\(\int \frac {5-x}{(3+2 x)^{3/2} (2+5 x+3 x^2)^3} \, dx\) [2572]

Optimal result

Integrand size = 27, antiderivative size = 115 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^3} \, dx=\frac {2667}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2}+\frac {1888+2229 x}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )}+402 \text {arctanh}\left (\sqrt {3+2 x}\right )-\frac {12717}{25} \sqrt {\frac {3}{5}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right ) \]

[Out]

402*arctanh((3+2*x)^(1/2))-12717/125*arctanh(1/5*15^(1/2)*(3+2*x)^(1/2))*15^(1/2)+2667/25/(3+2*x)^(1/2)-3/10*(
37+47*x)/(3*x^2+5*x+2)^2/(3+2*x)^(1/2)+1/10*(1888+2229*x)/(3*x^2+5*x+2)/(3+2*x)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 115, 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, 842, 840, 1180, 213} \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^3} \, dx=402 \text {arctanh}\left (\sqrt {2 x+3}\right )-\frac {12717}{25} \sqrt {\frac {3}{5}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )-\frac {3 (47 x+37)}{10 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^2}+\frac {2229 x+1888}{10 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )}+\frac {2667}{25 \sqrt {2 x+3}} \]

[In]

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

[Out]

2667/(25*Sqrt[3 + 2*x]) - (3*(37 + 47*x))/(10*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^2) + (1888 + 2229*x)/(10*Sqrt[3
+ 2*x]*(2 + 5*x + 3*x^2)) + 402*ArcTanh[Sqrt[3 + 2*x]] - (12717*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/25

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 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 840

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

Rule 842

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e
*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(d +
e*x)^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^2)), 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] && FractionQ[m] && LtQ[m, -1]

Rule 1180

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

Rubi steps \begin{align*} \text {integral}& = -\frac {3 (37+47 x)}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2}-\frac {1}{10} \int \frac {1328+987 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2} \, dx \\ & = -\frac {3 (37+47 x)}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2}+\frac {1888+2229 x}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )}+\frac {1}{50} \int \frac {43485+33435 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx \\ & = \frac {2667}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2}+\frac {1888+2229 x}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )}+\frac {1}{250} \int \frac {90255+40005 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx \\ & = \frac {2667}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2}+\frac {1888+2229 x}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )}+\frac {1}{125} \text {Subst}\left (\int \frac {60495+40005 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right ) \\ & = \frac {2667}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2}+\frac {1888+2229 x}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )}-1206 \text {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )+\frac {38151}{25} \text {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right ) \\ & = \frac {2667}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2}+\frac {1888+2229 x}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )}+402 \tanh ^{-1}\left (\sqrt {3+2 x}\right )-\frac {12717}{25} \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.75 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^3} \, dx=402 \text {arctanh}\left (\sqrt {3+2 x}\right )+\frac {1}{250} \left (\frac {5 \left (39661+175465 x+281403 x^2+193455 x^3+48006 x^4\right )}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2}-25434 \sqrt {15} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )\right ) \]

[In]

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

[Out]

402*ArcTanh[Sqrt[3 + 2*x]] + ((5*(39661 + 175465*x + 281403*x^2 + 193455*x^3 + 48006*x^4))/(Sqrt[3 + 2*x]*(2 +
5*x + 3*x^2)^2) - 25434*Sqrt[15]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/250

