\(\int \frac {(5-x) (2+3 x^2)^{5/2}}{(3+2 x)^6} \, dx\) [230]

Optimal result

Integrand size = 24, antiderivative size = 133 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx=-\frac {9 (8575+2643 x) \sqrt {2+3 x^2}}{19600 (3+2 x)}+\frac {(6637+8193 x) \left (2+3 x^2\right )^{3/2}}{9800 (3+2 x)^3}+\frac {(23+76 x) \left (2+3 x^2\right )^{5/2}}{140 (3+2 x)^5}+\frac {63}{32} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )+\frac {789723 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{39200 \sqrt {35}} \] Output:

-9*(8575+2643*x)*(3*x^2+2)^(1/2)/(58800+39200*x)+1/9800*(6637+8193*x)*(3*x 
^2+2)^(3/2)/(3+2*x)^3+1/140*(23+76*x)*(3*x^2+2)^(5/2)/(3+2*x)^5+63/32*arcs 
inh(1/2*x*6^(1/2))*3^(1/2)+789723/1372000*35^(1/2)*arctanh(1/35*(4-9*x)*35 
^(1/2)/(3*x^2+2)^(1/2))
 

Mathematica [A] (verified)

Time = 1.74 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.92 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx=-\frac {\sqrt {2+3 x^2} \left (5999363+17940463 x+20911298 x^2+11367738 x^3+2740188 x^4+88200 x^5\right )}{19600 (3+2 x)^5}-\frac {789723 \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{19600 \sqrt {35}}-\frac {63}{32} \sqrt {3} \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right ) \] Input:

Integrate[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^6,x]
 

Output:

-1/19600*(Sqrt[2 + 3*x^2]*(5999363 + 17940463*x + 20911298*x^2 + 11367738* 
x^3 + 2740188*x^4 + 88200*x^5))/(3 + 2*x)^5 - (789723*ArcTanh[(3*Sqrt[3] + 
 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]])/(19600*Sqrt[35]) - (63*Sqrt[3 
]*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/32
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {680, 27, 680, 27, 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.

\(\displaystyle \int \frac {(5-x) \left (3 x^2+2\right )^{5/2}}{(2 x+3)^6} \, dx\)

\(\Big \downarrow \) 680

\(\displaystyle \frac {(76 x+23) \left (3 x^2+2\right )^{5/2}}{140 (2 x+3)^5}-\frac {\int -\frac {24 (52-73 x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^4}dx}{1120}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{140} \int \frac {(52-73 x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^4}dx+\frac {(76 x+23) \left (3 x^2+2\right )^{5/2}}{140 (2 x+3)^5}\)

\(\Big \downarrow \) 680

\(\displaystyle \frac {3}{140} \left (\frac {(8193 x+6637) \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^3}-\frac {1}{560} \int -\frac {24 (646-2643 x) \sqrt {3 x^2+2}}{(2 x+3)^2}dx\right )+\frac {(76 x+23) \left (3 x^2+2\right )^{5/2}}{140 (2 x+3)^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{140} \left (\frac {3}{70} \int \frac {(646-2643 x) \sqrt {3 x^2+2}}{(2 x+3)^2}dx+\frac {(8193 x+6637) \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^3}\right )+\frac {(76 x+23) \left (3 x^2+2\right )^{5/2}}{140 (2 x+3)^5}\)

