\(\int \frac {(5-x) (2+5 x+3 x^2)^{7/2}}{3+2 x} \, dx\) [960]

Optimal result

Integrand size = 27, antiderivative size = 159 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{3+2 x} \, dx=\frac {5 (1229315-2568342 x) \sqrt {2+5 x+3 x^2}}{2654208}+\frac {5 (6205-127338 x) \left (2+5 x+3 x^2\right )^{3/2}}{331776}-\frac {(589+7446 x) \left (2+5 x+3 x^2\right )^{5/2}}{6912}+\frac {1}{672} (277-42 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {65251715 \text {arctanh}\left (\frac {\sqrt {3} (1+x)}{\sqrt {2+5 x+3 x^2}}\right )}{2654208 \sqrt {3}}+\frac {1625}{256} \sqrt {5} \text {arctanh}\left (\frac {\sqrt {5} (1+x)}{\sqrt {2+5 x+3 x^2}}\right ) \] Output:

5/2654208*(1229315-2568342*x)*(3*x^2+5*x+2)^(1/2)+5/331776*(6205-127338*x) 
*(3*x^2+5*x+2)^(3/2)-1/6912*(589+7446*x)*(3*x^2+5*x+2)^(5/2)+1/672*(277-42 
*x)*(3*x^2+5*x+2)^(7/2)-65251715/7962624*arctanh(3^(1/2)*(1+x)/(3*x^2+5*x+ 
2)^(1/2))*3^(1/2)+1625/256*5^(1/2)*arctanh(5^(1/2)*(1+x)/(3*x^2+5*x+2)^(1/ 
2))
 

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.73 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{3+2 x} \, dx=\frac {-3 \sqrt {2+5 x+3 x^2} \left (-101435865-185981750 x-722869752 x^2-1224844848 x^3-1167854976 x^4-529784064 x^5-50015232 x^6+31352832 x^7\right )[phone] \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )-456762005 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{55738368} \] Input:

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

Output:

(-3*Sqrt[2 + 5*x + 3*x^2]*(-101435865 - 185981750*x - 722869752*x^2 - 1224 
844848*x^3 - 1167854976*x^4 - 529784064*x^5 - 50015232*x^6 + 31352832*x^7) 
 + 353808000*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)] - 45676200 
5*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/55738368
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.16, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {1231, 1231, 27, 1231, 27, 1231, 27, 1269, 1092, 219, 1154, 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.

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

\(\Big \downarrow \) 1231

\(\displaystyle \frac {1}{672} (277-42 x) \left (3 x^2+5 x+2\right )^{7/2}-\frac {1}{192} \int \frac {(2482 x+2163) \left (3 x^2+5 x+2\right )^{5/2}}{2 x+3}dx\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {1}{192} \left (\frac {1}{144} \int -\frac {10 (42446 x+35589) \left (3 x^2+5 x+2\right )^{3/2}}{2 x+3}dx-\frac {1}{36} (7446 x+589) \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {1}{672} (277-42 x) \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{192} \left (-\frac {5}{72} \int \frac {(42446 x+35589) \left (3 x^2+5 x+2\right )^{3/2}}{2 x+3}dx-\frac {1}{36} (7446 x+589) \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {1}{672} (277-42 x) \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {1}{192} \left (-\frac {5}{72} \left (-\frac {1}{96} \int -\frac {6 (856114 x+722571) \sqrt {3 x^2+5 x+2}}{2 x+3}dx-\frac {1}{24} (6205-127338 x) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {1}{36} (7446 x+589) \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {1}{672} (277-42 x) \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{192} \left (-\frac {5}{72} \left (\frac {1}{16} \int \frac {(856114 x+722571) \sqrt {3 x^2+5 x+2}}{2 x+3}dx-\frac {1}{24} (6205-127338 x) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {1}{36} (7446 x+589) \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {1}{672} (277-42 x) \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {1}{192} \left (-\frac {5}{72} \left (\frac {1}{16} \left (-\frac {1}{48} \int -\frac {2 (26100686 x+22303029)}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {1}{12} \sqrt {3 x^2+5 x+2} (1229315-2568342 x)\right )-\frac {1}{24} (6205-127338 x) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {1}{36} (7446 x+589) \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {1}{672} (277-42 x) \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{192} \left (-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{24} \int \frac {26100686 x+22303029}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {1}{12} (1229315-2568342 x) \sqrt {3 x^2+5 x+2}\right )-\frac {1}{24} (6205-127338 x) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {1}{36} (7446 x+589) \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {1}{672} (277-42 x) \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{192} \left (-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{24} \left (13050343 \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-16848000 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {1}{12} (1229315-2568342 x) \sqrt {3 x^2+5 x+2}\right )-\frac {1}{24} (6205-127338 x) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {1}{36} (7446 x+589) \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {1}{672} (277-42 x) \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {1}{192} \left (-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{24} \left (26100686 \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-16848000 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {1}{12} (1229315-2568342 x) \sqrt {3 x^2+5 x+2}\right )-\frac {1}{24} (6205-127338 x) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {1}{36} (7446 x+589) \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {1}{672} (277-42 x) \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{192} \left (-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{24} \left (\frac {13050343 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}-16848000 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {1}{12} (1229315-2568342 x) \sqrt {3 x^2+5 x+2}\right )-\frac {1}{24} (6205-127338 x) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {1}{36} (7446 x+589) \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {1}{672} (277-42 x) \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {1}{192} \left (-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{24} \left (33696000 \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )+\frac {13050343 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}\right )-\frac {1}{12} (1229315-2568342 x) \sqrt {3 x^2+5 x+2}\right )-\frac {1}{24} (6205-127338 x) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {1}{36} (7446 x+589) \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {1}{672} (277-42 x) \left (3 x^2+5 x+2\right )^{7/2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{192} \left (-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{24} \left (\frac {13050343 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}-3369600 \sqrt {5} \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )\right )-\frac {1}{12} (1229315-2568342 x) \sqrt {3 x^2+5 x+2}\right )-\frac {1}{24} (6205-127338 x) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {1}{36} (7446 x+589) \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {1}{672} (277-42 x) \left (3 x^2+5 x+2\right )^{7/2}\)

