\(\int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^2} \, dx\) [201]

Optimal result

Integrand size = 24, antiderivative size = 73 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^2} \, dx=-\frac {(8+x) \sqrt {2+3 x^2}}{2 (3+2 x)}+2 \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )+\frac {19 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{\sqrt {35}} \] Output:

-1/2*(8+x)*(3*x^2+2)^(1/2)/(3+2*x)+2*arcsinh(1/2*x*6^(1/2))*3^(1/2)+19/35* 
35^(1/2)*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))
 

Mathematica [A] (verified)

Time = 0.95 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.30 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^2} \, dx=-\frac {(8+x) \sqrt {2+3 x^2}}{6+4 x}-\frac {38 \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{\sqrt {35}}-2 \sqrt {3} \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right ) \] Input:

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

Output:

-(((8 + x)*Sqrt[2 + 3*x^2])/(6 + 4*x)) - (38*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3 
]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]])/Sqrt[35] - 2*Sqrt[3]*Log[-(Sqrt[3]*x) 
+ Sqrt[2 + 3*x^2]]
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {681, 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.

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

\(\Big \downarrow \) 681

\(\displaystyle -\frac {1}{8} \int \frac {8 (1-12 x)}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {\sqrt {3 x^2+2} (x+8)}{2 (2 x+3)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\int \frac {1-12 x}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {\sqrt {3 x^2+2} (x+8)}{2 (2 x+3)}\)

\(\Big \downarrow \) 719

\(\displaystyle 6 \int \frac {1}{\sqrt {3 x^2+2}}dx-19 \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {\sqrt {3 x^2+2} (x+8)}{2 (2 x+3)}\)

\(\Big \downarrow \) 222

\(\displaystyle -19 \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx+2 \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-\frac {\sqrt {3 x^2+2} (x+8)}{2 (2 x+3)}\)

\(\Big \downarrow \) 488

\(\displaystyle 19 \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}+2 \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-\frac {\sqrt {3 x^2+2} (x+8)}{2 (2 x+3)}\)

\(\Big \downarrow \) 219

\(\displaystyle 2 \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )+\frac {19 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{\sqrt {35}}-\frac {\sqrt {3 x^2+2} (x+8)}{2 (2 x+3)}\)

Input:

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

Output:

-1/2*((8 + x)*Sqrt[2 + 3*x^2])/(3 + 2*x) + 2*Sqrt[3]*ArcSinh[Sqrt[3/2]*x] 
+ (19*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/Sqrt[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 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 681

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) 
 + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/ 
(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Sim 
p[g*(2*a*e + 2*a*e*m) + (g*(2*c*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x] 
, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, -1] || 
EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ[m + 2 
*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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 = 0.94 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.05

method	result	size
risch	\(-\frac {3 x^{3}+24 x^{2}+2 x +16}{2 \left (2 x +3\right ) \sqrt {3 x^{2}+2}}+2 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}+\frac {19 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{35}\)	\(77\)
default	\(-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{70 \left (x +\frac {3}{2}\right )}-\frac {19 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{35}+2 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}+\frac {19 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{35}+\frac {39 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{70}\)	\(98\)
trager	\(-\frac {\left (8+x \right ) \sqrt {3 x^{2}+2}}{2 \left (2 x +3\right )}-\frac {19 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )+35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{35}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )\)	\(98\)

Input:

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

Output:

-1/2*(3*x^3+24*x^2+2*x+16)/(2*x+3)/(3*x^2+2)^(1/2)+2*arcsinh(1/2*6^(1/2)*x 
)*3^(1/2)+19/35*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x- 
19)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.49 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^2} \, dx=\frac {70 \, \sqrt {3} {\left (2 \, x + 3\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 19 \, \sqrt {35} {\left (2 \, x + 3\right )} \log \left (\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} - 93 \, x^{2} + 36 \, x - 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \, \sqrt {3 \, x^{2} + 2} {\left (x + 8\right )}}{70 \, {\left (2 \, x + 3\right )}} \] Input:

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

Output:

1/70*(70*sqrt(3)*(2*x + 3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 1 
9*sqrt(35)*(2*x + 3)*log((sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) - 93*x^2 + 36 
*x - 43)/(4*x^2 + 12*x + 9)) - 35*sqrt(3*x^2 + 2)*(x + 8))/(2*x + 3)
 