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.74

method	result	size
risch	\(\frac {48006 x^{4}+193455 x^{3}+281403 x^{2}+175465 x +39661}{50 \left (3 x^{2}+5 x +2\right )^{2} \sqrt {3+2 x}}+201 \ln \left (\sqrt {3+2 x}+1\right )-\frac {12717 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{125}-201 \ln \left (\sqrt {3+2 x}-1\right )\)	\(85\)
trager	\(\frac {48006 x^{4}+193455 x^{3}+281403 x^{2}+175465 x +39661}{50 \left (3 x^{2}+5 x +2\right )^{2} \sqrt {3+2 x}}+201 \ln \left (\frac {\sqrt {3+2 x}+2+x}{1+x}\right )-\frac {81 \operatorname {RootOf}\left (\textit {\_Z}^{2}-369735\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-369735\right ) x +7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-369735\right )+2355 \sqrt {3+2 x}}{2+3 x}\right )}{250}\)	\(106\)
derivativedivides	\(\frac {32}{\sqrt {3+2 x}-1}+\frac {3}{\left (\sqrt {3+2 x}-1\right )^{2}}-201 \ln \left (\sqrt {3+2 x}-1\right )+\frac {\frac {51759 \left (3+2 x \right )^{\frac {3}{2}}}{125}-\frac {18171 \sqrt {3+2 x}}{25}}{\left (6 x +4\right )^{2}}-\frac {12717 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{125}+\frac {32}{\sqrt {3+2 x}+1}-\frac {3}{\left (\sqrt {3+2 x}+1\right )^{2}}+201 \ln \left (\sqrt {3+2 x}+1\right )-\frac {416}{125 \sqrt {3+2 x}}\)	\(133\)
default	\(\frac {32}{\sqrt {3+2 x}-1}+\frac {3}{\left (\sqrt {3+2 x}-1\right )^{2}}-201 \ln \left (\sqrt {3+2 x}-1\right )+\frac {\frac {51759 \left (3+2 x \right )^{\frac {3}{2}}}{125}-\frac {18171 \sqrt {3+2 x}}{25}}{\left (6 x +4\right )^{2}}-\frac {12717 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{125}+\frac {32}{\sqrt {3+2 x}+1}-\frac {3}{\left (\sqrt {3+2 x}+1\right )^{2}}+201 \ln \left (\sqrt {3+2 x}+1\right )-\frac {416}{125 \sqrt {3+2 x}}\)	\(133\)
pseudoelliptic	\(-\frac {457812 \left (\sqrt {3+2 x}\, \sqrt {15}\, \left (x +\frac {2}{3}\right )^{2} \left (1+x \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right )+\frac {8375 \left (x +\frac {2}{3}\right )^{2} \left (1+x \right )^{2} \sqrt {3+2 x}\, \ln \left (\sqrt {3+2 x}-1\right )}{4239}-\frac {8375 \left (x +\frac {2}{3}\right )^{2} \left (1+x \right )^{2} \sqrt {3+2 x}\, \ln \left (\sqrt {3+2 x}+1\right )}{4239}-\frac {4445 x^{4}}{4239}-\frac {35825 x^{3}}{8478}-\frac {156335 x^{2}}{25434}-\frac {877325 x}{228906}-\frac {198305}{228906}\right )}{125 \sqrt {3+2 x}\, \left (\sqrt {3+2 x}-1\right )^{2} \left (2+3 x \right )^{2} \left (\sqrt {3+2 x}+1\right )^{2}}\)	\(151\)

[In]

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

[Out]

1/50*(48006*x^4+193455*x^3+281403*x^2+175465*x+39661)/(3*x^2+5*x+2)^2/(3+2*x)^(1/2)+201*ln((3+2*x)^(1/2)+1)-12
717/125*arctanh(1/5*15^(1/2)*(3+2*x)^(1/2))*15^(1/2)-201*ln((3+2*x)^(1/2)-1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (90) = 180\).

