\(\int \frac {5-x}{(3+2 x)^5 (2+5 x+3 x^2)^{3/2}} \, dx\) [2515]

Optimal result

Integrand size = 27, antiderivative size = 169 \[ \int \frac {5-x}{(3+2 x)^5 \left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\frac {6 (37+47 x)}{5 (3+2 x)^4 \sqrt {2+5 x+3 x^2}}-\frac {817 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^4}-\frac {11596 \sqrt {2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac {973 \sqrt {2+5 x+3 x^2}}{30 (3+2 x)^2}-\frac {25458 \sqrt {2+5 x+3 x^2}}{625 (3+2 x)}+\frac {82039 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{2500 \sqrt {5}} \]

[Out]

82039/12500*arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^2+5*x+2)^(1/2))*5^(1/2)-6/5*(37+47*x)/(3+2*x)^4/(3*x^2+5*x+2)^(1
/2)-817/25*(3*x^2+5*x+2)^(1/2)/(3+2*x)^4-11596/375*(3*x^2+5*x+2)^(1/2)/(3+2*x)^3-973/30*(3*x^2+5*x+2)^(1/2)/(3
+2*x)^2-25458/625*(3*x^2+5*x+2)^(1/2)/(3+2*x)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {836, 848, 820, 738, 212} \[ \int \frac {5-x}{(3+2 x)^5 \left (2+5 x+3 x^2\right )^{3/2}} \, dx=\frac {82039 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{2500 \sqrt {5}}-\frac {6 (47 x+37)}{5 (2 x+3)^4 \sqrt {3 x^2+5 x+2}}-\frac {25458 \sqrt {3 x^2+5 x+2}}{625 (2 x+3)}-\frac {973 \sqrt {3 x^2+5 x+2}}{30 (2 x+3)^2}-\frac {11596 \sqrt {3 x^2+5 x+2}}{375 (2 x+3)^3}-\frac {817 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^4} \]

[In]

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

[Out]

(-6*(37 + 47*x))/(5*(3 + 2*x)^4*Sqrt[2 + 5*x + 3*x^2]) - (817*Sqrt[2 + 5*x + 3*x^2])/(25*(3 + 2*x)^4) - (11596
*Sqrt[2 + 5*x + 3*x^2])/(375*(3 + 2*x)^3) - (973*Sqrt[2 + 5*x + 3*x^2])/(30*(3 + 2*x)^2) - (25458*Sqrt[2 + 5*x
+ 3*x^2])/(625*(3 + 2*x)) + (82039*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(2500*Sqrt[5])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 738

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

Rule 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Dist[
(b*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(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] && EqQ[S
implify[m + 2*p + 3], 0]

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 848

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

Rubi steps \begin{align*} \text {integral}& = -\frac {6 (37+47 x)}{5 (3+2 x)^4 \sqrt {2+5 x+3 x^2}}-\frac {2}{5} \int \frac {875+1128 x}{(3+2 x)^5 \sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {6 (37+47 x)}{5 (3+2 x)^4 \sqrt {2+5 x+3 x^2}}-\frac {817 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^4}+\frac {1}{50} \int \frac {-10463-14706 x}{(3+2 x)^4 \sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {6 (37+47 x)}{5 (3+2 x)^4 \sqrt {2+5 x+3 x^2}}-\frac {817 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^4}-\frac {11596 \sqrt {2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac {1}{750} \int \frac {87103+139152 x}{(3+2 x)^3 \sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {6 (37+47 x)}{5 (3+2 x)^4 \sqrt {2+5 x+3 x^2}}-\frac {817 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^4}-\frac {11596 \sqrt {2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac {973 \sqrt {2+5 x+3 x^2}}{30 (3+2 x)^2}+\frac {\int \frac {-330885-729750 x}{(3+2 x)^2 \sqrt {2+5 x+3 x^2}} \, dx}{7500} \\ & = -\frac {6 (37+47 x)}{5 (3+2 x)^4 \sqrt {2+5 x+3 x^2}}-\frac {817 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^4}-\frac {11596 \sqrt {2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac {973 \sqrt {2+5 x+3 x^2}}{30 (3+2 x)^2}-\frac {25458 \sqrt {2+5 x+3 x^2}}{625 (3+2 x)}+\frac {82039 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{2500} \\ & = -\frac {6 (37+47 x)}{5 (3+2 x)^4 \sqrt {2+5 x+3 x^2}}-\frac {817 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^4}-\frac {11596 \sqrt {2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac {973 \sqrt {2+5 x+3 x^2}}{30 (3+2 x)^2}-\frac {25458 \sqrt {2+5 x+3 x^2}}{625 (3+2 x)}-\frac {82039 \text {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{1250} \\ & = -\frac {6 (37+47 x)}{5 (3+2 x)^4 \sqrt {2+5 x+3 x^2}}-\frac {817 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^4}-\frac {11596 \sqrt {2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac {973 \sqrt {2+5 x+3 x^2}}{30 (3+2 x)^2}-\frac {25458 \sqrt {2+5 x+3 x^2}}{625 (3+2 x)}+\frac {82039 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{2500 \sqrt {5}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.56 \[ \int \frac {5-x}{(3+2 x)^5 \left (2+5 x+3 x^2\right )^{3/2}} \, dx=\frac {-\frac {5 \sqrt {2+5 x+3 x^2} \left (11545002+48537379 x+78737669 x^2+62190544 x^3+24066204 x^4+3665952 x^5\right )}{(1+x) (3+2 x)^4 (2+3 x)}+246117 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )}{18750} \]