\(\Big \downarrow \) 681

\(\displaystyle \frac {3}{140} \left (\frac {3}{70} \left (-\frac {1}{8} \int \frac {12 (1762-8575 x)}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {\sqrt {3 x^2+2} (2643 x+8575)}{2 (2 x+3)}\right )+\frac {(8193 x+6637) \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^3}\right )+\frac {(76 x+23) \left (3 x^2+2\right )^{5/2}}{140 (2 x+3)^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{140} \left (\frac {3}{70} \left (-\frac {3}{2} \int \frac {1762-8575 x}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {\sqrt {3 x^2+2} (2643 x+8575)}{2 (2 x+3)}\right )+\frac {(8193 x+6637) \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^3}\right )+\frac {(76 x+23) \left (3 x^2+2\right )^{5/2}}{140 (2 x+3)^5}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {3}{140} \left (\frac {3}{70} \left (-\frac {3}{2} \left (\frac {29249}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {8575}{2} \int \frac {1}{\sqrt {3 x^2+2}}dx\right )-\frac {\sqrt {3 x^2+2} (2643 x+8575)}{2 (2 x+3)}\right )+\frac {(8193 x+6637) \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^3}\right )+\frac {(76 x+23) \left (3 x^2+2\right )^{5/2}}{140 (2 x+3)^5}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {3}{140} \left (\frac {3}{70} \left (-\frac {3}{2} \left (\frac {29249}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {8575 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}\right )-\frac {\sqrt {3 x^2+2} (2643 x+8575)}{2 (2 x+3)}\right )+\frac {(8193 x+6637) \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^3}\right )+\frac {(76 x+23) \left (3 x^2+2\right )^{5/2}}{140 (2 x+3)^5}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {3}{140} \left (\frac {3}{70} \left (-\frac {3}{2} \left (-\frac {29249}{2} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {8575 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}\right )-\frac {\sqrt {3 x^2+2} (2643 x+8575)}{2 (2 x+3)}\right )+\frac {(8193 x+6637) \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^3}\right )+\frac {(76 x+23) \left (3 x^2+2\right )^{5/2}}{140 (2 x+3)^5}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {3}{140} \left (\frac {3}{70} \left (-\frac {3}{2} \left (-\frac {8575 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}-\frac {29249 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{2 \sqrt {35}}\right )-\frac {\sqrt {3 x^2+2} (2643 x+8575)}{2 (2 x+3)}\right )+\frac {(8193 x+6637) \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^3}\right )+\frac {(76 x+23) \left (3 x^2+2\right )^{5/2}}{140 (2 x+3)^5}\)

Input:

Int[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^6,x]
 

Output:

((23 + 76*x)*(2 + 3*x^2)^(5/2))/(140*(3 + 2*x)^5) + (3*(((6637 + 8193*x)*( 
2 + 3*x^2)^(3/2))/(210*(3 + 2*x)^3) + (3*(-1/2*((8575 + 2643*x)*Sqrt[2 + 3 
*x^2])/(3 + 2*x) - (3*((-8575*ArcSinh[Sqrt[3/2]*x])/(2*Sqrt[3]) - (29249*A 
rcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(2*Sqrt[35])))/2))/70))/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 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 = 0.97 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.73

method	result	size
risch	\(-\frac {264600 x^{7}+8220564 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]}{19600 \left (2 x +3\right )^{5} \sqrt {3 x^{2}+2}}+\frac {63 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{32}+\frac {789723 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{1372000}\)	\(97\)
trager	\(-\frac {\left (88200 x^{5}+2740188 x^{4}[phone] x^{3}[phone] x^{2}[phone] x +5999363\right ) \sqrt {3 x^{2}+2}}{19600 \left (2 x +3\right )^{5}}-\frac {789723 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )+35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{1372000}+\frac {63 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{32}\)	\(120\)
default	\(-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{5600 \left (x +\frac {3}{2}\right )^{5}}-\frac {789723 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{1372000}-\frac {789723 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{262609375}-\frac {263241 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{6002500}+\frac {248967 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{686000}+\frac {1131399 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{525218750}+\frac {789723 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{1372000}+\frac {63 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{32}-\frac {377133 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{525218750 \left (x +\frac {3}{2}\right )}-\frac {11 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{24500 \left (x +\frac {3}{2}\right )^{4}}-\frac {521 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{857500 \left (x +\frac {3}{2}\right )^{3}}-\frac {2241 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{30012500 \left (x +\frac {3}{2}\right )^{2}}+\frac {267723 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{12005000}\)	\(248\)

Input:

int((5-x)*(3*x^2+2)^(5/2)/(2*x+3)^6,x,method=_RETURNVERBOSE)
 

Output:

-1/19600*(264600*x^7+8220564*x^6+34279614*x^5+68214270*x^4+76556865*x^3+59 
820685*x^2+35880926*x[phone])/(2*x+3)^5/(3*x^2+2)^(1/2)+63/32*arcsinh(1/ 
2*6^(1/2)*x)*3^(1/2)+789723/1372000*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 = 191, normalized size of antiderivative = 1.44 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx=\frac {2701125 \, \sqrt {3} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 789723 \, \sqrt {35} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} - 93 \, x^{2} + 36 \, x - 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 140 \, {\left (88200 \, x^{5} + 2740188 \, x^{4} + 11367738 \, x^{3} + 20911298 \, x^{2} + 17940463 \, x + 5999363\right )} \sqrt {3 \, x^{2} + 2}}{2744000 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \] Input:

integrate((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^6,x, algorithm="fricas")
 

