\(\int \frac {5-x}{(3+2 x)^3 \sqrt {2+5 x+3 x^2}} \, dx\) [1009]

Optimal result

Integrand size = 27, antiderivative size = 84 \[ \int \frac {5-x}{(3+2 x)^3 \sqrt {2+5 x+3 x^2}} \, dx=-\frac {13 \sqrt {2+5 x+3 x^2}}{10 (3+2 x)^2}-\frac {73 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)}+\frac {389 \text {arctanh}\left (\frac {\sqrt {5} (1+x)}{\sqrt {2+5 x+3 x^2}}\right )}{50 \sqrt {5}} \] Output:

-13/10*(3*x^2+5*x+2)^(1/2)/(3+2*x)^2-73*(3*x^2+5*x+2)^(1/2)/(75+50*x)+389/ 
250*5^(1/2)*arctanh(5^(1/2)*(1+x)/(3*x^2+5*x+2)^(1/2))
 

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.75 \[ \int \frac {5-x}{(3+2 x)^3 \sqrt {2+5 x+3 x^2}} \, dx=\frac {1}{250} \left (-\frac {5 (503+292 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^2}+389 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )\right ) \] Input:

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

Output:

((-5*(503 + 292*x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^2 + 389*Sqrt[5]*ArcTan 
h[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)])/250
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1237, 27, 1228, 1154, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {5-x}{(2 x+3)^3 \sqrt {3 x^2+5 x+2}} \, dx\)

\(\Big \downarrow \) 1237

\(\displaystyle -\frac {1}{10} \int -\frac {29-78 x}{2 (2 x+3)^2 \sqrt {3 x^2+5 x+2}}dx-\frac {13 \sqrt {3 x^2+5 x+2}}{10 (2 x+3)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{20} \int \frac {29-78 x}{(2 x+3)^2 \sqrt {3 x^2+5 x+2}}dx-\frac {13 \sqrt {3 x^2+5 x+2}}{10 (2 x+3)^2}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {1}{20} \left (\frac {389}{5} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {292 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)}\right )-\frac {13 \sqrt {3 x^2+5 x+2}}{10 (2 x+3)^2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {1}{20} \left (-\frac {778}{5} \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )-\frac {292 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)}\right )-\frac {13 \sqrt {3 x^2+5 x+2}}{10 (2 x+3)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{20} \left (\frac {389 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{5 \sqrt {5}}-\frac {292 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)}\right )-\frac {13 \sqrt {3 x^2+5 x+2}}{10 (2 x+3)^2}\)

Input:

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

Output:

(-13*Sqrt[2 + 5*x + 3*x^2])/(10*(3 + 2*x)^2) + ((-292*Sqrt[2 + 5*x + 3*x^2 
])/(5*(3 + 2*x)) + (389*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2] 
)])/(5*Sqrt[5]))/20
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-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]
 

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(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] - Simp[(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 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 1237

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(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] + Simp[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] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

Maple [A] (verified)

Time = 1.62 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.81

method	result	size
risch	\(-\frac {876 x^{3}+2969 x^{2}+3099 x +1006}{50 \left (2 x +3\right )^{2} \sqrt {3 x^{2}+5 x +2}}-\frac {389 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{500}\)	\(68\)
default	\(-\frac {13 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{40 \left (x +\frac {3}{2}\right )^{2}}-\frac {73 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{50 \left (x +\frac {3}{2}\right )}-\frac {389 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{500}\)	\(74\)
trager	\(-\frac {\left (292 x +503\right ) \sqrt {3 x^{2}+5 x +2}}{50 \left (2 x +3\right )^{2}}-\frac {389 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (-\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )-10 \sqrt {3 x^{2}+5 x +2}}{2 x +3}\right )}{500}\)	\(78\)

Input:

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

Output:

-1/50*(876*x^3+2969*x^2+3099*x+1006)/(2*x+3)^2/(3*x^2+5*x+2)^(1/2)-389/500 
*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)

Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.13 \[ \int \frac {5-x}{(3+2 x)^3 \sqrt {2+5 x+3 x^2}} \, dx=\frac {389 \, \sqrt {5} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 20 \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (292 \, x + 503\right )}}{1000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \] Input:

