\(\int \frac {(5-x) (3+2 x)^{7/2}}{(2+5 x+3 x^2)^3} \, dx\) [844]

Optimal result

Integrand size = 27, antiderivative size = 100 \[ \int \frac {(5-x) (3+2 x)^{7/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=-\frac {\sqrt {3+2 x} (2449+2611 x)}{54 \left (2+5 x+3 x^2\right )^2}+\frac {\sqrt {3+2 x} (30722+37329 x)}{54 \left (2+5 x+3 x^2\right )}+1582 \text {arctanh}\left (\sqrt {3+2 x}\right )-1225 \sqrt {\frac {5}{3}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right ) \] Output:

-1/54*(3+2*x)^(1/2)*(2449+2611*x)/(3*x^2+5*x+2)^2+(3+2*x)^(1/2)*(30722+373 
29*x)/(162*x^2+270*x+108)+1582*arctanh((3+2*x)^(1/2))-1225/3*15^(1/2)*arct 
anh(1/5*15^(1/2)*(3+2*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.80 \[ \int \frac {(5-x) (3+2 x)^{7/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=\frac {\sqrt {3+2 x} \left (6555+25073 x+30979 x^2+12443 x^3\right )}{6 \left (2+5 x+3 x^2\right )^2}+1582 \text {arctanh}\left (\sqrt {3+2 x}\right )-1225 \sqrt {\frac {5}{3}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right ) \] Input:

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

Output:

(Sqrt[3 + 2*x]*(6555 + 25073*x + 30979*x^2 + 12443*x^3))/(6*(2 + 5*x + 3*x 
^2)^2) + 1582*ArcTanh[Sqrt[3 + 2*x]] - 1225*Sqrt[5/3]*ArcTanh[Sqrt[3/5]*Sq 
rt[3 + 2*x]]
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1233, 27, 1233, 27, 1197, 1480, 220}

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.

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

\(\Big \downarrow \) 1233

\(\displaystyle \frac {1}{6} \int -\frac {7 (2 x+3)^{3/2} (21 x+94)}{\left (3 x^2+5 x+2\right )^2}dx-\frac {(2 x+3)^{5/2} (139 x+121)}{6 \left (3 x^2+5 x+2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {7}{6} \int \frac {(2 x+3)^{3/2} (21 x+94)}{\left (3 x^2+5 x+2\right )^2}dx-\frac {(139 x+121) (2 x+3)^{5/2}}{6 \left (3 x^2+5 x+2\right )^2}\)

\(\Big \downarrow \) 1233

\(\displaystyle -\frac {7}{6} \left (\frac {1}{3} \int -\frac {9 (197 x+423)}{\sqrt {2 x+3} \left (3 x^2+5 x+2\right )}dx-\frac {\sqrt {2 x+3} (619 x+546)}{3 x^2+5 x+2}\right )-\frac {(139 x+121) (2 x+3)^{5/2}}{6 \left (3 x^2+5 x+2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {7}{6} \left (-3 \int \frac {197 x+423}{\sqrt {2 x+3} \left (3 x^2+5 x+2\right )}dx-\frac {\sqrt {2 x+3} (619 x+546)}{3 x^2+5 x+2}\right )-\frac {(139 x+121) (2 x+3)^{5/2}}{6 \left (3 x^2+5 x+2\right )^2}\)

\(\Big \downarrow \) 1197

\(\displaystyle -\frac {7}{6} \left (-6 \int \frac {197 (2 x+3)+255}{3 (2 x+3)^2-8 (2 x+3)+5}d\sqrt {2 x+3}-\frac {\sqrt {2 x+3} (619 x+546)}{3 x^2+5 x+2}\right )-\frac {(139 x+121) (2 x+3)^{5/2}}{6 \left (3 x^2+5 x+2\right )^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle -\frac {7}{6} \left (-6 \left (875 \int \frac {1}{3 (2 x+3)-5}d\sqrt {2 x+3}-678 \int \frac {1}{3 (2 x+3)-3}d\sqrt {2 x+3}\right )-\frac {\sqrt {2 x+3} (619 x+546)}{3 x^2+5 x+2}\right )-\frac {(139 x+121) (2 x+3)^{5/2}}{6 \left (3 x^2+5 x+2\right )^2}\)

