2-7*x-6)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.18, size = 40, normalized size = 0.26 \begin {gather*} \frac {{\left (90 \, x^{2} + 328 \, x - 799\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{50 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/50*(90*x^2 + 328*x - 799)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(2*x - 1)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________