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

Output:

1/1000*(389*sqrt(5)*(4*x^2 + 12*x + 9)*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*sqrt(3*x^2 + 
5*x + 2)*(292*x + 503))/(4*x^2 + 12*x + 9)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {5-x}{(3+2 x)^3 \sqrt {2+5 x+3 x^2}} \, dx=- \int \frac {x}{8 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 36 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 54 x \sqrt {3 x^{2} + 5 x + 2} + 27 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{8 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 36 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 54 x \sqrt {3 x^{2} + 5 x + 2} + 27 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \] Input:

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

Output:

-Integral(x/(8*x**3*sqrt(3*x**2 + 5*x + 2) + 36*x**2*sqrt(3*x**2 + 5*x + 2 
) + 54*x*sqrt(3*x**2 + 5*x + 2) + 27*sqrt(3*x**2 + 5*x + 2)), x) - Integra 
l(-5/(8*x**3*sqrt(3*x**2 + 5*x + 2) + 36*x**2*sqrt(3*x**2 + 5*x + 2) + 54* 
x*sqrt(3*x**2 + 5*x + 2) + 27*sqrt(3*x**2 + 5*x + 2)), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.07 \[ \int \frac {5-x}{(3+2 x)^3 \sqrt {2+5 x+3 x^2}} \, dx=-\frac {389}{500} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {13 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{10 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {73 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{25 \, {\left (2 \, x + 3\right )}} \] Input:

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

Output:

-389/500*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs( 
2*x + 3) - 2) - 13/10*sqrt(3*x^2 + 5*x + 2)/(4*x^2 + 12*x + 9) - 73/25*sqr 
t(3*x^2 + 5*x + 2)/(2*x + 3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (68) = 136\).

Time = 0.41 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.45 \[ \int \frac {5-x}{(3+2 x)^3 \sqrt {2+5 x+3 x^2}} \, dx=\frac {389}{500} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac {778 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 3551 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 13217 \, \sqrt {3} x + 4971 \, \sqrt {3} - 13217 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{50 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{2}} \] Input:

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

Output:

389/500*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^ 
2 + 5*x + 2))/abs(-4*sqrt(3)*x + 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5* 
x + 2))) - 1/50*(778*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 3551*sqrt(3)* 
(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 13217*sqrt(3)*x + 4971*sqrt(3) - 1 
3217*sqrt(3*x^2 + 5*x + 2))/(2*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 6*s 
qrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^2
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.77 \[ \int \frac {5-x}{(3+2 x)^3 \sqrt {2+5 x+3 x^2}} \, dx=\frac {-90520 \sqrt {3 x^{2}+5 x +2}\, x -155930 \sqrt {3 x^{2}+5 x +2}+48236 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}-\sqrt {15}+6 x +9\right ) x^{2}+144708 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}-\sqrt {15}+6 x +9\right ) x +108531 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}-\sqrt {15}+6 x +9\right )-48236 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+\sqrt {15}+6 x +9\right ) x^{2}-144708 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+\sqrt {15}+6 x +9\right ) x -108531 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+\sqrt {15}+6 x +9\right )+76460 \sqrt {3}\, x^{2}+229380 \sqrt {3}\, x +172035 \sqrt {3}}{62000 x^{2}+186000 x +139500} \] Input:

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

Output:

( - 90520*sqrt(3*x**2 + 5*x + 2)*x - 155930*sqrt(3*x**2 + 5*x + 2) + 48236 
*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) - sqrt(15) + 6*x + 9)*x**2 + 
 144708*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) - sqrt(15) + 6*x + 9) 
*x + 108531*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) - sqrt(15) + 6*x 
+ 9) - 48236*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) + sqrt(15) + 6*x 
 + 9)*x**2 - 144708*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) + sqrt(15 
) + 6*x + 9)*x - 108531*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) + sqr 
t(15) + 6*x + 9) + 76460*sqrt(3)*x**2 + 229380*sqrt(3)*x + 172035*sqrt(3)) 
/(15500*(4*x**2 + 12*x + 9))