\(\Big \downarrow \) 220

\(\displaystyle -\frac {7}{6} \left (-6 \left (226 \text {arctanh}\left (\sqrt {2 x+3}\right )-175 \sqrt {\frac {5}{3}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )\right )-\frac {\sqrt {2 x+3} (619 x+546)}{3 x^2+5 x+2}\right )-\frac {(139 x+121) (2 x+3)^{5/2}}{6 \left (3 x^2+5 x+2\right )^2}\)

Input:

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

Output:

-1/6*((3 + 2*x)^(5/2)*(121 + 139*x))/(2 + 5*x + 3*x^2)^2 - (7*(-((Sqrt[3 + 
 2*x]*(546 + 619*x))/(2 + 5*x + 3*x^2)) - 6*(226*ArcTanh[Sqrt[3 + 2*x]] - 
175*Sqrt[5/3]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])))/6
 

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 220

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 
1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && 
 (LtQ[a, 0] || GtQ[b, 0])
 

rule 1197

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)), x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - 
b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr 
eeQ[{a, b, c, d, e, f, g}, x]
 

rule 1233

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2) 
^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c 
*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( 
p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim 
p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f 
*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( 
m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* 
p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && 
GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | 
|  !ILtQ[m + 2*p + 3, 0])
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

Maple [A] (verified)

Time = 1.58 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.80

method	result	size
risch	\(\frac {\left (12443 x^{3}+30979 x^{2}+25073 x +6555\right ) \sqrt {2 x +3}}{6 \left (3 x^{2}+5 x +2\right )^{2}}+791 \ln \left (\sqrt {2 x +3}+1\right )-791 \ln \left (\sqrt {2 x +3}-1\right )-\frac {1225 \sqrt {15}\, \operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{3}\)	\(80\)
trager	\(\frac {\left (12443 x^{3}+30979 x^{2}+25073 x +6555\right ) \sqrt {2 x +3}}{6 \left (3 x^{2}+5 x +2\right )^{2}}+\frac {1225 \operatorname {RootOf}\left (\textit {\_Z}^{2}-15\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-15\right ) x +7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-15\right )-15 \sqrt {2 x +3}}{3 x +2}\right )}{6}-791 \ln \left (\frac {-2-x +\sqrt {2 x +3}}{x +1}\right )\)	\(104\)
derivativedivides	\(\frac {\frac {7475 \left (2 x +3\right )^{\frac {3}{2}}}{3}-4625 \sqrt {2 x +3}}{\left (6 x +4\right )^{2}}-\frac {1225 \sqrt {15}\, \operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{3}+\frac {3}{\left (\sqrt {2 x +3}-1\right )^{2}}+\frac {92}{\sqrt {2 x +3}-1}-791 \ln \left (\sqrt {2 x +3}-1\right )-\frac {3}{\left (\sqrt {2 x +3}+1\right )^{2}}+\frac {92}{\sqrt {2 x +3}+1}+791 \ln \left (\sqrt {2 x +3}+1\right )\)	\(124\)
default	\(\frac {\frac {7475 \left (2 x +3\right )^{\frac {3}{2}}}{3}-4625 \sqrt {2 x +3}}{\left (6 x +4\right )^{2}}-\frac {1225 \sqrt {15}\, \operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{3}+\frac {3}{\left (\sqrt {2 x +3}-1\right )^{2}}+\frac {92}{\sqrt {2 x +3}-1}-791 \ln \left (\sqrt {2 x +3}-1\right )-\frac {3}{\left (\sqrt {2 x +3}+1\right )^{2}}+\frac {92}{\sqrt {2 x +3}+1}+791 \ln \left (\sqrt {2 x +3}+1\right )\)	\(124\)
pseudoelliptic	\(\frac {-14700 \sqrt {15}\, \left (x +1\right )^{2} \left (x +\frac {2}{3}\right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )-28476 \left (x +1\right )^{2} \left (x +\frac {2}{3}\right )^{2} \ln \left (\sqrt {2 x +3}-1\right )+28476 \left (x +1\right )^{2} \left (x +\frac {2}{3}\right )^{2} \ln \left (\sqrt {2 x +3}+1\right )+\frac {2 \left (12443 x^{3}+30979 x^{2}+25073 x +6555\right ) \sqrt {2 x +3}}{3}}{\left (3 x +2\right )^{2} \left (\sqrt {2 x +3}-1\right )^{2} \left (\sqrt {2 x +3}+1\right )^{2}}\)	\(128\)

