\(\int (5-x) (3+2 x)^3 (2+5 x+3 x^2)^{3/2} \, dx\) [927]

Optimal result

Integrand size = 27, antiderivative size = 153 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {12277 (5+6 x) \sqrt {2+5 x+3 x^2}}{165888}+\frac {12277 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{20736}+\frac {67}{126} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{24} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}+\frac {(75451+33210 x) \left (2+5 x+3 x^2\right )^{5/2}}{15120}+\frac {12277 \text {arctanh}\left (\frac {\sqrt {3} (1+x)}{\sqrt {2+5 x+3 x^2}}\right )}{165888 \sqrt {3}} \] Output:

-12277/165888*(5+6*x)*(3*x^2+5*x+2)^(1/2)+12277/20736*(5+6*x)*(3*x^2+5*x+2 
)^(3/2)+67/126*(3+2*x)^2*(3*x^2+5*x+2)^(5/2)-1/24*(3+2*x)^3*(3*x^2+5*x+2)^ 
(5/2)+1/15120*(75451+33210*x)*(3*x^2+5*x+2)^(5/2)+12277/497664*arctanh(3^( 
1/2)*(1+x)/(3*x^2+5*x+2)^(1/2))*3^(1/2)
 

Mathematica [A] (verified)

Time = 0.81 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.56 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2} \, dx=\frac {-3 \sqrt {2+5 x+3 x^2} \left (-233137461-1276112350 x-2762417688 x^2-2993047920 x^3-1650151296 x^4-368236800 x^5+25297920 x^6+17418240 x^7\right )+429695 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{17418240} \] Input:

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

Output:

(-3*Sqrt[2 + 5*x + 3*x^2]*(-233137461 - 1276112350*x - 2762417688*x^2 - 29 
93047920*x^3 - 1650151296*x^4 - 368236800*x^5 + 25297920*x^6 + 17418240*x^ 
7) + 429695*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/17418240
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1236, 27, 1236, 1225, 1087, 1087, 1092, 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.

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {1}{24} \int \frac {1}{2} (2 x+3)^2 (536 x+819) \left (3 x^2+5 x+2\right )^{3/2}dx-\frac {1}{24} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{48} \int (2 x+3)^2 (536 x+819) \left (3 x^2+5 x+2\right )^{3/2}dx-\frac {1}{24} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {1}{48} \left (\frac {1}{21} \int (2 x+3) (19926 x+27209) \left (3 x^2+5 x+2\right )^{3/2}dx+\frac {536}{21} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {1}{24} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {1}{48} \left (\frac {1}{21} \left (\frac {85939}{6} \int \left (3 x^2+5 x+2\right )^{3/2}dx+\frac {1}{15} (33210 x+75451) \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {536}{21} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {1}{24} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{48} \left (\frac {1}{21} \left (\frac {85939}{6} \left (\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{16} \int \sqrt {3 x^2+5 x+2}dx\right )+\frac {1}{15} (33210 x+75451) \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {536}{21} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {1}{24} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{48} \left (\frac {1}{21} \left (\frac {85939}{6} \left (\frac {1}{16} \left (\frac {1}{24} \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1}{15} (33210 x+75451) \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {536}{21} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {1}{24} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {1}{48} \left (\frac {1}{21} \left (\frac {85939}{6} \left (\frac {1}{16} \left (\frac {1}{12} \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1}{15} (33210 x+75451) \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {536}{21} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {1}{24} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{48} \left (\frac {1}{21} \left (\frac {85939}{6} \left (\frac {1}{16} \left (\frac {\text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{24 \sqrt {3}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1}{15} (33210 x+75451) \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {536}{21} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {1}{24} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{5/2}\)

Input:

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

Output:

-1/24*((3 + 2*x)^3*(2 + 5*x + 3*x^2)^(5/2)) + ((536*(3 + 2*x)^2*(2 + 5*x + 
 3*x^2)^(5/2))/21 + (((75451 + 33210*x)*(2 + 5*x + 3*x^2)^(5/2))/15 + (859 
39*(((5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/24 + (-1/12*((5 + 6*x)*Sqrt[2 + 5* 
x + 3*x^2]) + ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])]/(24*Sqr 
t[3]))/16))/6)/21)/48
 

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 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1225

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 
 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), 
 x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p 
+ 3))/(2*c^2*(2*p + 3))   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, p}, x] &&  !LeQ[p, -1]
 

rule 1236

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

Maple [A] (verified)

Time = 1.63 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.52

method	result	size
risch	\(-\frac {\left (17418240 x^{7}[phone] x^{6}-368236800 x^{5}-1650151296 x^{4}-2993047920 x^{3}-2762417688 x^{2}-1276112350 x -233137461\right ) \sqrt {3 x^{2}+5 x +2}}{5806080}+\frac {12277 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{995328}\)	\(80\)
trager	\(\left (-3 x^{7}-\frac {61}{14} x^{6}+\frac {10655}{168} x^{5}+\frac {1432423}{5040} x^{4}+\frac {4157011}{8064} x^{3}+\frac {115100737}{241920} x^{2}+\frac {127611235}{580608} x +\frac {77712487}{1935360}\right ) \sqrt {3 x^{2}+5 x +2}-\frac {12277 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{995328}\)	\(91\)
default	\(\frac {12277 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{20736}-\frac {12277 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{165888}+\frac {12277 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{995328}+\frac {130801 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{15120}+\frac {1063 x \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{168}+\frac {79 x^{2} \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{126}-\frac {x^{3} \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{3}\)	\(132\)

