\(\int \frac {5-x}{(3+2 x) (2+3 x^2)^{5/2}} \, dx\) [259]

Optimal result

Integrand size = 24, antiderivative size = 73 \[ \int \frac {5-x}{(3+2 x) \left (2+3 x^2\right )^{5/2}} \, dx=\frac {26+41 x}{210 \left (2+3 x^2\right )^{3/2}}+\frac {312+2137 x}{7350 \sqrt {2+3 x^2}}-\frac {104 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{1225 \sqrt {35}} \] Output:

1/210*(26+41*x)/(3*x^2+2)^(3/2)+1/7350*(312+2137*x)/(3*x^2+2)^(1/2)-104/42 
875*35^(1/2)*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))
 

Mathematica [A] (verified)

Time = 1.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.86 \[ \int \frac {5-x}{(3+2 x) \left (2+3 x^2\right )^{5/2}} \, dx=\frac {\frac {35 \left (1534+5709 x+936 x^2+6411 x^3\right )}{\left (2+3 x^2\right )^{3/2}}-624 \sqrt {35} \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{257250} \] Input:

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

Output:

((35*(1534 + 5709*x + 936*x^2 + 6411*x^3))/(2 + 3*x^2)^(3/2) - 624*Sqrt[35 
]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/257250
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {686, 27, 686, 27, 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.

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

\(\Big \downarrow \) 686

\(\displaystyle \frac {41 x+26}{210 \left (3 x^2+2\right )^{3/2}}-\frac {1}{630} \int -\frac {6 (82 x+201)}{(2 x+3) \left (3 x^2+2\right )^{3/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{105} \int \frac {82 x+201}{(2 x+3) \left (3 x^2+2\right )^{3/2}}dx+\frac {41 x+26}{210 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 686

\(\displaystyle \frac {1}{105} \left (\frac {2137 x+312}{70 \sqrt {3 x^2+2}}-\frac {1}{210} \int -\frac {1872}{(2 x+3) \sqrt {3 x^2+2}}dx\right )+\frac {41 x+26}{210 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{105} \left (\frac {312}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx+\frac {2137 x+312}{70 \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {1}{105} \left (\frac {2137 x+312}{70 \sqrt {3 x^2+2}}-\frac {312}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{105} \left (\frac {2137 x+312}{70 \sqrt {3 x^2+2}}-\frac {312 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}\right )+\frac {41 x+26}{210 \left (3 x^2+2\right )^{3/2}}\)

Input:

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

Output:

(26 + 41*x)/(210*(2 + 3*x^2)^(3/2)) + ((312 + 2137*x)/(70*Sqrt[2 + 3*x^2]) 
 - (312*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(35*Sqrt[35]))/105
 

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 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 [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.88 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.03

method	result	size
trager	\(\frac {6411 x^{3}+936 x^{2}+5709 x +1534}{7350 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {104 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (-\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )-35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{42875}\)	\(75\)
default	\(-\frac {x}{12 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {x}{12 \sqrt {3 x^{2}+2}}+\frac {13}{105 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {39 x}{140 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {1833 x}{4900 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}+\frac {52}{1225 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}-\frac {104 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{42875}\)	\(122\)

Input:

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

Output:

1/7350*(6411*x^3+936*x^2+5709*x+1534)/(3*x^2+2)^(3/2)-104/42875*RootOf(_Z^ 
2-35)*ln(-(9*RootOf(_Z^2-35)*x-4*RootOf(_Z^2-35)-35*(3*x^2+2)^(1/2))/(2*x+ 
3))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.41 \[ \int \frac {5-x}{(3+2 x) \left (2+3 x^2\right )^{5/2}} \, dx=\frac {312 \, \sqrt {35} {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) + 35 \, {\left (6411 \, x^{3} + 936 \, x^{2} + 5709 \, x + 1534\right )} \sqrt {3 \, x^{2} + 2}}{257250 \, {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \] Input:

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

Output:

