\(\int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^3} \, dx\) [202]

Optimal result

Integrand size = 24, antiderivative size = 79 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^3} \, dx=\frac {(53+187 x) \sqrt {2+3 x^2}}{140 (3+2 x)^2}-\frac {1}{8} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-\frac {471 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{280 \sqrt {35}} \] Output:

1/140*(53+187*x)*(3*x^2+2)^(1/2)/(3+2*x)^2-1/8*arcsinh(1/2*x*6^(1/2))*3^(1 
/2)-471/9800*35^(1/2)*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))
 

Mathematica [A] (verified)

Time = 0.93 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.30 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^3} \, dx=\frac {(53+187 x) \sqrt {2+3 x^2}}{140 (3+2 x)^2}+\frac {471 \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{140 \sqrt {35}}+\frac {1}{8} \sqrt {3} \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right ) \] Input:

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

Output:

((53 + 187*x)*Sqrt[2 + 3*x^2])/(140*(3 + 2*x)^2) + (471*ArcTanh[(3*Sqrt[3] 
 + 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]])/(140*Sqrt[35]) + (Sqrt[3]*L 
og[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/8
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {680, 27, 719, 222, 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.

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

\(\Big \downarrow \) 680

\(\displaystyle \frac {(187 x+53) \sqrt {3 x^2+2}}{140 (2 x+3)^2}-\frac {1}{560} \int -\frac {12 (26-35 x)}{(2 x+3) \sqrt {3 x^2+2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{140} \int \frac {26-35 x}{(2 x+3) \sqrt {3 x^2+2}}dx+\frac {\sqrt {3 x^2+2} (187 x+53)}{140 (2 x+3)^2}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {3}{140} \left (\frac {157}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {35}{2} \int \frac {1}{\sqrt {3 x^2+2}}dx\right )+\frac {\sqrt {3 x^2+2} (187 x+53)}{140 (2 x+3)^2}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {3}{140} \left (\frac {157}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {35 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}\right )+\frac {\sqrt {3 x^2+2} (187 x+53)}{140 (2 x+3)^2}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {3}{140} \left (-\frac {157}{2} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {35 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}\right )+\frac {\sqrt {3 x^2+2} (187 x+53)}{140 (2 x+3)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {3}{140} \left (-\frac {35 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}-\frac {157 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{2 \sqrt {35}}\right )+\frac {\sqrt {3 x^2+2} (187 x+53)}{140 (2 x+3)^2}\)

Input:

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

Output:

((53 + 187*x)*Sqrt[2 + 3*x^2])/(140*(3 + 2*x)^2) + (3*((-35*ArcSinh[Sqrt[3 
/2]*x])/(2*Sqrt[3]) - (157*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/ 
(2*Sqrt[35])))/140
 

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 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

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 680

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

rule 719

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

Maple [A] (verified)

Time = 0.98 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.97

method	result	size
risch	\(\frac {561 x^{3}+159 x^{2}+374 x +106}{140 \left (2 x +3\right )^{2} \sqrt {3 x^{2}+2}}-\frac {\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{8}-\frac {471 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{9800}\)	\(77\)
trager	\(\frac {\left (53+187 x \right ) \sqrt {3 x^{2}+2}}{140 \left (2 x +3\right )^{2}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{8}+\frac {471 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )+35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{9800}\)	\(100\)
default	\(-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{280 \left (x +\frac {3}{2}\right )^{2}}-\frac {47 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{4900 \left (x +\frac {3}{2}\right )}+\frac {471 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{9800}-\frac {471 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{9800}+\frac {141 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{4900}-\frac {\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{8}\)	\(119\)

Input:

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

Output:

1/140*(561*x^3+159*x^2+374*x+106)/(2*x+3)^2/(3*x^2+2)^(1/2)-1/8*arcsinh(1/ 
2*6^(1/2)*x)*3^(1/2)-471/9800*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*( 
x+3/2)^2-36*x-19)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (61) = 122\).