Time = 0.30 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.70 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^3} \, dx=\frac {12717 \, \sqrt {5} \sqrt {3} {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} \log \left (-\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 50250 \, {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) - 50250 \, {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) + 5 \, {\left (48006 \, x^{4} + 193455 \, x^{3} + 281403 \, x^{2} + 175465 \, x + 39661\right )} \sqrt {2 \, x + 3}}{250 \, {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )}} \]

[In]

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

[Out]

1/250*(12717*sqrt(5)*sqrt(3)*(18*x^5 + 87*x^4 + 164*x^3 + 151*x^2 + 68*x + 12)*log(-(sqrt(5)*sqrt(3)*sqrt(2*x
+ 3) - 3*x - 7)/(3*x + 2)) + 50250*(18*x^5 + 87*x^4 + 164*x^3 + 151*x^2 + 68*x + 12)*log(sqrt(2*x + 3) + 1) -
50250*(18*x^5 + 87*x^4 + 164*x^3 + 151*x^2 + 68*x + 12)*log(sqrt(2*x + 3) - 1) + 5*(48006*x^4 + 193455*x^3 + 2
81403*x^2 + 175465*x + 39661)*sqrt(2*x + 3))/(18*x^5 + 87*x^4 + 164*x^3 + 151*x^2 + 68*x + 12)

Sympy [A] (verification not implemented)

Time = 115.77 (sec) , antiderivative size = 420, normalized size of antiderivative = 3.65 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^3} \, dx=\frac {28917 \sqrt {15} \left (\log {\left (\sqrt {2 x + 3} - \frac {\sqrt {15}}{3} \right )} - \log {\left (\sqrt {2 x + 3} + \frac {\sqrt {15}}{3} \right )}\right )}{625} - \frac {31752 \left (\begin {cases} \frac {\sqrt {15} \left (- \frac {\log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1\right )}\right )}{75} & \text {for}\: \sqrt {2 x + 3} > - \frac {\sqrt {15}}{3} \wedge \sqrt {2 x + 3} < \frac {\sqrt {15}}{3} \end {cases}\right )}{25} + \frac {3672 \left (\begin {cases} \frac {\sqrt {15} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1\right )^{2}}\right )}{375} & \text {for}\: \sqrt {2 x + 3} > - \frac {\sqrt {15}}{3} \wedge \sqrt {2 x + 3} < \frac {\sqrt {15}}{3} \end {cases}\right )}{5} - 201 \log {\left (\sqrt {2 x + 3} - 1 \right )} + 201 \log {\left (\sqrt {2 x + 3} + 1 \right )} + \frac {32}{\sqrt {2 x + 3} + 1} - \frac {3}{\left (\sqrt {2 x + 3} + 1\right )^{2}} + \frac {32}{\sqrt {2 x + 3} - 1} + \frac {3}{\left (\sqrt {2 x + 3} - 1\right )^{2}} - \frac {416}{125 \sqrt {2 x + 3}} \]

[In]

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

[Out]

28917*sqrt(15)*(log(sqrt(2*x + 3) - sqrt(15)/3) - log(sqrt(2*x + 3) + sqrt(15)/3))/625 - 31752*Piecewise((sqrt
(15)*(-log(sqrt(15)*sqrt(2*x + 3)/5 - 1)/4 + log(sqrt(15)*sqrt(2*x + 3)/5 + 1)/4 - 1/(4*(sqrt(15)*sqrt(2*x + 3
)/5 + 1)) - 1/(4*(sqrt(15)*sqrt(2*x + 3)/5 - 1)))/75, (sqrt(2*x + 3) > -sqrt(15)/3) & (sqrt(2*x + 3) < sqrt(15
)/3)))/25 + 3672*Piecewise((sqrt(15)*(3*log(sqrt(15)*sqrt(2*x + 3)/5 - 1)/16 - 3*log(sqrt(15)*sqrt(2*x + 3)/5
+ 1)/16 + 3/(16*(sqrt(15)*sqrt(2*x + 3)/5 + 1)) + 1/(16*(sqrt(15)*sqrt(2*x + 3)/5 + 1)**2) + 3/(16*(sqrt(15)*s
qrt(2*x + 3)/5 - 1)) - 1/(16*(sqrt(15)*sqrt(2*x + 3)/5 - 1)**2))/375, (sqrt(2*x + 3) > -sqrt(15)/3) & (sqrt(2*
x + 3) < sqrt(15)/3)))/5 - 201*log(sqrt(2*x + 3) - 1) + 201*log(sqrt(2*x + 3) + 1) + 32/(sqrt(2*x + 3) + 1) -
3/(sqrt(2*x + 3) + 1)**2 + 32/(sqrt(2*x + 3) - 1) + 3/(sqrt(2*x + 3) - 1)**2 - 416/(125*sqrt(2*x + 3))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.24 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^3} \, dx=\frac {12717}{250} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) + \frac {24003 \, {\left (2 \, x + 3\right )}^{4} - 94581 \, {\left (2 \, x + 3\right )}^{3} + 117873 \, {\left (2 \, x + 3\right )}^{2} - 88030 \, x - 134125}{25 \, {\left (9 \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - 48 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + 94 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - 80 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + 25 \, \sqrt {2 \, x + 3}\right )}} + 201 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 201 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \]