Input:

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

Output:

-1/5806080*(17418240*x^7+25297920*x^6-368236800*x^5-1650151296*x^4-2993047 
920*x^3-2762417688*x^2-1276112350*x-233137461)*(3*x^2+5*x+2)^(1/2)+12277/9 
95328*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.58 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {1}{5806080} \, {\left (17418240 \, x^{7} + 25297920 \, x^{6} - 368236800 \, x^{5} - 1650151296 \, x^{4} - 2993047920 \, x^{3} - 2762417688 \, x^{2} - 1276112350 \, x - 233137461\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {12277}{1990656} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \] Input:

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

Output:

-1/5806080*(17418240*x^7 + 25297920*x^6 - 368236800*x^5 - 1650151296*x^4 - 
 2993047920*x^3 - 2762417688*x^2 - 1276112350*x - 233137461)*sqrt(3*x^2 + 
5*x + 2) + 12277/1990656*sqrt(3)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x 
+ 5) + 72*x^2 + 120*x + 49)
 

Sympy [A] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.62 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2} \, dx=\sqrt {3 x^{2} + 5 x + 2} \left (- 3 x^{7} - \frac {61 x^{6}}{14} + \frac {10655 x^{5}}{168} + \frac {1432423 x^{4}}{5040} + \frac {4157011 x^{3}}{8064} + \frac {115100737 x^{2}}{241920} + \frac {127611235 x}{580608} + \frac {77712487}{1935360}\right ) + \frac {12277 \sqrt {3} \log {\left (6 x + 2 \sqrt {3} \sqrt {3 x^{2} + 5 x + 2} + 5 \right )}}{995328} \] Input:

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

Output:

sqrt(3*x**2 + 5*x + 2)*(-3*x**7 - 61*x**6/14 + 10655*x**5/168 + 1432423*x* 
*4/5040 + 4157011*x**3/8064 + 115100737*x**2/241920 + 127611235*x/580608 + 
 77712487/1935360) + 12277*sqrt(3)*log(6*x + 2*sqrt(3)*sqrt(3*x**2 + 5*x + 
 2) + 5)/995328
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.98 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {1}{3} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x^{3} + \frac {79}{126} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x^{2} + \frac {1063}{168} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {130801}{15120} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {12277}{3456} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {61385}{20736} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {12277}{27648} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {12277}{995328} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {61385}{165888} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \] Input:

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

Output:

-1/3*(3*x^2 + 5*x + 2)^(5/2)*x^3 + 79/126*(3*x^2 + 5*x + 2)^(5/2)*x^2 + 10 
63/168*(3*x^2 + 5*x + 2)^(5/2)*x + 130801/15120*(3*x^2 + 5*x + 2)^(5/2) + 
12277/3456*(3*x^2 + 5*x + 2)^(3/2)*x + 61385/20736*(3*x^2 + 5*x + 2)^(3/2) 
 - 12277/27648*sqrt(3*x^2 + 5*x + 2)*x + 12277/995328*sqrt(3)*log(2*sqrt(3 
)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 61385/165888*sqrt(3*x^2 + 5*x + 2)
 

Giac [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.55 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {1}{5806080} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (30 \, {\left (12 \, {\left (42 \, x + 61\right )} x - 10655\right )} x - 1432423\right )} x - 20785055\right )} x - 115100737\right )} x - 638056175\right )} x - 233137461\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {12277}{995328} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \] Input:

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

Output:

-1/5806080*(2*(12*(6*(8*(30*(12*(42*x + 61)*x - 10655)*x - 1432423)*x - 20 
785055)*x - 115100737)*x - 638056175)*x - 233137461)*sqrt(3*x^2 + 5*x + 2) 
 - 12277/995328*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 
 2)) - 5))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.97 \[ \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2} \, dx=-3 \sqrt {3 x^{2}+5 x +2}\, x^{7}-\frac {61 \sqrt {3 x^{2}+5 x +2}\, x^{6}}{14}+\frac {10655 \sqrt {3 x^{2}+5 x +2}\, x^{5}}{168}+\frac {1432423 \sqrt {3 x^{2}+5 x +2}\, x^{4}}{5040}+\frac {4157011 \sqrt {3 x^{2}+5 x +2}\, x^{3}}{8064}+\frac {115100737 \sqrt {3 x^{2}+5 x +2}\, x^{2}}{241920}+\frac {127611235 \sqrt {3 x^{2}+5 x +2}\, x}{580608}+\frac {77712487 \sqrt {3 x^{2}+5 x +2}}{1935360}+\frac {12277 \sqrt {3}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+6 x +5\right )}{995328} \] Input:

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

Output:

( - 104509440*sqrt(3*x**2 + 5*x + 2)*x**7 - 151787520*sqrt(3*x**2 + 5*x + 
2)*x**6 + 2209420800*sqrt(3*x**2 + 5*x + 2)*x**5 + 9900907776*sqrt(3*x**2 
+ 5*x + 2)*x**4 + 17958287520*sqrt(3*x**2 + 5*x + 2)*x**3 + 16574506128*sq 
rt(3*x**2 + 5*x + 2)*x**2 + 7656674100*sqrt(3*x**2 + 5*x + 2)*x + 13988247 
66*sqrt(3*x**2 + 5*x + 2) + 429695*sqrt(3)*log(2*sqrt(3*x**2 + 5*x + 2)*sq 
rt(3) + 6*x + 5))/34836480