\(\int \frac {(5-x) (2+5 x+3 x^2)^{7/2}}{(3+2 x)^3} \, dx\) [962]

Optimal result

Integrand size = 27, antiderivative size = 166 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^3} \, dx=\frac {7 (167495-349806 x) \sqrt {2+5 x+3 x^2}}{36864}+\frac {7 (805-17394 x) \left (2+5 x+3 x^2\right )^{3/2}}{4608}-\frac {7}{240} (20+483 x) \left (2+5 x+3 x^2\right )^{5/2}+\frac {(1057+748 x) \left (2+5 x+3 x^2\right )^{7/2}}{40 (3+2 x)^2}-\frac {12443893 \text {arctanh}\left (\frac {\sqrt {3} (1+x)}{\sqrt {2+5 x+3 x^2}}\right )}{36864 \sqrt {3}}+\frac {44625}{512} \sqrt {5} \text {arctanh}\left (\frac {\sqrt {5} (1+x)}{\sqrt {2+5 x+3 x^2}}\right ) \] Output:

7/36864*(167495-349806*x)*(3*x^2+5*x+2)^(1/2)+7/4608*(805-17394*x)*(3*x^2+ 
5*x+2)^(3/2)-7/240*(20+483*x)*(3*x^2+5*x+2)^(5/2)+1/40*(1057+748*x)*(3*x^2 
+5*x+2)^(7/2)/(3+2*x)^2-12443893/110592*arctanh(3^(1/2)*(1+x)/(3*x^2+5*x+2 
)^(1/2))*3^(1/2)+44625/512*5^(1/2)*arctanh(5^(1/2)*(1+x)/(3*x^2+5*x+2)^(1/ 
2))
 

Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.74 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^3} \, dx=\frac {-\frac {3 \sqrt {2+5 x+3 x^2} \left (-91912653-89867034 x-19284852 x^2-12848072 x^3-15112992 x^4-6830784 x^5-926208 x^6+414720 x^7\right )}{(3+2 x)^2}[phone] \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )-62219465 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{552960} \] Input:

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

Output:

((-3*Sqrt[2 + 5*x + 3*x^2]*(-91912653 - 89867034*x - 19284852*x^2 - 128480 
72*x^3 - 15112992*x^4 - 6830784*x^5 - 926208*x^6 + 414720*x^7))/(3 + 2*x)^ 
2 + 48195000*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)] - 62219465 
*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/552960
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.19, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {1230, 27, 1230, 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.

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

\(\Big \downarrow \) 1230

\(\displaystyle -\frac {7}{96} \int -\frac {4 (121 x+101) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^2}dx-\frac {(x+21) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7}{24} \int \frac {(121 x+101) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^2}dx-\frac {(x+21) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^2}\)

