\(\int \frac {5-x}{(3+2 x)^2 (2+3 x^2)^{3/2}} \, dx\) [250]

Optimal result

Integrand size = 24, antiderivative size = 75 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=-\frac {13}{35 (3+2 x) \sqrt {2+3 x^2}}+\frac {632+57 x}{2450 \sqrt {2+3 x^2}}-\frac {632 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{1225 \sqrt {35}} \] Output:

-13/35/(3+2*x)/(3*x^2+2)^(1/2)+1/2450*(632+57*x)/(3*x^2+2)^(1/2)-632/42875 
*35^(1/2)*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))
 

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.87 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {\frac {35 \left (986+1435 x+114 x^2\right )}{(3+2 x) \sqrt {2+3 x^2}}-1264 \sqrt {35} \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{85750} \] Input:

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

Output:

((35*(986 + 1435*x + 114*x^2))/((3 + 2*x)*Sqrt[2 + 3*x^2]) - 1264*Sqrt[35] 
*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/85750
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {686, 27, 679, 488, 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.

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

\(\Big \downarrow \) 686

\(\displaystyle \frac {41 x+26}{70 (2 x+3) \sqrt {3 x^2+2}}-\frac {1}{210} \int -\frac {6 (41 x+52)}{(2 x+3)^2 \sqrt {3 x^2+2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{35} \int \frac {41 x+52}{(2 x+3)^2 \sqrt {3 x^2+2}}dx+\frac {41 x+26}{70 (2 x+3) \sqrt {3 x^2+2}}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {1}{35} \left (\frac {632}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx+\frac {19 \sqrt {3 x^2+2}}{35 (2 x+3)}\right )+\frac {41 x+26}{70 (2 x+3) \sqrt {3 x^2+2}}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {1}{35} \left (\frac {19 \sqrt {3 x^2+2}}{35 (2 x+3)}-\frac {632}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}\right )+\frac {41 x+26}{70 (2 x+3) \sqrt {3 x^2+2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{35} \left (\frac {19 \sqrt {3 x^2+2}}{35 (2 x+3)}-\frac {632 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}\right )+\frac {41 x+26}{70 (2 x+3) \sqrt {3 x^2+2}}\)

Input:

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

Output:

(26 + 41*x)/(70*(3 + 2*x)*Sqrt[2 + 3*x^2]) + ((19*Sqrt[2 + 3*x^2])/(35*(3 
+ 2*x)) - (632*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(35*Sqrt[35] 
))/35
 

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 488

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

rule 679

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 
)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) 
 Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, 
 p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 686

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

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.80

method	result	size
risch	\(\frac {114 x^{2}+1435 x +986}{2450 \left (2 x +3\right ) \sqrt {3 x^{2}+2}}-\frac {632 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{42875}\)	\(60\)
default	\(-\frac {13}{70 \left (x +\frac {3}{2}\right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}+\frac {316}{1225 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}+\frac {57 x}{2450 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}-\frac {632 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{42875}\)	\(86\)
trager	\(\frac {\left (114 x^{2}+1435 x +986\right ) \sqrt {3 x^{2}+2}}{14700 x^{3}+22050 x^{2}+9800 x +14700}+\frac {632 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )+35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{42875}\)	\(86\)

Input:

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

Output:

1/2450*(114*x^2+1435*x+986)/(2*x+3)/(3*x^2+2)^(1/2)-632/42875*35^(1/2)*arc 
tanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.39 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {632 \, \sqrt {35} {\left (6 \, x^{3} + 9 \, x^{2} + 4 \, x + 6\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) + 35 \, {\left (114 \, x^{2} + 1435 \, x + 986\right )} \sqrt {3 \, x^{2} + 2}}{85750 \, {\left (6 \, x^{3} + 9 \, x^{2} + 4 \, x + 6\right )}} \] Input:

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

Output:

1/85750*(632*sqrt(35)*(6*x^3 + 9*x^2 + 4*x + 6)*log(-(sqrt(35)*sqrt(3*x^2 
+ 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) + 35*(114*x^2 + 1 
435*x + 986)*sqrt(3*x^2 + 2))/(6*x^3 + 9*x^2 + 4*x + 6)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=- \int \frac {x}{12 x^{4} \sqrt {3 x^{2} + 2} + 36 x^{3} \sqrt {3 x^{2} + 2} + 35 x^{2} \sqrt {3 x^{2} + 2} + 24 x \sqrt {3 x^{2} + 2} + 18 \sqrt {3 x^{2} + 2}}\, dx - \int \left (- \frac {5}{12 x^{4} \sqrt {3 x^{2} + 2} + 36 x^{3} \sqrt {3 x^{2} + 2} + 35 x^{2} \sqrt {3 x^{2} + 2} + 24 x \sqrt {3 x^{2} + 2} + 18 \sqrt {3 x^{2} + 2}}\right )\, dx \] Input:

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

Output:

-Integral(x/(12*x**4*sqrt(3*x**2 + 2) + 36*x**3*sqrt(3*x**2 + 2) + 35*x**2 
*sqrt(3*x**2 + 2) + 24*x*sqrt(3*x**2 + 2) + 18*sqrt(3*x**2 + 2)), x) - Int 
egral(-5/(12*x**4*sqrt(3*x**2 + 2) + 36*x**3*sqrt(3*x**2 + 2) + 35*x**2*sq 
rt(3*x**2 + 2) + 24*x*sqrt(3*x**2 + 2) + 18*sqrt(3*x**2 + 2)), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.15 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {632}{42875} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {57 \, x}{2450 \, \sqrt {3 \, x^{2} + 2}} + \frac {316}{1225 \, \sqrt {3 \, x^{2} + 2}} - \frac {13}{35 \, {\left (2 \, \sqrt {3 \, x^{2} + 2} x + 3 \, \sqrt {3 \, x^{2} + 2}\right )}} \] Input:

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

Output:

632/42875*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2* 
x + 3)) + 57/2450*x/sqrt(3*x^2 + 2) + 316/1225/sqrt(3*x^2 + 2) - 13/35/(2* 
sqrt(3*x^2 + 2)*x + 3*sqrt(3*x^2 + 2))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (60) = 120\).

Time = 0.13 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.24 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=-\frac {1}{85750} \, \sqrt {35} {\left (19 \, \sqrt {35} \sqrt {3} - 1264 \, \log \left (\sqrt {35} \sqrt {3} - 9\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + \frac {\frac {\frac {1093}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} - \frac {1820}{{\left (2 \, x + 3\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2 \, x + 3} + \frac {57}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2450 \, \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3}} - \frac {632 \, \sqrt {35} \log \left (\sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} - 9\right )}{42875 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} \] Input:

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

Output:

-1/85750*sqrt(35)*(19*sqrt(35)*sqrt(3) - 1264*log(sqrt(35)*sqrt(3) - 9))*s 
gn(1/(2*x + 3)) + 1/2450*((1093/sgn(1/(2*x + 3)) - 1820/((2*x + 3)*sgn(1/( 
2*x + 3))))/(2*x + 3) + 57/sgn(1/(2*x + 3)))/sqrt(-18/(2*x + 3) + 35/(2*x 
+ 3)^2 + 3) - 632/42875*sqrt(35)*log(sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2* 
x + 3)^2 + 3) + sqrt(35)/(2*x + 3)) - 9)/sgn(1/(2*x + 3))
 

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.09 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {632\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{42875}-\frac {632\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{42875}+\frac {71\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{4900\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {71\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{4900\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {26\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1225\,\left (x+\frac {3}{2}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,199{}\mathrm {i}}{14700\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,199{}\mathrm {i}}{14700\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \] Input:

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

Output:

(632*35^(1/2)*log(x + 3/2))/42875 - (632*35^(1/2)*log(x - (3^(1/2)*35^(1/2 
)*(x^2 + 2/3)^(1/2))/9 - 4/9))/42875 + (71*3^(1/2)*(x^2 + 2/3)^(1/2))/(490 
0*(x - (6^(1/2)*1i)/3)) + (71*3^(1/2)*(x^2 + 2/3)^(1/2))/(4900*(x + (6^(1/ 
2)*1i)/3)) - (26*3^(1/2)*(x^2 + 2/3)^(1/2))/(1225*(x + 3/2)) - (3^(1/2)*6^ 
(1/2)*(x^2 + 2/3)^(1/2)*199i)/(14700*(x - (6^(1/2)*1i)/3)) + (3^(1/2)*6^(1 
/2)*(x^2 + 2/3)^(1/2)*199i)/(14700*(x + (6^(1/2)*1i)/3))
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.55 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {3990 \sqrt {3 x^{2}+2}\, x^{2}+50225 \sqrt {3 x^{2}+2}\, x +34510 \sqrt {3 x^{2}+2}+7584 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{3}+11376 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{2}+5056 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x +7584 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right )-7584 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{3}-11376 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{2}-5056 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x -7584 \sqrt {35}\, \mathrm {log}\left (2 x +3\right )}{514500 x^{3}+771750 x^{2}+343000 x +514500} \] Input:

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

Output:

(3990*sqrt(3*x**2 + 2)*x**2 + 50225*sqrt(3*x**2 + 2)*x + 34510*sqrt(3*x**2 
 + 2) + 7584*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**3 + 1137 
6*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**2 + 5056*sqrt(35)*l 
og(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x + 7584*sqrt(35)*log(sqrt(3*x**2 
+ 2)*sqrt(35) + 9*x - 4) - 7584*sqrt(35)*log(2*x + 3)*x**3 - 11376*sqrt(35 
)*log(2*x + 3)*x**2 - 5056*sqrt(35)*log(2*x + 3)*x - 7584*sqrt(35)*log(2*x 
 + 3))/(85750*(6*x**3 + 9*x**2 + 4*x + 6))