1/257250*(312*sqrt(35)*(9*x^4 + 12*x^2 + 4)*log(-(sqrt(35)*sqrt(3*x^2 + 2) 
*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) + 35*(6411*x^3 + 936* 
x^2 + 5709*x + 1534)*sqrt(3*x^2 + 2))/(9*x^4 + 12*x^2 + 4)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.11 \[ \int \frac {5-x}{(3+2 x) \left (2+3 x^2\right )^{5/2}} \, dx=\frac {104}{42875} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {2137 \, x}{7350 \, \sqrt {3 \, x^{2} + 2}} + \frac {52}{1225 \, \sqrt {3 \, x^{2} + 2}} + \frac {41 \, x}{210 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} + \frac {13}{105 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \] Input:

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

Output:

104/42875*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2* 
x + 3)) + 2137/7350*x/sqrt(3*x^2 + 2) + 52/1225/sqrt(3*x^2 + 2) + 41/210*x 
/(3*x^2 + 2)^(3/2) + 13/105/(3*x^2 + 2)^(3/2)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.27 \[ \int \frac {5-x}{(3+2 x) \left (2+3 x^2\right )^{5/2}} \, dx=\frac {104}{42875} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) + \frac {3 \, {\left ({\left (2137 \, x + 312\right )} x + 1903\right )} x + 1534}{7350 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \] Input:

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

Output:

104/42875*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3 
*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) + 1/7 
350*(3*((2137*x + 312)*x + 1903)*x + 1534)/(3*x^2 + 2)^(3/2)
 

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.99 \[ \int \frac {5-x}{(3+2 x) \left (2+3 x^2\right )^{5/2}} \, dx=\frac {\sqrt {35}\,\left (104\,\ln \left (x+\frac {3}{2}\right )-104\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )\right )}{42875}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {123}{560}+\frac {\sqrt {6}\,39{}\mathrm {i}}{560}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (-\frac {41}{280}+\frac {\sqrt {6}\,13{}\mathrm {i}}{280}\right )\,1{}\mathrm {i}}{2\,{\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {\frac {123}{560}+\frac {\sqrt {6}\,39{}\mathrm {i}}{560}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (\frac {41}{280}+\frac {\sqrt {6}\,13{}\mathrm {i}}{280}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-3744+\sqrt {6}\,7113{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1058400\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (3744+\sqrt {6}\,7113{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1058400\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \] Input:

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

Output:

(35^(1/2)*(104*log(x + 3/2) - 104*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1 
/2))/9 - 4/9)))/42875 - (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*39i)/560 - 1 
23/560)/(x - (6^(1/2)*1i)/3) - (6^(1/2)*((6^(1/2)*13i)/280 - 41/280)*1i)/( 
2*(x - (6^(1/2)*1i)/3)^2)))/27 + (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*39i 
)/560 + 123/560)/(x + (6^(1/2)*1i)/3) + (6^(1/2)*((6^(1/2)*13i)/280 + 41/2 
80)*1i)/(2*(x + (6^(1/2)*1i)/3)^2)))/27 - (3^(1/2)*6^(1/2)*(6^(1/2)*7113i 
- 3744)*(x^2 + 2/3)^(1/2)*1i)/(1058400*(x + (6^(1/2)*1i)/3)) - (3^(1/2)*6^ 
(1/2)*(6^(1/2)*7113i + 3744)*(x^2 + 2/3)^(1/2)*1i)/(1058400*(x - (6^(1/2)* 
1i)/3))
 