\(\Big \downarrow \) 1230

\(\displaystyle \frac {7}{24} \left (\frac {(121 x+584) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}-\frac {1}{8} \int \frac {2 (2899 x+2436) \left (3 x^2+5 x+2\right )^{3/2}}{2 x+3}dx\right )-\frac {(x+21) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7}{24} \left (\frac {(121 x+584) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}-\frac {1}{4} \int \frac {(2899 x+2436) \left (3 x^2+5 x+2\right )^{3/2}}{2 x+3}dx\right )-\frac {(x+21) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^2}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {7}{24} \left (\frac {1}{4} \left (\frac {1}{96} \int -\frac {3 (116602 x+98403) \sqrt {3 x^2+5 x+2}}{2 x+3}dx+\frac {1}{48} (805-17394 x) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {(121 x+584) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}\right )-\frac {(x+21) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7}{24} \left (\frac {1}{4} \left (\frac {1}{48} (805-17394 x) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{32} \int \frac {(116602 x+98403) \sqrt {3 x^2+5 x+2}}{2 x+3}dx\right )+\frac {(121 x+584) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}\right )-\frac {(x+21) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^2}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {7}{24} \left (\frac {1}{4} \left (\frac {1}{32} \left (\frac {1}{48} \int -\frac {2 (3555398 x+3038097)}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx+\frac {1}{12} \sqrt {3 x^2+5 x+2} (167495-349806 x)\right )+\frac {1}{48} (805-17394 x) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {(121 x+584) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}\right )-\frac {(x+21) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7}{24} \left (\frac {1}{4} \left (\frac {1}{32} \left (\frac {1}{12} (167495-349806 x) \sqrt {3 x^2+5 x+2}-\frac {1}{24} \int \frac {3555398 x+3038097}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )+\frac {1}{48} (805-17394 x) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {(121 x+584) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}\right )-\frac {(x+21) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {7}{24} \left (\frac {1}{4} \left (\frac {1}{32} \left (\frac {1}{24} \left (2295000 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-1777699 \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx\right )+\frac {1}{12} \sqrt {3 x^2+5 x+2} (167495-349806 x)\right )+\frac {1}{48} (805-17394 x) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {(121 x+584) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}\right )-\frac {(x+21) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {7}{24} \left (\frac {1}{4} \left (\frac {1}{32} \left (\frac {1}{24} \left (2295000 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-3555398 \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}\right )+\frac {1}{12} \sqrt {3 x^2+5 x+2} (167495-349806 x)\right )+\frac {1}{48} (805-17394 x) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {(121 x+584) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}\right )-\frac {(x+21) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {7}{24} \left (\frac {1}{4} \left (\frac {1}{32} \left (\frac {1}{24} \left (2295000 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {1777699 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}\right )+\frac {1}{12} \sqrt {3 x^2+5 x+2} (167495-349806 x)\right )+\frac {1}{48} (805-17394 x) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {(121 x+584) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}\right )-\frac {(x+21) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {7}{24} \left (\frac {1}{4} \left (\frac {1}{32} \left (\frac {1}{24} \left (-4590000 \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )-\frac {1777699 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}\right )+\frac {1}{12} \sqrt {3 x^2+5 x+2} (167495-349806 x)\right )+\frac {1}{48} (805-17394 x) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {(121 x+584) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}\right )-\frac {(x+21) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {7}{24} \left (\frac {1}{4} \left (\frac {1}{32} \left (\frac {1}{24} \left (459000 \sqrt {5} \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-\frac {1777699 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}\right )+\frac {1}{12} \sqrt {3 x^2+5 x+2} (167495-349806 x)\right )+\frac {1}{48} (805-17394 x) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {(121 x+584) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}\right )-\frac {(x+21) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^2}\)

Input:

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

Output:

-1/12*((21 + x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^2 + (7*(((584 + 121*x)* 
(2 + 5*x + 3*x^2)^(5/2))/(10*(3 + 2*x)) + (((805 - 17394*x)*(2 + 5*x + 3*x 
^2)^(3/2))/48 + (((167495 - 349806*x)*Sqrt[2 + 5*x + 3*x^2])/12 + ((-17776 
99*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/Sqrt[3] + 459000* 
Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/24)/32)/4))/ 
24
 

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 1230

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (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 + b*x + 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 
+ b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m 
+ 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, 
 x], x], x] /; FreeQ[{a, b, 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 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 = 1.75 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.77

method	result	size
risch	\(-\frac {1244160 x^{9}-705024 x^{8}-24293952 x^{7}-81345312 x^{6}-127770744 x^{5}-152320900 x^{4}-391721506 x^{3}-763642833 x^{2}-639297333 x -183825306}{184320 \left (2 x +3\right )^{2} \sqrt {3 x^{2}+5 x +2}}-\frac {12443893 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{221184}-\frac {44625 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{1024}\)	\(127\)
trager	\(-\frac {\left (414720 x^{7}-926208 x^{6}-6830784 x^{5}-15112992 x^{4}-12848072 x^{3}-19284852 x^{2}-89867034 x -91912653\right ) \sqrt {3 x^{2}+5 x +2}}{184320 \left (2 x +3\right )^{2}}+\frac {44625 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{2 x +3}\right )}{1024}-\frac {12443893 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{221184}\)	\(148\)
default	\(-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{40 \left (x +\frac {3}{2}\right )^{2}}+\frac {27 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{10 \left (x +\frac {3}{2}\right )}+\frac {51 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{8}-\frac {1127 \left (6 x +5\right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{480}-\frac {20293 \left (6 x +5\right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{4608}-\frac {408107 \left (6 x +5\right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{36864}-\frac {12443893 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}\right ) \sqrt {3}}{221184}+\frac {357 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{32}+\frac {2975 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{128}+\frac {44625 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{1024}-\frac {44625 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{1024}-\frac {27 \left (6 x +5\right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{20}\)	\(253\)

