3.25.51 \(\int (5-x) (2+5 x+3 x^2)^{7/2} \, dx\) [2451]

3.25.51.1 Optimal result

Integrand size = 20, antiderivative size = 149 \[ \int (5-x) \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\frac {1225 (5+6 x) \sqrt {2+5 x+3 x^2}}{7962624}+\frac {1225 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{995328}-\frac {245 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{20736}+\frac {35}{288} (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{27} \left (2+5 x+3 x^2\right )^{9/2}+\frac {1225 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{15925248 \sqrt {3}} \]

output

1225/995328*(5+6*x)*(3*x^2+5*x+2)^(3/2)-245/20736*(5+6*x)*(3*x^2+5*x+2)^(5 
/2)+35/288*(5+6*x)*(3*x^2+5*x+2)^(7/2)-1/27*(3*x^2+5*x+2)^(9/2)+1225/47775 
744*arctanh(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x+2)^(1/2))*3^(1/2)-1225/7962624* 
(5+6*x)*(3*x^2+5*x+2)^(1/2)
 

3.25.51.2 Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.61 \[ \int (5-x) \left (2+5 x+3 x^2\right )^{7/2} \, dx=\frac {-3 \sqrt {2+5 x+3 x^2} \left (-32198883-278256050 x-1014795048 x^2-2013572880 x^3-2320737408 x^4-1507127040 x^5-452625408 x^6+2488320 x^7+23887872 x^8\right )+1225 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{23887872} \]

input

Integrate[(5 - x)*(2 + 5*x + 3*x^2)^(7/2),x]
 

output

(-3*Sqrt[2 + 5*x + 3*x^2]*(-32198883 - 278256050*x - 1014795048*x^2 - 2013 
572880*x^3 - 2320737408*x^4 - 1507127040*x^5 - 452625408*x^6 + 2488320*x^7 
 + 23887872*x^8) + 1225*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)] 
)/23887872
 

3.25.51.3 Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1160, 1087, 1087, 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.

\(\displaystyle \int (5-x) \left (3 x^2+5 x+2\right )^{7/2} \, dx\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {35}{6} \int \left (3 x^2+5 x+2\right )^{7/2}dx-\frac {1}{27} \left (3 x^2+5 x+2\right )^{9/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {35}{6} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \int \left (3 x^2+5 x+2\right )^{5/2}dx\right )-\frac {1}{27} \left (3 x^2+5 x+2\right )^{9/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {35}{6} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \int \left (3 x^2+5 x+2\right )^{3/2}dx\right )\right )-\frac {1}{27} \left (3 x^2+5 x+2\right )^{9/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {35}{6} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{16} \int \sqrt {3 x^2+5 x+2}dx\right )\right )\right )-\frac {1}{27} \left (3 x^2+5 x+2\right )^{9/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {35}{6} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{24} \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )\right )-\frac {1}{27} \left (3 x^2+5 x+2\right )^{9/2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {35}{6} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{12} \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )\right )-\frac {1}{27} \left (3 x^2+5 x+2\right )^{9/2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {35}{6} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {\text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{24 \sqrt {3}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )\right )-\frac {1}{27} \left (3 x^2+5 x+2\right )^{9/2}\)

input

Int[(5 - x)*(2 + 5*x + 3*x^2)^(7/2),x]
 

output

-1/27*(2 + 5*x + 3*x^2)^(9/2) + (35*(((5 + 6*x)*(2 + 5*x + 3*x^2)^(7/2))/4 
8 - (7*(((5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/36 - (5*(((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*Sqrt[3]))/16))/72))/96))/6
 

3.25.51.3.1 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 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 1160

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

3.25.51.4 Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.57

method	result	size
risch	\(-\frac {\left (23887872 x^{8}+2488320 x^{7}-452625408 x^{6}-1507127040 x^{5}-2320737408 x^{4}-2013572880 x^{3}-1014795048 x^{2}-278256050 x -32198883\right ) \sqrt {3 x^{2}+5 x +2}}{7962624}+\frac {1225 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{47775744}\)	\(85\)
trager	\(\left (-3 x^{8}-\frac {5}{16} x^{7}+\frac {1819}{32} x^{6}+\frac {218045}{1152} x^{5}+\frac {2014529}{6912} x^{4}+\frac {13983145}{55296} x^{3}+\frac {42283127}{331776} x^{2}+\frac {139128025}{3981312} x +\frac {10732961}{2654208}\right ) \sqrt {3 x^{2}+5 x +2}+\frac {1225 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{47775744}\)	\(96\)
default	\(\frac {35 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{288}-\frac {245 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{20736}+\frac {1225 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{995328}-\frac {1225 \left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{7962624}+\frac {1225 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{47775744}-\frac {\left (3 x^{2}+5 x +2\right )^{\frac {9}{2}}}{27}\)	\(121\)

input

int((5-x)*(3*x^2+5*x+2)^(7/2),x,method=_RETURNVERBOSE)
 

output

-1/7962624*(23887872*x^8+2488320*x^7-452625408*x^6-1507127040*x^5-23207374 
08*x^4-2013572880*x^3-1014795048*x^2-278256050*x-32198883)*(3*x^2+5*x+2)^( 
1/2)+1225/47775744*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)
 