[In]

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

[Out]

((-5*Sqrt[2 + 5*x + 3*x^2]*(11545002 + 48537379*x + 78737669*x^2 + 62190544*x^3 + 24066204*x^4 + 3665952*x^5))
/((1 + x)*(3 + 2*x)^4*(2 + 3*x)) + 246117*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)])/18750

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.46

method	result	size
risch	\(-\frac {3665952 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]}{3750 \left (3+2 x \right )^{4} \sqrt {3 x^{2}+5 x +2}}-\frac {82039 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{12500}\)	\(78\)
trager	\(-\frac {3665952 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]}{3750 \left (3+2 x \right )^{4} \sqrt {3 x^{2}+5 x +2}}-\frac {82039 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{12500}\)	\(97\)
default	\(-\frac {14}{75 \left (x +\frac {3}{2}\right )^{3} \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}-\frac {9619}{12000 \left (x +\frac {3}{2}\right )^{2} \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}-\frac {6931}{1500 \left (x +\frac {3}{2}\right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}+\frac {82039}{5000 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}-\frac {12729 \left (5+6 x \right )}{1250 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}-\frac {82039 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{12500}-\frac {13}{320 \left (x +\frac {3}{2}\right )^{4} \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}\)	\(153\)

[In]

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

[Out]

-1/3750*(3665952*x^5+24066204*x^4+62190544*x^3+78737669*x^2+48537379*x+11545002)/(3+2*x)^4/(3*x^2+5*x+2)^(1/2)
-82039/12500*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.92 \[ \int \frac {5-x}{(3+2 x)^5 \left (2+5 x+3 x^2\right )^{3/2}} \, dx=\frac {246117 \, \sqrt {5} {\left (48 \, x^{6} + 368 \, x^{5} + 1160 \, x^{4} + 1920 \, x^{3} + 1755 \, x^{2} + 837 \, x + 162\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 20 \, {\left (3665952 \, x^{5} + 24066204 \, x^{4} + 62190544 \, x^{3} + 78737669 \, x^{2} + 48537379 \, x + 11545002\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{75000 \, {\left (48 \, x^{6} + 368 \, x^{5} + 1160 \, x^{4} + 1920 \, x^{3} + 1755 \, x^{2} + 837 \, x + 162\right )}} \]

[In]

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

[Out]

1/75000*(246117*sqrt(5)*(48*x^6 + 368*x^5 + 1160*x^4 + 1920*x^3 + 1755*x^2 + 837*x + 162)*log((4*sqrt(5)*sqrt(
3*x^2 + 5*x + 2)*(8*x + 7) + 124*x^2 + 212*x + 89)/(4*x^2 + 12*x + 9)) - 20*(3665952*x^5 + 24066204*x^4 + 6219
0544*x^3 + 78737669*x^2 + 48537379*x + 11545002)*sqrt(3*x^2 + 5*x + 2))/(48*x^6 + 368*x^5 + 1160*x^4 + 1920*x^
3 + 1755*x^2 + 837*x + 162)

Sympy [F]

\[ \int \frac {5-x}{(3+2 x)^5 \left (2+5 x+3 x^2\right )^{3/2}} \, dx=- \int \frac {x}{96 x^{7} \sqrt {3 x^{2} + 5 x + 2} + 880 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 3424 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 7320 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 9270 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 6939 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 2835 x \sqrt {3 x^{2} + 5 x + 2} + 486 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{96 x^{7} \sqrt {3 x^{2} + 5 x + 2} + 880 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 3424 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 7320 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 9270 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 6939 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 2835 x \sqrt {3 x^{2} + 5 x + 2} + 486 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]

[In]

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

[Out]