Input:

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

Output:

1/6*(12443*x^3+30979*x^2+25073*x+6555)/(3*x^2+5*x+2)^2*(2*x+3)^(1/2)+791*l 
n((2*x+3)^(1/2)+1)-791*ln((2*x+3)^(1/2)-1)-1225/3*15^(1/2)*arctanh(1/5*15^ 
(1/2)*(2*x+3)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (81) = 162\).

Time = 0.09 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.64 \[ \int \frac {(5-x) (3+2 x)^{7/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=\frac {3675 \, \sqrt {\frac {5}{3}} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (-\frac {3 \, \sqrt {\frac {5}{3}} \sqrt {2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 4746 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) - 4746 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) + {\left (12443 \, x^{3} + 30979 \, x^{2} + 25073 \, x + 6555\right )} \sqrt {2 \, x + 3}}{6 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \] Input:

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

Output:

1/6*(3675*sqrt(5/3)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(-(3*sqrt(5/3) 
*sqrt(2*x + 3) - 3*x - 7)/(3*x + 2)) + 4746*(9*x^4 + 30*x^3 + 37*x^2 + 20* 
x + 4)*log(sqrt(2*x + 3) + 1) - 4746*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)* 
log(sqrt(2*x + 3) - 1) + (12443*x^3 + 30979*x^2 + 25073*x + 6555)*sqrt(2*x 
 + 3))/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)
 

Sympy [A] (verification not implemented)

Time = 101.01 (sec) , antiderivative size = 408, normalized size of antiderivative = 4.08 \[ \int \frac {(5-x) (3+2 x)^{7/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=\frac {4765 \sqrt {15} \left (\log {\left (\sqrt {2 x + 3} - \frac {\sqrt {15}}{3} \right )} - \log {\left (\sqrt {2 x + 3} + \frac {\sqrt {15}}{3} \right )}\right )}{27} - \frac {62000 \left (\begin {cases} \frac {\sqrt {15} \left (- \frac {\log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1\right )}\right )}{75} & \text {for}\: \sqrt {2 x + 3} > - \frac {\sqrt {15}}{3} \wedge \sqrt {2 x + 3} < \frac {\sqrt {15}}{3} \end {cases}\right )}{9} + \frac {85000 \left (\begin {cases} \frac {\sqrt {15} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1\right )^{2}}\right )}{375} & \text {for}\: \sqrt {2 x + 3} > - \frac {\sqrt {15}}{3} \wedge \sqrt {2 x + 3} < \frac {\sqrt {15}}{3} \end {cases}\right )}{9} - 791 \log {\left (\sqrt {2 x + 3} - 1 \right )} + 791 \log {\left (\sqrt {2 x + 3} + 1 \right )} + \frac {92}{\sqrt {2 x + 3} + 1} - \frac {3}{\left (\sqrt {2 x + 3} + 1\right )^{2}} + \frac {92}{\sqrt {2 x + 3} - 1} + \frac {3}{\left (\sqrt {2 x + 3} - 1\right )^{2}} \] Input:

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

Output:

4765*sqrt(15)*(log(sqrt(2*x + 3) - sqrt(15)/3) - log(sqrt(2*x + 3) + sqrt( 
15)/3))/27 - 62000*Piecewise((sqrt(15)*(-log(sqrt(15)*sqrt(2*x + 3)/5 - 1) 
/4 + log(sqrt(15)*sqrt(2*x + 3)/5 + 1)/4 - 1/(4*(sqrt(15)*sqrt(2*x + 3)/5 
+ 1)) - 1/(4*(sqrt(15)*sqrt(2*x + 3)/5 - 1)))/75, (sqrt(2*x + 3) > -sqrt(1 
5)/3) & (sqrt(2*x + 3) < sqrt(15)/3)))/9 + 85000*Piecewise((sqrt(15)*(3*lo 
g(sqrt(15)*sqrt(2*x + 3)/5 - 1)/16 - 3*log(sqrt(15)*sqrt(2*x + 3)/5 + 1)/1 
6 + 3/(16*(sqrt(15)*sqrt(2*x + 3)/5 + 1)) + 1/(16*(sqrt(15)*sqrt(2*x + 3)/ 
5 + 1)**2) + 3/(16*(sqrt(15)*sqrt(2*x + 3)/5 - 1)) - 1/(16*(sqrt(15)*sqrt( 
2*x + 3)/5 - 1)**2))/375, (sqrt(2*x + 3) > -sqrt(15)/3) & (sqrt(2*x + 3) < 
 sqrt(15)/3)))/9 - 791*log(sqrt(2*x + 3) - 1) + 791*log(sqrt(2*x + 3) + 1) 
 + 92/(sqrt(2*x + 3) + 1) - 3/(sqrt(2*x + 3) + 1)**2 + 92/(sqrt(2*x + 3) - 
 1) + 3/(sqrt(2*x + 3) - 1)**2
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.34 \[ \int \frac {(5-x) (3+2 x)^{7/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=\frac {1225}{6} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) + \frac {12443 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 50029 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 64505 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 26775 \, \sqrt {2 \, x + 3}}{3 \, {\left (9 \, {\left (2 \, x + 3\right )}^{4} - 48 \, {\left (2 \, x + 3\right )}^{3} + 94 \, {\left (2 \, x + 3\right )}^{2} - 160 \, x - 215\right )}} + 791 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 791 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \] Input:

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

Output:

1225/6*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 
 3))) + 1/3*(12443*(2*x + 3)^(7/2) - 50029*(2*x + 3)^(5/2) + 64505*(2*x + 
3)^(3/2) - 26775*sqrt(2*x + 3))/(9*(2*x + 3)^4 - 48*(2*x + 3)^3 + 94*(2*x 
+ 3)^2 - 160*x - 215) + 791*log(sqrt(2*x + 3) + 1) - 791*log(sqrt(2*x + 3) 
 - 1)
                                                                                    
                                                                                    
 

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.20 \[ \int \frac {(5-x) (3+2 x)^{7/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=\frac {1225}{6} \, \sqrt {15} \log \left (\frac {{\left | -2 \, \sqrt {15} + 6 \, \sqrt {2 \, x + 3} \right |}}{2 \, {\left (\sqrt {15} + 3 \, \sqrt {2 \, x + 3}\right )}}\right ) + \frac {12443 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 50029 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 64505 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 26775 \, \sqrt {2 \, x + 3}}{3 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 791 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 791 \, \log \left ({\left | \sqrt {2 \, x + 3} - 1 \right |}\right ) \] Input:

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

Output:

1225/6*sqrt(15)*log(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*s 
qrt(2*x + 3))) + 1/3*(12443*(2*x + 3)^(7/2) - 50029*(2*x + 3)^(5/2) + 6450 
5*(2*x + 3)^(3/2) - 26775*sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16*x - 19)^2 + 7 
91*log(sqrt(2*x + 3) + 1) - 791*log(abs(sqrt(2*x + 3) - 1))
 

