\(\int \frac {(5-x) (2+3 x^2)^{3/2}}{3+2 x} \, dx\) [212]

Optimal result

Integrand size = 24, antiderivative size = 92 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{3+2 x} \, dx=\frac {1}{16} (455-123 x) \sqrt {2+3 x^2}+\frac {1}{24} (26-3 x) \left (2+3 x^2\right )^{3/2}-\frac {1529}{32} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-\frac {455}{32} \sqrt {35} \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right ) \] Output:

1/16*(455-123*x)*(3*x^2+2)^(1/2)+1/24*(26-3*x)*(3*x^2+2)^(3/2)-1529/32*arc 
sinh(1/2*x*6^(1/2))*3^(1/2)-455/32*35^(1/2)*arctanh(1/35*(4-9*x)*35^(1/2)/ 
(3*x^2+2)^(1/2))
 

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.13 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{3+2 x} \, dx=\frac {1}{96} \left (-2 \sqrt {2+3 x^2} \left (-1469+381 x-156 x^2+18 x^3\right )+2730 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )+4587 \sqrt {3} \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )\right ) \] Input:

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

Output:

(-2*Sqrt[2 + 3*x^2]*(-1469 + 381*x - 156*x^2 + 18*x^3) + 2730*Sqrt[35]*Arc 
Tanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]] + 4587*Sqrt[3 
]*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/96
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {682, 27, 682, 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.

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

\(\Big \downarrow \) 682

\(\displaystyle \frac {1}{48} \int \frac {12 (43-123 x) \sqrt {3 x^2+2}}{2 x+3}dx+\frac {1}{24} (26-3 x) \left (3 x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \int \frac {(43-123 x) \sqrt {3 x^2+2}}{2 x+3}dx+\frac {1}{24} (26-3 x) \left (3 x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 682

\(\displaystyle \frac {1}{4} \left (\frac {1}{24} \int \frac {6 (1082-4587 x)}{(2 x+3) \sqrt {3 x^2+2}}dx+\frac {1}{4} \sqrt {3 x^2+2} (455-123 x)\right )+\frac {1}{24} (26-3 x) \left (3 x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (\frac {1}{4} \int \frac {1082-4587 x}{(2 x+3) \sqrt {3 x^2+2}}dx+\frac {1}{4} \sqrt {3 x^2+2} (455-123 x)\right )+\frac {1}{24} (26-3 x) \left (3 x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {1}{4} \left (\frac {1}{4} \left (\frac {15925}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {4587}{2} \int \frac {1}{\sqrt {3 x^2+2}}dx\right )+\frac {1}{4} \sqrt {3 x^2+2} (455-123 x)\right )+\frac {1}{24} (26-3 x) \left (3 x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{4} \left (\frac {1}{4} \left (\frac {15925}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {1529}{2} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )\right )+\frac {1}{4} \sqrt {3 x^2+2} (455-123 x)\right )+\frac {1}{24} (26-3 x) \left (3 x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {1}{4} \left (\frac {1}{4} \left (-\frac {15925}{2} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {1529}{2} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )\right )+\frac {1}{4} \sqrt {3 x^2+2} (455-123 x)\right )+\frac {1}{24} (26-3 x) \left (3 x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{4} \left (\frac {1}{4} \left (-\frac {1529}{2} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-\frac {455}{2} \sqrt {35} \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )\right )+\frac {1}{4} \sqrt {3 x^2+2} (455-123 x)\right )+\frac {1}{24} (26-3 x) \left (3 x^2+2\right )^{3/2}\)

Input:

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

Output:

((26 - 3*x)*(2 + 3*x^2)^(3/2))/24 + (((455 - 123*x)*Sqrt[2 + 3*x^2])/4 + ( 
(-1529*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/2 - (455*Sqrt[35]*ArcTanh[(4 - 9*x)/( 
Sqrt[35]*Sqrt[2 + 3*x^2])])/2)/4)/4
 

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 682

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p 
+ 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p 
+ 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)))   Int[(d + e*x) 
^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* 
d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x 
], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  ! 
RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Intege 
rQ[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 = 0.93 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.76

method	result	size
risch	\(-\frac {\left (18 x^{3}-156 x^{2}+381 x -1469\right ) \sqrt {3 x^{2}+2}}{48}-\frac {1529 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{32}-\frac {455 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{32}\)	\(70\)
trager	\(\left (-\frac {3}{8} x^{3}+\frac {13}{4} x^{2}-\frac {127}{16} x +\frac {1469}{48}\right ) \sqrt {3 x^{2}+2}+\frac {455 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )+35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{32}-\frac {11 \operatorname {RootOf}\left (\textit {\_Z}^{2}-57963\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-57963\right ) \sqrt {3 x^{2}+2}+417 x \right )}{32}\)	\(102\)
default	\(-\frac {x \left (3 x^{2}+2\right )^{\frac {3}{2}}}{8}-\frac {3 x \sqrt {3 x^{2}+2}}{8}-\frac {1529 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{32}+\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{12}-\frac {117 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{16}+\frac {455 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{32}-\frac {455 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{32}\)	\(117\)

Input:

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

Output:

-1/48*(18*x^3-156*x^2+381*x-1469)*(3*x^2+2)^(1/2)-1529/32*arcsinh(1/2*6^(1 
/2)*x)*3^(1/2)-455/32*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 = 102, normalized size of antiderivative = 1.11 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{3+2 x} \, dx=-\frac {1}{48} \, {\left (18 \, x^{3} - 156 \, x^{2} + 381 \, x - 1469\right )} \sqrt {3 \, x^{2} + 2} + \frac {1529}{64} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + \frac {455}{64} \, \sqrt {35} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) \] Input:

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

Output:

-1/48*(18*x^3 - 156*x^2 + 381*x - 1469)*sqrt(3*x^2 + 2) + 1529/64*sqrt(3)* 
log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 455/64*sqrt(35)*log(-(sqrt(35 
)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9))
 

Sympy [F]

\[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{3+2 x} \, dx=- \int \left (- \frac {10 \sqrt {3 x^{2} + 2}}{2 x + 3}\right )\, dx - \int \frac {2 x \sqrt {3 x^{2} + 2}}{2 x + 3}\, dx - \int \left (- \frac {15 x^{2} \sqrt {3 x^{2} + 2}}{2 x + 3}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 2}}{2 x + 3}\, dx \] Input:

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

Output:

-Integral(-10*sqrt(3*x**2 + 2)/(2*x + 3), x) - Integral(2*x*sqrt(3*x**2 + 
2)/(2*x + 3), x) - Integral(-15*x**2*sqrt(3*x**2 + 2)/(2*x + 3), x) - Inte 
gral(3*x**3*sqrt(3*x**2 + 2)/(2*x + 3), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.01 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{3+2 x} \, dx=-\frac {1}{8} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {13}{12} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {123}{16} \, \sqrt {3 \, x^{2} + 2} x - \frac {1529}{32} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {455}{32} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {455}{16} \, \sqrt {3 \, x^{2} + 2} \] Input:

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

Output:

-1/8*(3*x^2 + 2)^(3/2)*x + 13/12*(3*x^2 + 2)^(3/2) - 123/16*sqrt(3*x^2 + 2 
)*x - 1529/32*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 455/32*sqrt(35)*arcsinh(3/2 
*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 455/16*sqrt(3*x^2 + 
2)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.26 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{3+2 x} \, dx=-\frac {1}{48} \, {\left (3 \, {\left (2 \, {\left (3 \, x - 26\right )} x + 127\right )} x - 1469\right )} \sqrt {3 \, x^{2} + 2} + \frac {1529}{32} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {455}{32} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) \] Input:

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

Output:

-1/48*(3*(2*(3*x - 26)*x + 127)*x - 1469)*sqrt(3*x^2 + 2) + 1529/32*sqrt(3 
)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 455/32*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)))
 

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.83 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{3+2 x} \, dx=\frac {\sqrt {35}\,\left (31850\,\ln \left (x+\frac {3}{2}\right )-31850\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )\right )}{2240}-\frac {1529\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{32}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {9\,x^3}{8}-\frac {39\,x^2}{4}+\frac {381\,x}{16}-\frac {1469}{16}\right )}{3} \] Input:

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

Output:

(35^(1/2)*(31850*log(x + 3/2) - 31850*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3 
)^(1/2))/9 - 4/9)))/2240 - (1529*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/32 
- (3^(1/2)*(x^2 + 2/3)^(1/2)*((381*x)/16 - (39*x^2)/4 + (9*x^3)/8 - 1469/1 
6))/3
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.80 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{3+2 x} \, dx=\frac {455 \sqrt {35}\, \mathit {atan} \left (\frac {2 \sqrt {3 x^{2}+2}\, i +2 \sqrt {3}\, i x}{\sqrt {35}-3 \sqrt {3}}\right ) i}{32}-\frac {3 \sqrt {3 x^{2}+2}\, x^{3}}{8}+\frac {13 \sqrt {3 x^{2}+2}\, x^{2}}{4}-\frac {127 \sqrt {3 x^{2}+2}\, x}{16}+\frac {1469 \sqrt {3 x^{2}+2}}{48}+\frac {455 \sqrt {35}\, \mathrm {log}\left (4 \sqrt {3 x^{2}+2}\, \sqrt {3}\, x +3 \sqrt {105}+12 x^{2}-27\right )}{64}-\frac {455 \sqrt {35}\, \mathrm {log}\left (\frac {2 \sqrt {3 x^{2}+2}+\sqrt {35}+2 \sqrt {3}\, x +3 \sqrt {3}}{\sqrt {2}}\right )}{32}-\frac {1529 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right )}{32} \] Input:

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

Output:

(2730*sqrt(35)*atan((2*sqrt(3*x**2 + 2)*i + 2*sqrt(3)*i*x)/(sqrt(35) - 3*s 
qrt(3)))*i - 72*sqrt(3*x**2 + 2)*x**3 + 624*sqrt(3*x**2 + 2)*x**2 - 1524*s 
qrt(3*x**2 + 2)*x + 5876*sqrt(3*x**2 + 2) + 1365*sqrt(35)*log(4*sqrt(3*x** 
2 + 2)*sqrt(3)*x + 3*sqrt(105) + 12*x**2 - 27) - 2730*sqrt(35)*log((2*sqrt 
(3*x**2 + 2) + sqrt(35) + 2*sqrt(3)*x + 3*sqrt(3))/sqrt(2)) - 9174*sqrt(3) 
*log((sqrt(3*x**2 + 2) + sqrt(3)*x)/sqrt(2)))/192