Input:

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

Output:

((277 - 42*x)*(2 + 5*x + 3*x^2)^(7/2))/672 + (-1/36*((589 + 7446*x)*(2 + 5 
*x + 3*x^2)^(5/2)) - (5*(-1/24*((6205 - 127338*x)*(2 + 5*x + 3*x^2)^(3/2)) 
 + (-1/12*((1229315 - 2568342*x)*Sqrt[2 + 5*x + 3*x^2]) + ((13050343*ArcTa 
nh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/Sqrt[3] - 3369600*Sqrt[5] 
*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/24)/16))/72)/192
 

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 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 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1231

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

rule 1269

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

Maple [A] (verified)

Time = 1.84 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.69

method	result	size
risch	\(-\frac {\left (31352832 x^{7}-50015232 x^{6}-529784064 x^{5}-1167854976 x^{4}-1224844848 x^{3}-722869752 x^{2}-185981750 x -101435865\right ) \sqrt {3 x^{2}+5 x +2}}{18579456}-\frac {65251715 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{15925248}-\frac {1625 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{512}\)	\(110\)
trager	\(\left (-\frac {27}{16} x^{7}+\frac {603}{224} x^{6}+\frac {25549}{896} x^{5}+\frac {337921}{5376} x^{4}+\frac {2835289}{43008} x^{3}+\frac {30119573}{774144} x^{2}+\frac {92990875}{9289728} x +\frac {33811955}{6193152}\right ) \sqrt {3 x^{2}+5 x +2}-\frac {1625 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (-\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )-10 \sqrt {3 x^{2}+5 x +2}}{2 x +3}\right )}{512}+\frac {65251715 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{15925248}\)	\(141\)
default	\(-\frac {\left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{96}+\frac {7 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{6912}-\frac {35 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{331776}+\frac {35 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{2654208}-\frac {35 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{15925248}+\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{28}-\frac {13 \left (6 x +5\right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{72}-\frac {1105 \left (6 x +5\right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{3456}-\frac {22295 \left (6 x +5\right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{27648}-\frac {679705 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}\right ) \sqrt {3}}{165888}+\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{16}+\frac {325 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{192}+\frac {1625 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{512}-\frac {1625 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{512}\)	\(295\)

Input:

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

Output:

-1/18579456*(31352832*x^7-50015232*x^6-529784064*x^5-1167854976*x^4-122484 
4848*x^3-722869752*x^2-185981750*x-101435865)*(3*x^2+5*x+2)^(1/2)-65251715 
/15925248*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)-1625/512*5 
^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.87 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{3+2 x} \, dx=-\frac {1}{18579456} \, {\left (31352832 \, x^{7} - 50015232 \, x^{6} - 529784064 \, x^{5} - 1167854976 \, x^{4} - 1224844848 \, x^{3} - 722869752 \, x^{2} - 185981750 \, x - 101435865\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {65251715}{31850496} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + \frac {1625}{1024} \, \sqrt {5} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) \] Input:

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

Output:

-1/18579456*(31352832*x^7 - 50015232*x^6 - 529784064*x^5 - 1167854976*x^4 
- 1224844848*x^3 - 722869752*x^2 - 185981750*x - 101435865)*sqrt(3*x^2 + 5 
*x + 2) + 65251715/31850496*sqrt(3)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*( 
6*x + 5) + 72*x^2 + 120*x + 49) + 1625/1024*sqrt(5)*log((4*sqrt(5)*sqrt(3* 
x^2 + 5*x + 2)*(8*x + 7) + 124*x^2 + 212*x + 89)/(4*x^2 + 12*x + 9))
 

