3.27.26 \(\int \frac {5-x}{\sqrt {3+2 x} (2+5 x+3 x^2)^{5/2}} \, dx\) [2626]

Optimal. Leaf size=175 \[ -\frac {2 \sqrt {3+2 x} (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 \sqrt {3+2 x} (2152+2607 x)}{25 \sqrt {2+5 x+3 x^2}}-\frac {3476 \sqrt {3} \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{25 \sqrt {2+5 x+3 x^2}}+\frac {916 \sqrt {3} \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{5 \sqrt {2+5 x+3 x^2}} \]

[Out]

-2/5*(37+47*x)*(3+2*x)^(1/2)/(3*x^2+5*x+2)^(3/2)+4/25*(2152+2607*x)*(3+2*x)^(1/2)/(3*x^2+5*x+2)^(1/2)-3476/25*
EllipticE(3^(1/2)*(1+x)^(1/2),1/3*I*6^(1/2))*(-3*x^2-5*x-2)^(1/2)*3^(1/2)/(3*x^2+5*x+2)^(1/2)+916/5*EllipticF(
3^(1/2)*(1+x)^(1/2),1/3*I*6^(1/2))*(-3*x^2-5*x-2)^(1/2)*3^(1/2)/(3*x^2+5*x+2)^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {836, 857, 732, 435, 430} \begin {gather*} \frac {916 \sqrt {3} \sqrt {-3 x^2-5 x-2} F\left (\text {ArcSin}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{5 \sqrt {3 x^2+5 x+2}}-\frac {3476 \sqrt {3} \sqrt {-3 x^2-5 x-2} E\left (\text {ArcSin}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{25 \sqrt {3 x^2+5 x+2}}-\frac {2 \sqrt {2 x+3} (47 x+37)}{5 \left (3 x^2+5 x+2\right )^{3/2}}+\frac {4 \sqrt {2 x+3} (2607 x+2152)}{25 \sqrt {3 x^2+5 x+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[3 + 2*x]*(37 + 47*x))/(5*(2 + 5*x + 3*x^2)^(3/2)) + (4*Sqrt[3 + 2*x]*(2152 + 2607*x))/(25*Sqrt[2 + 5*
x + 3*x^2]) - (3476*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(25*Sqrt[2 +
5*x + 3*x^2]) + (916*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(5*Sqrt[2 +
5*x + 3*x^2])

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 732

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2*Rt[b^2 - 4*a*c, 2]*
(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*
e - e*Rt[b^2 - 4*a*c, 2])))^m)), Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2*c*d - b*e - e*Rt[b^2 - 4*a*c, 2
])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b
, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

Rule 836

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {5-x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac {2 \sqrt {3+2 x} (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {2}{15} \int \frac {696+423 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {2 \sqrt {3+2 x} (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 \sqrt {3+2 x} (2152+2607 x)}{25 \sqrt {2+5 x+3 x^2}}+\frac {4}{75} \int \frac {-6579-7821 x}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 \sqrt {3+2 x} (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 \sqrt {3+2 x} (2152+2607 x)}{25 \sqrt {2+5 x+3 x^2}}-\frac {5214}{25} \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx+\frac {1374}{5} \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 \sqrt {3+2 x} (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 \sqrt {3+2 x} (2152+2607 x)}{25 \sqrt {2+5 x+3 x^2}}-\frac {\left (3476 \sqrt {3} \sqrt {-2-5 x-3 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 x^2}{3}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {6+6 x}}{\sqrt {2}}\right )}{25 \sqrt {2+5 x+3 x^2}}+\frac {\left (916 \sqrt {3} \sqrt {-2-5 x-3 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 x^2}{3}}} \, dx,x,\frac {\sqrt {6+6 x}}{\sqrt {2}}\right )}{5 \sqrt {2+5 x+3 x^2}}\\ &=-\frac {2 \sqrt {3+2 x} (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 \sqrt {3+2 x} (2152+2607 x)}{25 \sqrt {2+5 x+3 x^2}}-\frac {3476 \sqrt {3} \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{25 \sqrt {2+5 x+3 x^2}}+\frac {916 \sqrt {3} \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{5 \sqrt {2+5 x+3 x^2}}\\ \end {align*}