Output:

1/2744000*(2701125*sqrt(3)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x 
+ 243)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 789723*sqrt(35)*(32*x 
^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*log((sqrt(35)*sqrt(3*x^2 
+ 2)*(9*x - 4) - 93*x^2 + 36*x - 43)/(4*x^2 + 12*x + 9)) - 140*(88200*x^5 
+ 2740188*x^4 + 11367738*x^3 + 20911298*x^2 + 17940463*x + 5999363)*sqrt(3 
*x^2 + 2))/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx=\text {Timed out} \] Input:

integrate((5-x)*(3*x**2+2)**(5/2)/(3+2*x)**6,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (107) = 214\).

Time = 0.14 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.83 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx=\frac {6723}{30012500} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{175 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {44 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{6125 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {1042 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{214375 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {2241 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{7503125 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {267723}{12005000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x - \frac {263241}{6002500} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {377133 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{30012500 \, {\left (2 \, x + 3\right )}} + \frac {248967}{686000} \, \sqrt {3 \, x^{2} + 2} x + \frac {63}{32} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {789723}{1372000} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) - \frac {789723}{686000} \, \sqrt {3 \, x^{2} + 2} \] Input:

integrate((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^6,x, algorithm="maxima")
 

Output:

6723/30012500*(3*x^2 + 2)^(5/2) - 13/175*(3*x^2 + 2)^(7/2)/(32*x^5 + 240*x 
^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 44/6125*(3*x^2 + 2)^(7/2)/(16*x^4 
 + 96*x^3 + 216*x^2 + 216*x + 81) - 1042/214375*(3*x^2 + 2)^(7/2)/(8*x^3 + 
 36*x^2 + 54*x + 27) - 2241/7503125*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 9) + 
 267723/12005000*(3*x^2 + 2)^(3/2)*x - 263241/6002500*(3*x^2 + 2)^(3/2) - 
377133/30012500*(3*x^2 + 2)^(5/2)/(2*x + 3) + 248967/686000*sqrt(3*x^2 + 2 
)*x + 63/32*sqrt(3)*arcsinh(1/2*sqrt(6)*x) - 789723/1372000*sqrt(35)*arcsi 
nh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 789723/686000* 
sqrt(3*x^2 + 2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (107) = 214\).

Time = 0.15 (sec) , antiderivative size = 355, normalized size of antiderivative = 2.67 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx=-\frac {63}{32} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) - \frac {789723}{1372000} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) - \frac {9}{64} \, \sqrt {3 \, x^{2} + 2} - \frac {3 \, \sqrt {3} {\left (1034487 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} + 28143036 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} + 94364251 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} + 328235733 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} - 120044232 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} - 774358774 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 578739476 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 495467552 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 66595728 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 11086336\right )}}{156800 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{5}} \] Input:

integrate((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^6,x, algorithm="giac")
 

Output:

-63/32*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 789723/1372000*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))) - 9/64*sqrt(3*x^2 + 2) 
- 3/156800*sqrt(3)*(1034487*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 2814 
3036*(sqrt(3)*x - sqrt(3*x^2 + 2))^8 + 94364251*sqrt(3)*(sqrt(3)*x - sqrt( 
3*x^2 + 2))^7 + 328235733*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 - 120044232*sqrt 
(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 - 774358774*(sqrt(3)*x - sqrt(3*x^2 + 
2))^4 + 578739476*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 495467552*(sqr 
t(3)*x - sqrt(3*x^2 + 2))^2 + 66595728*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2 
)) - 11086336)/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - s 
qrt(3*x^2 + 2)) - 2)^5
 

Mupad [B] (verification not implemented)