Input:

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

Output:

-1/184320*(1244160*x^9-705024*x^8-24293952*x^7-81345312*x^6-127770744*x^5- 
152320900*x^4-391721506*x^3-763642833*x^2-639297333*x-183825306)/(2*x+3)^2 
/(3*x^2+5*x+2)^(1/2)-12443893/221184*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2 
)^(1/2))*3^(1/2)-44625/1024*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 = 0.10 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.04 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^3} \, dx=\frac {62219465 \, \sqrt {3} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 48195000 \, \sqrt {5} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 12 \, {\left (414720 \, x^{7} - 926208 \, x^{6} - 6830784 \, x^{5} - 15112992 \, x^{4} - 12848072 \, x^{3} - 19284852 \, x^{2} - 89867034 \, x - 91912653\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{2211840 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \] Input:

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

Output:

1/2211840*(62219465*sqrt(3)*(4*x^2 + 12*x + 9)*log(-4*sqrt(3)*sqrt(3*x^2 + 
 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 48195000*sqrt(5)*(4*x^2 + 12* 
x + 9)*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)) - 12*(414720*x^7 - 926208*x^6 - 6830784*x^5 - 1511 
2992*x^4 - 12848072*x^3 - 19284852*x^2 - 89867034*x - 91912653)*sqrt(3*x^2 
 + 5*x + 2))/(4*x^2 + 12*x + 9)
 

Sympy [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^3} \, dx=- \int \left (- \frac {40 \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\, dx \] Input:

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

Output:

-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - 
Integral(-292*x*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) 
- Integral(-870*x**2*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27) 
, x) - Integral(-1339*x**3*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x 
 + 27), x) - Integral(-1090*x**4*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 
+ 54*x + 27), x) - Integral(-396*x**5*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36* 
x**2 + 54*x + 27), x) - Integral(27*x**7*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 
36*x**2 + 54*x + 27), x)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.31 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^3} \, dx=\frac {39}{40} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{10 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {1127}{80} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {7}{12} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{4 \, {\left (2 \, x + 3\right )}} - \frac {20293}{768} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {5635}{4608} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {408107}{6144} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {12443893}{221184} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {44625}{1024} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {1172465}{36864} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \] Input:

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

Output:

39/40*(3*x^2 + 5*x + 2)^(7/2) - 13/10*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12* 
x + 9) - 1127/80*(3*x^2 + 5*x + 2)^(5/2)*x - 7/12*(3*x^2 + 5*x + 2)^(5/2) 
+ 27/4*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) - 20293/768*(3*x^2 + 5*x + 2)^(3/ 
2)*x + 5635/4608*(3*x^2 + 5*x + 2)^(3/2) - 408107/6144*sqrt(3*x^2 + 5*x + 
2)*x - 12443893/221184*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5 
/2) - 44625/1024*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 
5/2/abs(2*x + 3) - 2) + 1172465/36864*sqrt(3*x^2 + 5*x + 2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (134) = 268\).