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Mathematica [A]
time = 30.26, size = 196, normalized size = 1.12 \begin {gather*} \frac {-\frac {6952 \left (2+5 x+3 x^2\right )}{\sqrt {3+2 x}}+\frac {2 \sqrt {3+2 x} \left (8423+31713 x+38982 x^2+15642 x^3\right )}{2+5 x+3 x^2}-\frac {3476 (1+x) \sqrt {\frac {2+3 x}{3+2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )}{\sqrt {\frac {1+x}{15+10 x}}}+\frac {728 (1+x) \sqrt {\frac {2+3 x}{3+2 x}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )}{\sqrt {\frac {1+x}{15+10 x}}}}{25 \sqrt {2+5 x+3 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-6952*(2 + 5*x + 3*x^2))/Sqrt[3 + 2*x] + (2*Sqrt[3 + 2*x]*(8423 + 31713*x + 38982*x^2 + 15642*x^3))/(2 + 5*x
+ 3*x^2) - (3476*(1 + x)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/Sqrt[(1 +
x)/(15 + 10*x)] + (728*(1 + x)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/Sqr
t[(1 + x)/(15 + 10*x)])/(25*Sqrt[2 + 5*x + 3*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(307\) vs. \(2(143)=286\).
time = 0.08, size = 308, normalized size = 1.76

method	result	size
elliptic	\(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (\frac {\left (-\frac {74}{45}-\frac {94 x}{45}\right ) \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )^{2}}-\frac {2 \left (9+6 x \right ) \left (-\frac {4304}{75}-\frac {1738 x}{25}\right )}{\sqrt {\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right ) \left (9+6 x \right )}}-\frac {2924 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{125 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}-\frac {3476 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{2}-\EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )\right )}{125 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(231\)
default	\(\frac {2 \left (2607 \sqrt {15}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{2} \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}+828 \sqrt {15}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{2} \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}+4345 \sqrt {15}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x \sqrt {-20-30 x}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}+1380 \sqrt {15}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x \sqrt {-20-30 x}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}+1738 \sqrt {15}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )+552 \sqrt {15}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )+156420 x^{4}+624450 x^{3}+901860 x^{2}+559925 x +126345\right ) \sqrt {3 x^{2}+5 x +2}}{125 \left (2+3 x \right )^{2} \left (1+x \right )^{2} \sqrt {3+2 x}}\)	\(308\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/125*(2607*15^(1/2)*EllipticE(1/5*(45+30*x)^(1/2),1/3*15^(1/2))*x^2*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(
1/2)+828*15^(1/2)*EllipticF(1/5*(45+30*x)^(1/2),1/3*15^(1/2))*x^2*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2
)+4345*15^(1/2)*EllipticE(1/5*(45+30*x)^(1/2),1/3*15^(1/2))*x*(-20-30*x)^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)+13
80*15^(1/2)*EllipticF(1/5*(45+30*x)^(1/2),1/3*15^(1/2))*x*(-20-30*x)^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)+1738*1
5^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)*EllipticE(1/5*(45+30*x)^(1/2),1/3*15^(1/2))+552*15^(1/2)
*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)*EllipticF(1/5*(45+30*x)^(1/2),1/3*15^(1/2))+156420*x^4+624450*x
^3+901860*x^2+559925*x+126345)*(3*x^2+5*x+2)^(1/2)/(2+3*x)^2/(1+x)^2/(3+2*x)^(1/2)

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.69, size = 126, normalized size = 0.72 \begin {gather*} \frac {2 \, {\left (3353 \, \sqrt {6} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 15642 \, \sqrt {6} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 9 \, {\left (15642 \, x^{3} + 38982 \, x^{2} + 31713 \, x + 8423\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}\right )}}{225 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/225*(3353*sqrt(6)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*weierstrassPInverse(19/27, -28/729, x + 19/18) + 1564
2*sqrt(6)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*weierstrassZeta(19/27, -28/729, weierstrassPInverse(19/27, -28/
729, x + 19/18)) + 9*(15642*x^3 + 38982*x^2 + 31713*x + 8423)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3))/(9*x^4 + 30
*x^3 + 37*x^2 + 20*x + 4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x}{9 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{9 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(x/(9*x**4*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 37*x
**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(2*x + 3)*sqrt(3*
x**2 + 5*x + 2)), x) - Integral(-5/(9*x**4*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(2*x + 3)*sqrt(3
*x**2 + 5*x + 2) + 37*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) +
4*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)), x)

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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