Reduce [B] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 384, normalized size of antiderivative = 5.26 \[ \int \frac {5-x}{(3+2 x) \left (2+3 x^2\right )^{5/2}} \, dx=\frac {16848 \sqrt {35}\, \mathit {atan} \left (\frac {2 \sqrt {3 x^{2}+2}\, i +2 \sqrt {3}\, i x}{\sqrt {35}-3 \sqrt {3}}\right ) i \,x^{4}+22464 \sqrt {35}\, \mathit {atan} \left (\frac {2 \sqrt {3 x^{2}+2}\, i +2 \sqrt {3}\, i x}{\sqrt {35}-3 \sqrt {3}}\right ) i \,x^{2}+7488 \sqrt {35}\, \mathit {atan} \left (\frac {2 \sqrt {3 x^{2}+2}\, i +2 \sqrt {3}\, i x}{\sqrt {35}-3 \sqrt {3}}\right ) i +673155 \sqrt {3 x^{2}+2}\, x^{3}+98280 \sqrt {3 x^{2}+2}\, x^{2}+599445 \sqrt {3 x^{2}+2}\, x +161070 \sqrt {3 x^{2}+2}+8424 \sqrt {35}\, \mathrm {log}\left (4 \sqrt {3 x^{2}+2}\, \sqrt {3}\, x +3 \sqrt {105}+12 x^{2}-27\right ) x^{4}+11232 \sqrt {35}\, \mathrm {log}\left (4 \sqrt {3 x^{2}+2}\, \sqrt {3}\, x +3 \sqrt {105}+12 x^{2}-27\right ) x^{2}+3744 \sqrt {35}\, \mathrm {log}\left (4 \sqrt {3 x^{2}+2}\, \sqrt {3}\, x +3 \sqrt {105}+12 x^{2}-27\right )-16848 \sqrt {35}\, \mathrm {log}\left (\frac {2 \sqrt {3 x^{2}+2}+\sqrt {35}+2 \sqrt {3}\, x +3 \sqrt {3}}{\sqrt {2}}\right ) x^{4}-22464 \sqrt {35}\, \mathrm {log}\left (\frac {2 \sqrt {3 x^{2}+2}+\sqrt {35}+2 \sqrt {3}\, x +3 \sqrt {3}}{\sqrt {2}}\right ) x^{2}-7488 \sqrt {35}\, \mathrm {log}\left (\frac {2 \sqrt {3 x^{2}+2}+\sqrt {35}+2 \sqrt {3}\, x +3 \sqrt {3}}{\sqrt {2}}\right )-525735 \sqrt {3}\, x^{4}-700980 \sqrt {3}\, x^{2}-233660 \sqrt {3}}{6945750 x^{4}+9261000 x^{2}+3087000} \] Input:

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

Output:

(16848*sqrt(35)*atan((2*sqrt(3*x**2 + 2)*i + 2*sqrt(3)*i*x)/(sqrt(35) - 3* 
sqrt(3)))*i*x**4 + 22464*sqrt(35)*atan((2*sqrt(3*x**2 + 2)*i + 2*sqrt(3)*i 
*x)/(sqrt(35) - 3*sqrt(3)))*i*x**2 + 7488*sqrt(35)*atan((2*sqrt(3*x**2 + 2 
)*i + 2*sqrt(3)*i*x)/(sqrt(35) - 3*sqrt(3)))*i + 673155*sqrt(3*x**2 + 2)*x 
**3 + 98280*sqrt(3*x**2 + 2)*x**2 + 599445*sqrt(3*x**2 + 2)*x + 161070*sqr 
t(3*x**2 + 2) + 8424*sqrt(35)*log(4*sqrt(3*x**2 + 2)*sqrt(3)*x + 3*sqrt(10 
5) + 12*x**2 - 27)*x**4 + 11232*sqrt(35)*log(4*sqrt(3*x**2 + 2)*sqrt(3)*x 
+ 3*sqrt(105) + 12*x**2 - 27)*x**2 + 3744*sqrt(35)*log(4*sqrt(3*x**2 + 2)* 
sqrt(3)*x + 3*sqrt(105) + 12*x**2 - 27) - 16848*sqrt(35)*log((2*sqrt(3*x** 
2 + 2) + sqrt(35) + 2*sqrt(3)*x + 3*sqrt(3))/sqrt(2))*x**4 - 22464*sqrt(35 
)*log((2*sqrt(3*x**2 + 2) + sqrt(35) + 2*sqrt(3)*x + 3*sqrt(3))/sqrt(2))*x 
**2 - 7488*sqrt(35)*log((2*sqrt(3*x**2 + 2) + sqrt(35) + 2*sqrt(3)*x + 3*s 
qrt(3))/sqrt(2)) - 525735*sqrt(3)*x**4 - 700980*sqrt(3)*x**2 - 233660*sqrt 
(3))/(771750*(9*x**4 + 12*x**2 + 4))