Sympy [F]

\[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^2} \, dx=- \int \left (- \frac {5 \sqrt {3 x^{2} + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \frac {x \sqrt {3 x^{2} + 2}}{4 x^{2} + 12 x + 9}\, dx \] Input:

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

Output:

-Integral(-5*sqrt(3*x**2 + 2)/(4*x**2 + 12*x + 9), x) - Integral(x*sqrt(3* 
x**2 + 2)/(4*x**2 + 12*x + 9), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.04 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^2} \, dx=2 \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {19}{35} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) - \frac {1}{4} \, \sqrt {3 \, x^{2} + 2} - \frac {13 \, \sqrt {3 \, x^{2} + 2}}{4 \, {\left (2 \, x + 3\right )}} \] Input:

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

Output:

2*sqrt(3)*arcsinh(1/2*sqrt(6)*x) - 19/35*sqrt(35)*arcsinh(3/2*sqrt(6)*x/ab 
s(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 1/4*sqrt(3*x^2 + 2) - 13/4*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. 285 vs. \(2 (59) = 118\).

Time = 0.26 (sec) , antiderivative size = 285, normalized size of antiderivative = 3.90 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^2} \, dx=\frac {19}{35} \, \sqrt {35} \log \left (\sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} - 9\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 2 \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {35}}{2 \, x + 3} \right |}}{2 \, {\left (\sqrt {3} + \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {13}{8} \, \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + \frac {3 \, {\left (3 \, {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \sqrt {35} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{4 \, {\left ({\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{2} - 3\right )}} \] Input:

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

Output:

19/35*sqrt(35)*log(sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sq 
rt(35)/(2*x + 3)) - 9)*sgn(1/(2*x + 3)) - 2*sqrt(3)*log(1/2*abs(-2*sqrt(3) 
 + 2*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + 2*sqrt(35)/(2*x + 3))/(sqr 
t(3) + sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)))*sgn 
(1/(2*x + 3)) - 13/8*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3)*sgn(1/(2*x + 
 3)) + 3/4*(3*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 
3))*sgn(1/(2*x + 3)) - sqrt(35)*sgn(1/(2*x + 3)))/((sqrt(-18/(2*x + 3) + 3 
5/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^2 - 3)
 

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.10 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^2} \, dx=2\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{4}-\frac {19\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{35}+\frac {19\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{35}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{8\,\left (x+\frac {3}{2}\right )} \] Input:

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

Output:

2*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2) - (3^(1/2)*(x^2 + 2/3)^(1/2))/4 - ( 
19*35^(1/2)*log(x + 3/2))/35 + (19*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 
 + 2/3)^(1/2))/9 - 4/9))/35 - (13*3^(1/2)*(x^2 + 2/3)^(1/2))/(8*(x + 3/2))
 

Reduce [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.36 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^2} \, dx=\frac {-35 \sqrt {3 x^{2}+2}\, x -280 \sqrt {3 x^{2}+2}+76 \sqrt {35}\, \mathrm {log}\left (-\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x +114 \sqrt {35}\, \mathrm {log}\left (-\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right )-76 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x -114 \sqrt {35}\, \mathrm {log}\left (2 x +3\right )-140 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}-\sqrt {3}\, x \right ) x -210 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}-\sqrt {3}\, x \right )+140 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}+\sqrt {3}\, x \right ) x +210 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}+\sqrt {3}\, x \right )}{140 x +210} \] Input:

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

Output:

( - 35*sqrt(3*x**2 + 2)*x - 280*sqrt(3*x**2 + 2) + 76*sqrt(35)*log( - sqrt 
(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x + 114*sqrt(35)*log( - sqrt(3*x**2 + 2)* 
sqrt(35) + 9*x - 4) - 76*sqrt(35)*log(2*x + 3)*x - 114*sqrt(35)*log(2*x + 
3) - 140*sqrt(3)*log(sqrt(3*x**2 + 2) - sqrt(3)*x)*x - 210*sqrt(3)*log(sqr 
t(3*x**2 + 2) - sqrt(3)*x) + 140*sqrt(3)*log(sqrt(3*x**2 + 2) + sqrt(3)*x) 
*x + 210*sqrt(3)*log(sqrt(3*x**2 + 2) + sqrt(3)*x))/(70*(2*x + 3))