Time = 0.09 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.59 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^3} \, dx=\frac {1225 \, \sqrt {3} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 471 \, \sqrt {35} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) + 140 \, \sqrt {3 \, x^{2} + 2} {\left (187 \, x + 53\right )}}{19600 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \] Input:

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

Output:

1/19600*(1225*sqrt(3)*(4*x^2 + 12*x + 9)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3 
*x^2 - 1) + 471*sqrt(35)*(4*x^2 + 12*x + 9)*log(-(sqrt(35)*sqrt(3*x^2 + 2) 
*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) + 140*sqrt(3*x^2 + 2) 
*(187*x + 53))/(4*x^2 + 12*x + 9)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.25 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^3} \, dx=-\frac {1}{8} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {471}{9800} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {39}{280} \, \sqrt {3 \, x^{2} + 2} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{70 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {47 \, \sqrt {3 \, x^{2} + 2}}{280 \, {\left (2 \, x + 3\right )}} \] Input:

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

Output:

-1/8*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 471/9800*sqrt(35)*arcsinh(3/2*sqrt(6 
)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 39/280*sqrt(3*x^2 + 2) - 13 
/70*(3*x^2 + 2)^(3/2)/(4*x^2 + 12*x + 9) - 47/280*sqrt(3*x^2 + 2)/(2*x + 3 
)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (61) = 122\).

Time = 0.14 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.59 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^3} \, dx=\frac {1}{8} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {471}{9800} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) + \frac {3048 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} + 4301 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 7368 \, \sqrt {3} x + 1496 \, \sqrt {3} + 7368 \, \sqrt {3 \, x^{2} + 2}}{560 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{2}} \] Input:

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

Output:

1/8*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 471/9800*sqrt(35)*log(-abs 
(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - s 
qrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) + 1/560*(3048*(sqrt(3)*x - sqrt( 
3*x^2 + 2))^3 + 4301*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 7368*sqrt(3 
)*x + 1496*sqrt(3) + 7368*sqrt(3*x^2 + 2))/((sqrt(3)*x - sqrt(3*x^2 + 2))^ 
2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^2
 

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.16 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^3} \, dx=\frac {471\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{9800}-\frac {\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{8}-\frac {471\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{9800}+\frac {187\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{560\,\left (x+\frac {3}{2}\right )}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{32\,\left (x^2+3\,x+\frac {9}{4}\right )} \] Input:

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

Output:

(471*35^(1/2)*log(x + 3/2))/9800 - (3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/ 
8 - (471*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/9 
800 + (187*3^(1/2)*(x^2 + 2/3)^(1/2))/(560*(x + 3/2)) - (13*3^(1/2)*(x^2 + 
 2/3)^(1/2))/(32*(3*x + x^2 + 9/4))
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 255, normalized size of antiderivative = 3.23 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^3} \, dx=\frac {26180 \sqrt {3 x^{2}+2}\, x +7420 \sqrt {3 x^{2}+2}+3768 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{2}+11304 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x +8478 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right )-3768 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{2}-11304 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x -8478 \sqrt {35}\, \mathrm {log}\left (2 x +3\right )+4900 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}-\sqrt {3}\, x \right ) x^{2}+14700 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}-\sqrt {3}\, x \right ) x +11025 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}-\sqrt {3}\, x \right )-4900 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}+\sqrt {3}\, x \right ) x^{2}-14700 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}+\sqrt {3}\, x \right ) x -11025 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}+\sqrt {3}\, x \right )}{78400 x^{2}+235200 x +176400} \] Input:

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

Output:

(26180*sqrt(3*x**2 + 2)*x + 7420*sqrt(3*x**2 + 2) + 3768*sqrt(35)*log(sqrt 
(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**2 + 11304*sqrt(35)*log(sqrt(3*x**2 + 2 
)*sqrt(35) + 9*x - 4)*x + 8478*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9* 
x - 4) - 3768*sqrt(35)*log(2*x + 3)*x**2 - 11304*sqrt(35)*log(2*x + 3)*x - 
 8478*sqrt(35)*log(2*x + 3) + 4900*sqrt(3)*log(sqrt(3*x**2 + 2) - sqrt(3)* 
x)*x**2 + 14700*sqrt(3)*log(sqrt(3*x**2 + 2) - sqrt(3)*x)*x + 11025*sqrt(3 
)*log(sqrt(3*x**2 + 2) - sqrt(3)*x) - 4900*sqrt(3)*log(sqrt(3*x**2 + 2) + 
sqrt(3)*x)*x**2 - 14700*sqrt(3)*log(sqrt(3*x**2 + 2) + sqrt(3)*x)*x - 1102 
5*sqrt(3)*log(sqrt(3*x**2 + 2) + sqrt(3)*x))/(19600*(4*x**2 + 12*x + 9))