-Integral(x/(96*x**7*sqrt(3*x**2 + 5*x + 2) + 880*x**6*sqrt(3*x**2 + 5*x + 2) + 3424*x**5*sqrt(3*x**2 + 5*x +
2) + 7320*x**4*sqrt(3*x**2 + 5*x + 2) + 9270*x**3*sqrt(3*x**2 + 5*x + 2) + 6939*x**2*sqrt(3*x**2 + 5*x + 2) +
2835*x*sqrt(3*x**2 + 5*x + 2) + 486*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-5/(96*x**7*sqrt(3*x**2 + 5*x + 2)
+ 880*x**6*sqrt(3*x**2 + 5*x + 2) + 3424*x**5*sqrt(3*x**2 + 5*x + 2) + 7320*x**4*sqrt(3*x**2 + 5*x + 2) + 9270
*x**3*sqrt(3*x**2 + 5*x + 2) + 6939*x**2*sqrt(3*x**2 + 5*x + 2) + 2835*x*sqrt(3*x**2 + 5*x + 2) + 486*sqrt(3*x
**2 + 5*x + 2)), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 310 vs. \(2 (139) = 278\).

Time = 0.27 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.83 \[ \int \frac {5-x}{(3+2 x)^5 \left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\frac {82039}{12500} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {38187 \, x}{625 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {172541}{5000 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {13}{20 \, {\left (16 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{4} + 96 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{3} + 216 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{2} + 216 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + 81 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}} - \frac {112}{75 \, {\left (8 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{3} + 36 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{2} + 54 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + 27 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}} - \frac {9619}{3000 \, {\left (4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{2} + 12 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + 9 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}} - \frac {6931}{750 \, {\left (2 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + 3 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}} \]

[In]

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

[Out]

-82039/12500*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 38187/625*x/sqrt
(3*x^2 + 5*x + 2) - 172541/5000/sqrt(3*x^2 + 5*x + 2) - 13/20/(16*sqrt(3*x^2 + 5*x + 2)*x^4 + 96*sqrt(3*x^2 +
5*x + 2)*x^3 + 216*sqrt(3*x^2 + 5*x + 2)*x^2 + 216*sqrt(3*x^2 + 5*x + 2)*x + 81*sqrt(3*x^2 + 5*x + 2)) - 112/7
5/(8*sqrt(3*x^2 + 5*x + 2)*x^3 + 36*sqrt(3*x^2 + 5*x + 2)*x^2 + 54*sqrt(3*x^2 + 5*x + 2)*x + 27*sqrt(3*x^2 + 5
*x + 2)) - 9619/3000/(4*sqrt(3*x^2 + 5*x + 2)*x^2 + 12*sqrt(3*x^2 + 5*x + 2)*x + 9*sqrt(3*x^2 + 5*x + 2)) - 69
31/750/(2*sqrt(3*x^2 + 5*x + 2)*x + 3*sqrt(3*x^2 + 5*x + 2))

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.39 \[ \int \frac {5-x}{(3+2 x)^5 \left (2+5 x+3 x^2\right )^{3/2}} \, dx=\frac {1}{12500} \, \sqrt {5} {\left (50916 \, \sqrt {5} \sqrt {3} + 82039 \, \log \left (-\sqrt {5} \sqrt {3} + 4\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {\frac {\frac {5 \, {\left (\frac {\frac {10 \, {\left (\frac {448}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} + \frac {195}{{\left (2 \, x + 3\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{2 \, x + 3} + \frac {9619}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2 \, x + 3} + \frac {27724}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{2 \, x + 3} - \frac {857109}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2 \, x + 3} + \frac {458244}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{7500 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3}} - \frac {82039 \, \sqrt {5} \log \left ({\left | \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} - 4 \right |}\right )}{12500 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} \]

[In]

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

[Out]

1/12500*sqrt(5)*(50916*sqrt(5)*sqrt(3) + 82039*log(-sqrt(5)*sqrt(3) + 4))*sgn(1/(2*x + 3)) - 1/7500*((5*((10*(
448/sgn(1/(2*x + 3)) + 195/((2*x + 3)*sgn(1/(2*x + 3))))/(2*x + 3) + 9619/sgn(1/(2*x + 3)))/(2*x + 3) + 27724/
sgn(1/(2*x + 3)))/(2*x + 3) - 857109/sgn(1/(2*x + 3)))/(2*x + 3) + 458244/sgn(1/(2*x + 3)))/sqrt(-8/(2*x + 3)
+ 5/(2*x + 3)^2 + 3) - 82039/12500*sqrt(5)*log(abs(sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(
2*x + 3)) - 4))/sgn(1/(2*x + 3))

Mupad [F(-1)]

Timed out. \[ \int \frac {5-x}{(3+2 x)^5 \left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\int \frac {x-5}{{\left (2\,x+3\right )}^5\,{\left (3\,x^2+5\,x+2\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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