[In]

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

[Out]

12717/250*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3))) + 1/25*(24003*(2*x + 3)^4 -
94581*(2*x + 3)^3 + 117873*(2*x + 3)^2 - 88030*x - 134125)/(9*(2*x + 3)^(9/2) - 48*(2*x + 3)^(7/2) + 94*(2*x
+ 3)^(5/2) - 80*(2*x + 3)^(3/2) + 25*sqrt(2*x + 3)) + 201*log(sqrt(2*x + 3) + 1) - 201*log(sqrt(2*x + 3) - 1)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.12 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^3} \, dx=\frac {12717}{250} \, \sqrt {15} \log \left (\frac {{\left | -2 \, \sqrt {15} + 6 \, \sqrt {2 \, x + 3} \right |}}{2 \, {\left (\sqrt {15} + 3 \, \sqrt {2 \, x + 3}\right )}}\right ) - \frac {416}{125 \, \sqrt {2 \, x + 3}} + \frac {123759 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 492873 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 628469 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 253355 \, \sqrt {2 \, x + 3}}{125 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 201 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 201 \, \log \left ({\left | \sqrt {2 \, x + 3} - 1 \right |}\right ) \]

[In]

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

[Out]

12717/250*sqrt(15)*log(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3))) - 416/125/sqrt(2*x
+ 3) + 1/125*(123759*(2*x + 3)^(7/2) - 492873*(2*x + 3)^(5/2) + 628469*(2*x + 3)^(3/2) - 253355*sqrt(2*x + 3)
)/(3*(2*x + 3)^2 - 16*x - 19)^2 + 201*log(sqrt(2*x + 3) + 1) - 201*log(abs(sqrt(2*x + 3) - 1))

Mupad [B] (verification not implemented)

Time = 11.13 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.95 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^3} \, dx=402\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right )-\frac {12717\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{125}-\frac {\frac {17606\,x}{45}-\frac {13097\,{\left (2\,x+3\right )}^2}{25}+\frac {10509\,{\left (2\,x+3\right )}^3}{25}-\frac {2667\,{\left (2\,x+3\right )}^4}{25}+\frac {5365}{9}}{\frac {25\,\sqrt {2\,x+3}}{9}-\frac {80\,{\left (2\,x+3\right )}^{3/2}}{9}+\frac {94\,{\left (2\,x+3\right )}^{5/2}}{9}-\frac {16\,{\left (2\,x+3\right )}^{7/2}}{3}+{\left (2\,x+3\right )}^{9/2}} \]

[In]

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

[Out]

402*atanh((2*x + 3)^(1/2)) - (12717*15^(1/2)*atanh((15^(1/2)*(2*x + 3)^(1/2))/5))/125 - ((17606*x)/45 - (13097
*(2*x + 3)^2)/25 + (10509*(2*x + 3)^3)/25 - (2667*(2*x + 3)^4)/25 + 5365/9)/((25*(2*x + 3)^(1/2))/9 - (80*(2*x
+ 3)^(3/2))/9 + (94*(2*x + 3)^(5/2))/9 - (16*(2*x + 3)^(7/2))/3 + (2*x + 3)^(9/2))