\(\int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^4} \, dx\) [203]

Optimal result

Integrand size = 24, antiderivative size = 82 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^4} \, dx=-\frac {41 (4-9 x) \sqrt {2+3 x^2}}{2450 (3+2 x)^2}-\frac {13 \left (2+3 x^2\right )^{3/2}}{105 (3+2 x)^3}-\frac {123 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{1225 \sqrt {35}} \] Output:

-41/2450*(4-9*x)*(3*x^2+2)^(1/2)/(3+2*x)^2-13/105*(3*x^2+2)^(3/2)/(3+2*x)^ 
3-123/42875*35^(1/2)*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))
 

Mathematica [A] (verified)

Time = 1.36 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.95 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^4} \, dx=-\frac {\sqrt {2+3 x^2} \left (3296-2337 x+516 x^2\right )}{7350 (3+2 x)^3}+\frac {246 \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{1225 \sqrt {35}} \] Input:

Integrate[((5 - x)*Sqrt[2 + 3*x^2])/(3 + 2*x)^4,x]
 

Output:

-1/7350*(Sqrt[2 + 3*x^2]*(3296 - 2337*x + 516*x^2))/(3 + 2*x)^3 + (246*Arc 
Tanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]])/(1225*Sqrt[3 
5])
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {679, 486, 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.

\(\displaystyle \int \frac {(5-x) \sqrt {3 x^2+2}}{(2 x+3)^4} \, dx\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {41}{35} \int \frac {\sqrt {3 x^2+2}}{(2 x+3)^3}dx-\frac {13 \left (3 x^2+2\right )^{3/2}}{105 (2 x+3)^3}\)

\(\Big \downarrow \) 486

\(\displaystyle \frac {41}{35} \left (\frac {3}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {(4-9 x) \sqrt {3 x^2+2}}{70 (2 x+3)^2}\right )-\frac {13 \left (3 x^2+2\right )^{3/2}}{105 (2 x+3)^3}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {41}{35} \left (-\frac {3}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {13 \left (3 x^2+2\right )^{3/2}}{105 (2 x+3)^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {41}{35} \left (-\frac {3 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {13 \left (3 x^2+2\right )^{3/2}}{105 (2 x+3)^3}\)

Input:

Int[((5 - x)*Sqrt[2 + 3*x^2])/(3 + 2*x)^4,x]
 

Output:

(-13*(2 + 3*x^2)^(3/2))/(105*(3 + 2*x)^3) + (41*(-1/70*((4 - 9*x)*Sqrt[2 + 
 3*x^2])/(3 + 2*x)^2 - (3*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/( 
35*Sqrt[35])))/35
 

Defintions of rubi rules used

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 486

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(c + d*x)^(n + 1)*(a*d - b*c*x)*((a + b*x^2)^p/((n + 1)*(b*c^2 + a*d^2))), 
x] - Simp[2*a*b*(p/((n + 1)*(b*c^2 + a*d^2)))   Int[(c + d*x)^(n + 2)*(a + 
b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[n + 2*p + 2, 0] && 
GtQ[p, 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]
 

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.85

method	result	size
risch	\(-\frac {1548 x^{4}-7011 x^{3}+10920 x^{2}-4674 x +6592}{7350 \left (2 x +3\right )^{3} \sqrt {3 x^{2}+2}}-\frac {123 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{42875}\)	\(70\)
trager	\(-\frac {\left (516 x^{2}-2337 x +3296\right ) \sqrt {3 x^{2}+2}}{7350 \left (2 x +3\right )^{3}}-\frac {123 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (-\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )-35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{42875}\)	\(77\)
default	\(-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{840 \left (x +\frac {3}{2}\right )^{3}}-\frac {41 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{4900 \left (x +\frac {3}{2}\right )^{2}}-\frac {369 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{85750 \left (x +\frac {3}{2}\right )}+\frac {123 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{42875}-\frac {123 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{42875}+\frac {1107 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{85750}\)	\(128\)

Input:

int((5-x)*(3*x^2+2)^(1/2)/(2*x+3)^4,x,method=_RETURNVERBOSE)
 

Output:

-1/7350*(1548*x^4-7011*x^3+10920*x^2-4674*x+6592)/(2*x+3)^3/(3*x^2+2)^(1/2 
)-123/42875*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 = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.27 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^4} \, dx=\frac {369 \, \sqrt {35} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \, {\left (516 \, x^{2} - 2337 \, x + 3296\right )} \sqrt {3 \, x^{2} + 2}}{257250 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \] Input:

integrate((5-x)*(3*x^2+2)^(1/2)/(3+2*x)^4,x, algorithm="fricas")
 

Output:

1/257250*(369*sqrt(35)*(8*x^3 + 36*x^2 + 54*x + 27)*log(-(sqrt(35)*sqrt(3* 
x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(516*x^2 
 - 2337*x + 3296)*sqrt(3*x^2 + 2))/(8*x^3 + 36*x^2 + 54*x + 27)
 

Sympy [F]

\[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^4} \, dx=- \int \left (- \frac {5 \sqrt {3 x^{2} + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \frac {x \sqrt {3 x^{2} + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\, dx \] Input:

integrate((5-x)*(3*x**2+2)**(1/2)/(3+2*x)**4,x)
 

Output:

-Integral(-5*sqrt(3*x**2 + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), 
 x) - Integral(x*sqrt(3*x**2 + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 
81), x)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.40 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^4} \, dx=\frac {123}{42875} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {123}{4900} \, \sqrt {3 \, x^{2} + 2} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{105 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {41 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{1225 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {369 \, \sqrt {3 \, x^{2} + 2}}{4900 \, {\left (2 \, x + 3\right )}} \] Input:

integrate((5-x)*(3*x^2+2)^(1/2)/(3+2*x)^4,x, algorithm="maxima")
 

Output:

123/42875*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2* 
x + 3)) + 123/4900*sqrt(3*x^2 + 2) - 13/105*(3*x^2 + 2)^(3/2)/(8*x^3 + 36* 
x^2 + 54*x + 27) - 41/1225*(3*x^2 + 2)^(3/2)/(4*x^2 + 12*x + 9) - 369/4900 
*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. 232 vs. \(2 (67) = 134\).

Time = 0.14 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.83 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^4} \, dx=\frac {123}{42875} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) - \frac {\sqrt {3} {\left (1553 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} + 30 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 3870 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 25740 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 20 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 1376\right )}}{9800 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{3}} \] Input:

integrate((5-x)*(3*x^2+2)^(1/2)/(3+2*x)^4,x, algorithm="giac")
 

Output:

123/42875*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))) - 1/9 
800*sqrt(3)*(1553*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 30*(sqrt(3)*x 
- sqrt(3*x^2 + 2))^4 + 3870*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 2574 
0*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 20*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2 
)) - 1376)/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt( 
3*x^2 + 2)) - 2)^3
 

Mupad [B] (verification not implemented)

Time = 5.98 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.29 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^4} \, dx=\frac {123\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{42875}-\frac {123\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{42875}-\frac {43\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{4900\,\left (x+\frac {3}{2}\right )}+\frac {37\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{560\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{96\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \] Input:

int(-((3*x^2 + 2)^(1/2)*(x - 5))/(2*x + 3)^4,x)
 

Output:

(123*35^(1/2)*log(x + 3/2))/42875 - (123*35^(1/2)*log(x - (3^(1/2)*35^(1/2 
)*(x^2 + 2/3)^(1/2))/9 - 4/9))/42875 - (43*3^(1/2)*(x^2 + 2/3)^(1/2))/(490 
0*(x + 3/2)) + (37*3^(1/2)*(x^2 + 2/3)^(1/2))/(560*(3*x + x^2 + 9/4)) - (1 
3*3^(1/2)*(x^2 + 2/3)^(1/2))/(96*((27*x)/4 + (9*x^2)/2 + x^3 + 27/8))
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.33 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^4} \, dx=\frac {-18060 \sqrt {3 x^{2}+2}\, x^{2}+81795 \sqrt {3 x^{2}+2}\, x -115360 \sqrt {3 x^{2}+2}+5904 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{3}+26568 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{2}+39852 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x +19926 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right )-5904 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{3}-26568 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{2}-39852 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x -19926 \sqrt {35}\, \mathrm {log}\left (2 x +3\right )}{2058000 x^{3}+9261000 x^{2}[phone] x +6945750} \] Input:

int((5-x)*(3*x^2+2)^(1/2)/(3+2*x)^4,x)
 

Output:

( - 18060*sqrt(3*x**2 + 2)*x**2 + 81795*sqrt(3*x**2 + 2)*x - 115360*sqrt(3 
*x**2 + 2) + 5904*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**3 + 
 26568*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**2 + 39852*sqrt 
(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x + 19926*sqrt(35)*log(sqrt( 
3*x**2 + 2)*sqrt(35) + 9*x - 4) - 5904*sqrt(35)*log(2*x + 3)*x**3 - 26568* 
sqrt(35)*log(2*x + 3)*x**2 - 39852*sqrt(35)*log(2*x + 3)*x - 19926*sqrt(35 
)*log(2*x + 3))/(257250*(8*x**3 + 36*x**2 + 54*x + 27))