3.25.51.5 Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.62 \[ \int (5-x) \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\frac {1}{7962624} \, {\left (23887872 \, x^{8} + 2488320 \, x^{7} - 452625408 \, x^{6} - 1507127040 \, x^{5} - 2320737408 \, x^{4} - 2013572880 \, x^{3} - 1014795048 \, x^{2} - 278256050 \, x - 32198883\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {1225}{95551488} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \]

input

integrate((5-x)*(3*x^2+5*x+2)^(7/2),x, algorithm="fricas")
 

output

-1/7962624*(23887872*x^8 + 2488320*x^7 - 452625408*x^6 - 1507127040*x^5 - 
2320737408*x^4 - 2013572880*x^3 - 1014795048*x^2 - 278256050*x - 32198883) 
*sqrt(3*x^2 + 5*x + 2) + 1225/95551488*sqrt(3)*log(4*sqrt(3)*sqrt(3*x^2 + 
5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49)
 

3.25.51.6 Sympy [A] (verification not implemented)

Time = 0.75 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.68 \[ \int (5-x) \left (2+5 x+3 x^2\right )^{7/2} \, dx=\sqrt {3 x^{2} + 5 x + 2} \left (- 3 x^{8} - \frac {5 x^{7}}{16} + \frac {1819 x^{6}}{32} + \frac {218045 x^{5}}{1152} + \frac {2014529 x^{4}}{6912} + \frac {13983145 x^{3}}{55296} + \frac {42283127 x^{2}}{331776} + \frac {139128025 x}{3981312} + \frac {10732961}{2654208}\right ) + \frac {1225 \sqrt {3} \log {\left (6 x + 2 \sqrt {3} \sqrt {3 x^{2} + 5 x + 2} + 5 \right )}}{47775744} \]

input

integrate((5-x)*(3*x**2+5*x+2)**(7/2),x)
 

output

sqrt(3*x**2 + 5*x + 2)*(-3*x**8 - 5*x**7/16 + 1819*x**6/32 + 218045*x**5/1 
152 + 2014529*x**4/6912 + 13983145*x**3/55296 + 42283127*x**2/331776 + 139 
128025*x/3981312 + 10732961/2654208) + 1225*sqrt(3)*log(6*x + 2*sqrt(3)*sq 
rt(3*x**2 + 5*x + 2) + 5)/47775744
 

3.25.51.7 Maxima [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.07 \[ \int (5-x) \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\frac {1}{27} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} + \frac {35}{48} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {175}{288} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {245}{3456} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {1225}{20736} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {1225}{165888} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {6125}{995328} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {1225}{1327104} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {1225}{47775744} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {6125}{7962624} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]

input

integrate((5-x)*(3*x^2+5*x+2)^(7/2),x, algorithm="maxima")
 

output

-1/27*(3*x^2 + 5*x + 2)^(9/2) + 35/48*(3*x^2 + 5*x + 2)^(7/2)*x + 175/288* 
(3*x^2 + 5*x + 2)^(7/2) - 245/3456*(3*x^2 + 5*x + 2)^(5/2)*x - 1225/20736* 
(3*x^2 + 5*x + 2)^(5/2) + 1225/165888*(3*x^2 + 5*x + 2)^(3/2)*x + 6125/995 
328*(3*x^2 + 5*x + 2)^(3/2) - 1225/1327104*sqrt(3*x^2 + 5*x + 2)*x + 1225/ 
47775744*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 6125/796 
2624*sqrt(3*x^2 + 5*x + 2)
 

3.25.51.8 Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.60 \[ \int (5-x) \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\frac {1}{7962624} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (2 \, {\left (48 \, x + 5\right )} x - 1819\right )} x - 218045\right )} x - 2014529\right )} x - 13983145\right )} x - 42283127\right )} x - 139128025\right )} x - 32198883\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {1225}{47775744} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]

input

integrate((5-x)*(3*x^2+5*x+2)^(7/2),x, algorithm="giac")
 

output

-1/7962624*(2*(12*(6*(8*(6*(36*(2*(48*x + 5)*x - 1819)*x - 218045)*x - 201 
4529)*x - 13983145)*x - 42283127)*x - 139128025)*x - 32198883)*sqrt(3*x^2 
+ 5*x + 2) - 1225/47775744*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3* 
x^2 + 5*x + 2)) - 5))
 

3.25.51.9 Mupad [F(-1)]

Timed out. \[ \int (5-x) \left (2+5 x+3 x^2\right )^{7/2} \, dx=\int -\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2} \,d x \]

input

int(-(x - 5)*(5*x + 3*x^2 + 2)^(7/2),x)
 

output

int(-(x - 5)*(5*x + 3*x^2 + 2)^(7/2), x)