Time = 0.44 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.68 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^3} \, dx=-\frac {1}{184320} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (8 \, {\left (30 \, x - 157\right )} x - 725\right )} x - 67409\right )} x + 1173065\right )} x - 8219517\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {44625}{1024} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac {12443893}{221184} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac {25 \, {\left (5878 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 22241 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 75807 \, \sqrt {3} x + 27061 \, \sqrt {3} - 75807 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{512 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{2}} \] Input:

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

Output:

-1/184320*(2*(12*(18*(8*(30*x - 157)*x - 725)*x - 67409)*x + 1173065)*x - 
8219517)*sqrt(3*x^2 + 5*x + 2) + 44625/1024*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*sqr 
t(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) + 12443893/221184*sqrt(3)*log 
(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) + 25/512*(5878*( 
sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 22241*sqrt(3)*(sqrt(3)*x - sqrt(3*x 
^2 + 5*x + 2))^2 + 75807*sqrt(3)*x + 27061*sqrt(3) - 75807*sqrt(3*x^2 + 5* 
x + 2))/(2*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 6*sqrt(3)*(sqrt(3)*x - 
sqrt(3*x^2 + 5*x + 2)) + 11)^2
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.94 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.46 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^3} \, dx=\frac {-3702620160 \sqrt {3 x^{2}+5 x +2}\, x^{7}[phone] \sqrt {3 x^{2}+5 x +2}\, x^{6}[phone] \sqrt {3 x^{2}+5 x +2}\, x^{5}[phone] \sqrt {3 x^{2}+5 x +2}\, x^{4}[phone] \sqrt {3 x^{2}+5 x +2}\, x^{3}[phone] \sqrt {3 x^{2}+5 x +2}\, x^{2}[phone] \sqrt {3 x^{2}+5 x +2}\, x [phone] \sqrt {3 x^{2}+5 x +2}[phone] \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}-\sqrt {15}+6 x +9\right ) x^{2}[phone] \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}-\sqrt {15}+6 x +9\right ) x [phone] \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}-\sqrt {15}+6 x +9\right )-286856640000 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+\sqrt {15}+6 x +9\right ) x^{2}-860569920000 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+\sqrt {15}+6 x +9\right ) x -645427440000 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+\sqrt {15}+6 x +9\right )-370330255680 \sqrt {3}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+6 x +5\right ) x^{2}-1110990767040 \sqrt {3}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+6 x +5\right ) x -833243075280 \sqrt {3}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+6 x +5\right )-92607588220 \sqrt {3}\, x^{2}-277822764660 \sqrt {3}\, x -208367073495 \sqrt {3}}{6582435840 x^{2}[phone] x [phone]} \] Input:

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

Output:

( - 3702620160*sqrt(3*x**2 + 5*x + 2)*x**7 + 8269185024*sqrt(3*x**2 + 5*x 
+ 2)*x**6 + 60985239552*sqrt(3*x**2 + 5*x + 2)*x**5 + 134928792576*sqrt(3* 
x**2 + 5*x + 2)*x**4 + 114707586816*sqrt(3*x**2 + 5*x + 2)*x**3 + 17217515 
8656*sqrt(3*x**2 + 5*x + 2)*x**2 + 802332879552*sqrt(3*x**2 + 5*x + 2)*x + 
 820596165984*sqrt(3*x**2 + 5*x + 2) + 286856640000*sqrt(5)*log(2*sqrt(3*x 
**2 + 5*x + 2)*sqrt(3) - sqrt(15) + 6*x + 9)*x**2 + 860569920000*sqrt(5)*l 
og(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) - sqrt(15) + 6*x + 9)*x + 645427440000 
*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) - sqrt(15) + 6*x + 9) - 2868 
56640000*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) + sqrt(15) + 6*x + 9 
)*x**2 - 860569920000*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) + sqrt( 
15) + 6*x + 9)*x - 645427440000*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt( 
3) + sqrt(15) + 6*x + 9) - 370330255680*sqrt(3)*log(2*sqrt(3*x**2 + 5*x + 
2)*sqrt(3) + 6*x + 5)*x**2 - 1110990767040*sqrt(3)*log(2*sqrt(3*x**2 + 5*x 
 + 2)*sqrt(3) + 6*x + 5)*x - 833243075280*sqrt(3)*log(2*sqrt(3*x**2 + 5*x 
+ 2)*sqrt(3) + 6*x + 5) - 92607588220*sqrt(3)*x**2 - 277822764660*sqrt(3)* 
x - 208367073495*sqrt(3))/(1645608960*(4*x**2 + 12*x + 9))