Time = 5.94 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.55 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx=\frac {63\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{32}-\frac {9\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{64}-\frac {789723\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{1372000}+\frac {789723\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{1372000}+\frac {2303\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{512\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}-\frac {3185\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{2048\,\left (x^5+\frac {15\,x^4}{2}+\frac {45\,x^3}{2}+\frac {135\,x^2}{4}+\frac {405\,x}{16}+\frac {243}{32}\right )}-\frac {64959\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{19600\,\left (x+\frac {3}{2}\right )}+\frac {44127\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{8960\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {15397\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{2560\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \] Input:

int(-((3*x^2 + 2)^(5/2)*(x - 5))/(2*x + 3)^6,x)
 

Output:

(63*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/32 - (9*3^(1/2)*(x^2 + 2/3)^(1/2 
))/64 - (789723*35^(1/2)*log(x + 3/2))/1372000 + (789723*35^(1/2)*log(x - 
(3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/1372000 + (2303*3^(1/2)*(x^ 
2 + 2/3)^(1/2))/(512*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) - (318 
5*3^(1/2)*(x^2 + 2/3)^(1/2))/(2048*((405*x)/16 + (135*x^2)/4 + (45*x^3)/2 
+ (15*x^4)/2 + x^5 + 243/32)) - (64959*3^(1/2)*(x^2 + 2/3)^(1/2))/(19600*( 
x + 3/2)) + (44127*3^(1/2)*(x^2 + 2/3)^(1/2))/(8960*(3*x + x^2 + 9/4)) - ( 
15397*3^(1/2)*(x^2 + 2/3)^(1/2))/(2560*((27*x)/4 + (9*x^2)/2 + x^3 + 27/8) 
)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 568, normalized size of antiderivative = 4.27 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx =\text {Too large to display} \] Input:

int((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^6,x)
 

Output:

( - 12348000*sqrt(3*x**2 + 2)*x**5 - 383626320*sqrt(3*x**2 + 2)*x**4 - 159 
1483320*sqrt(3*x**2 + 2)*x**3 - 2927581720*sqrt(3*x**2 + 2)*x**2 - 2511664 
820*sqrt(3*x**2 + 2)*x - 839910820*sqrt(3*x**2 + 2) + 50542272*sqrt(35)*lo 
g( - sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**5 + 379067040*sqrt(35)*log( - 
 sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**4 + 1137201120*sqrt(35)*log( - sq 
rt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**3 + 1705801680*sqrt(35)*log( - sqrt( 
3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**2 + 1279351260*sqrt(35)*log( - sqrt(3*x 
**2 + 2)*sqrt(35) + 9*x - 4)*x + 383805378*sqrt(35)*log( - sqrt(3*x**2 + 2 
)*sqrt(35) + 9*x - 4) - 50542272*sqrt(35)*log(2*x + 3)*x**5 - 379067040*sq 
rt(35)*log(2*x + 3)*x**4 - 1137201120*sqrt(35)*log(2*x + 3)*x**3 - 1705801 
680*sqrt(35)*log(2*x + 3)*x**2 - 1279351260*sqrt(35)*log(2*x + 3)*x - 3838 
05378*sqrt(35)*log(2*x + 3) - 86436000*sqrt(3)*log(sqrt(3*x**2 + 2) - sqrt 
(3)*x)*x**5 - 648270000*sqrt(3)*log(sqrt(3*x**2 + 2) - sqrt(3)*x)*x**4 - 1 
944810000*sqrt(3)*log(sqrt(3*x**2 + 2) - sqrt(3)*x)*x**3 - 2917215000*sqrt 
(3)*log(sqrt(3*x**2 + 2) - sqrt(3)*x)*x**2 - 2187911250*sqrt(3)*log(sqrt(3 
*x**2 + 2) - sqrt(3)*x)*x - 656373375*sqrt(3)*log(sqrt(3*x**2 + 2) - sqrt( 
3)*x) + 86436000*sqrt(3)*log(sqrt(3*x**2 + 2) + sqrt(3)*x)*x**5 + 64827000 
0*sqrt(3)*log(sqrt(3*x**2 + 2) + sqrt(3)*x)*x**4 + 1944810000*sqrt(3)*log( 
sqrt(3*x**2 + 2) + sqrt(3)*x)*x**3 + 2917215000*sqrt(3)*log(sqrt(3*x**2 + 
2) + sqrt(3)*x)*x**2 + 2187911250*sqrt(3)*log(sqrt(3*x**2 + 2) + sqrt(3...