Sympy [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{3+2 x} \, dx=- \int \left (- \frac {40 \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\, dx \] Input:

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

Output:

-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x) - Integral(-292*x*sqrt( 
3*x**2 + 5*x + 2)/(2*x + 3), x) - Integral(-870*x**2*sqrt(3*x**2 + 5*x + 2 
)/(2*x + 3), x) - Integral(-1339*x**3*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x) 
 - Integral(-1090*x**4*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x) - Integral(-39 
6*x**5*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x) - Integral(27*x**7*sqrt(3*x**2 
 + 5*x + 2)/(2*x + 3), x)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.17 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{3+2 x} \, dx=-\frac {1}{16} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {277}{672} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {1241}{1152} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {589}{6912} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {106115}{55296} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {31025}{331776} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {2140285}{442368} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {65251715}{15925248} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {1625}{512} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {6146575}{2654208} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \] Input:

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

Output:

-1/16*(3*x^2 + 5*x + 2)^(7/2)*x + 277/672*(3*x^2 + 5*x + 2)^(7/2) - 1241/1 
152*(3*x^2 + 5*x + 2)^(5/2)*x - 589/6912*(3*x^2 + 5*x + 2)^(5/2) - 106115/ 
55296*(3*x^2 + 5*x + 2)^(3/2)*x + 31025/331776*(3*x^2 + 5*x + 2)^(3/2) - 2 
140285/442368*sqrt(3*x^2 + 5*x + 2)*x - 65251715/15925248*sqrt(3)*log(sqrt 
(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 1625/512*sqrt(5)*log(sqrt(5)*sqrt 
(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 6146575/2654208*s 
qrt(3*x^2 + 5*x + 2)
 

Giac [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.98 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{3+2 x} \, dx=-\frac {1}{18579456} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (42 \, x - 67\right )} x - 25549\right )} x - 337921\right )} x - 2835289\right )} x - 30119573\right )} x - 92990875\right )} x - 101435865\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {1625}{512} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac {65251715}{15925248} \, \sqrt {3} \log \left ({\left | -6 \, \sqrt {3} x - 5 \, \sqrt {3} + 6 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}\right ) \] Input:

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

Output:

-1/18579456*(2*(12*(18*(8*(6*(36*(42*x - 67)*x - 25549)*x - 337921)*x - 28 
35289)*x - 30119573)*x - 92990875)*x - 101435865)*sqrt(3*x^2 + 5*x + 2) + 
1625/512*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x 
^2 + 5*x + 2))/abs(-4*sqrt(3)*x + 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5 
*x + 2))) + 65251715/15925248*sqrt(3)*log(abs(-6*sqrt(3)*x - 5*sqrt(3) + 6 
*sqrt(3*x^2 + 5*x + 2)))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.29 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{3+2 x} \, dx=-\frac {27 \sqrt {3 x^{2}+5 x +2}\, x^{7}}{16}+\frac {603 \sqrt {3 x^{2}+5 x +2}\, x^{6}}{224}+\frac {25549 \sqrt {3 x^{2}+5 x +2}\, x^{5}}{896}+\frac {337921 \sqrt {3 x^{2}+5 x +2}\, x^{4}}{5376}+\frac {2835289 \sqrt {3 x^{2}+5 x +2}\, x^{3}}{43008}+\frac {30119573 \sqrt {3 x^{2}+5 x +2}\, x^{2}}{774144}+\frac {92990875 \sqrt {3 x^{2}+5 x +2}\, x}{9289728}+\frac {33811955 \sqrt {3 x^{2}+5 x +2}}{6193152}+\frac {1625 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}-\sqrt {15}+6 x +9\right )}{512}-\frac {1625 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+\sqrt {15}+6 x +9\right )}{512}-\frac {65251715 \sqrt {3}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+6 x +5\right )}{15925248} \] Input:

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

Output:

( - 188116992*sqrt(3*x**2 + 5*x + 2)*x**7 + 300091392*sqrt(3*x**2 + 5*x + 
2)*x**6 + 3178704384*sqrt(3*x**2 + 5*x + 2)*x**5 + 7007129856*sqrt(3*x**2 
+ 5*x + 2)*x**4 + 7349069088*sqrt(3*x**2 + 5*x + 2)*x**3 + 4337218512*sqrt 
(3*x**2 + 5*x + 2)*x**2 + 1115890500*sqrt(3*x**2 + 5*x + 2)*x + 608615190* 
sqrt(3*x**2 + 5*x + 2) + 353808000*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*sq 
rt(3) - sqrt(15) + 6*x + 9) - 353808000*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 
2)*sqrt(3) + sqrt(15) + 6*x + 9) - 456762005*sqrt(3)*log(2*sqrt(3*x**2 + 5 
*x + 2)*sqrt(3) + 6*x + 5))/111476736