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.01 \[ \int \frac {(5-x) (3+2 x)^{7/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=1582\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right )+\frac {\frac {2975\,\sqrt {2\,x+3}}{3}-\frac {64505\,{\left (2\,x+3\right )}^{3/2}}{27}+\frac {50029\,{\left (2\,x+3\right )}^{5/2}}{27}-\frac {12443\,{\left (2\,x+3\right )}^{7/2}}{27}}{\frac {160\,x}{9}-\frac {94\,{\left (2\,x+3\right )}^2}{9}+\frac {16\,{\left (2\,x+3\right )}^3}{3}-{\left (2\,x+3\right )}^4+\frac {215}{9}}-\frac {1225\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{3} \] Input:

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

Output:

1582*atanh((2*x + 3)^(1/2)) + ((2975*(2*x + 3)^(1/2))/3 - (64505*(2*x + 3) 
^(3/2))/27 + (50029*(2*x + 3)^(5/2))/27 - (12443*(2*x + 3)^(7/2))/27)/((16 
0*x)/9 - (94*(2*x + 3)^2)/9 + (16*(2*x + 3)^3)/3 - (2*x + 3)^4 + 215/9) - 
(1225*15^(1/2)*atanh((15^(1/2)*(2*x + 3)^(1/2))/5))/3
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 383, normalized size of antiderivative = 3.83 \[ \int \frac {(5-x) (3+2 x)^{7/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=\frac {12443 \sqrt {2 x +3}\, x^{3}+30979 \sqrt {2 x +3}\, x^{2}+25073 \sqrt {2 x +3}\, x +6555 \sqrt {2 x +3}+11025 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}-\sqrt {15}\right ) x^{4}+36750 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}-\sqrt {15}\right ) x^{3}+45325 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}-\sqrt {15}\right ) x^{2}+24500 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}-\sqrt {15}\right ) x +4900 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}-\sqrt {15}\right )-11025 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}+\sqrt {15}\right ) x^{4}-36750 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}+\sqrt {15}\right ) x^{3}-45325 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}+\sqrt {15}\right ) x^{2}-24500 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}+\sqrt {15}\right ) x -4900 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}+\sqrt {15}\right )-42714 \,\mathrm {log}\left (\sqrt {2 x +3}-1\right ) x^{4}-142380 \,\mathrm {log}\left (\sqrt {2 x +3}-1\right ) x^{3}-175602 \,\mathrm {log}\left (\sqrt {2 x +3}-1\right ) x^{2}-94920 \,\mathrm {log}\left (\sqrt {2 x +3}-1\right ) x -18984 \,\mathrm {log}\left (\sqrt {2 x +3}-1\right )+42714 \,\mathrm {log}\left (\sqrt {2 x +3}+1\right ) x^{4}+142380 \,\mathrm {log}\left (\sqrt {2 x +3}+1\right ) x^{3}+175602 \,\mathrm {log}\left (\sqrt {2 x +3}+1\right ) x^{2}+94920 \,\mathrm {log}\left (\sqrt {2 x +3}+1\right ) x +18984 \,\mathrm {log}\left (\sqrt {2 x +3}+1\right )}{54 x^{4}+180 x^{3}+222 x^{2}+120 x +24} \] Input:

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

Output:

(12443*sqrt(2*x + 3)*x**3 + 30979*sqrt(2*x + 3)*x**2 + 25073*sqrt(2*x + 3) 
*x + 6555*sqrt(2*x + 3) + 11025*sqrt(15)*log(3*sqrt(2*x + 3) - sqrt(15))*x 
**4 + 36750*sqrt(15)*log(3*sqrt(2*x + 3) - sqrt(15))*x**3 + 45325*sqrt(15) 
*log(3*sqrt(2*x + 3) - sqrt(15))*x**2 + 24500*sqrt(15)*log(3*sqrt(2*x + 3) 
 - sqrt(15))*x + 4900*sqrt(15)*log(3*sqrt(2*x + 3) - sqrt(15)) - 11025*sqr 
t(15)*log(3*sqrt(2*x + 3) + sqrt(15))*x**4 - 36750*sqrt(15)*log(3*sqrt(2*x 
 + 3) + sqrt(15))*x**3 - 45325*sqrt(15)*log(3*sqrt(2*x + 3) + sqrt(15))*x* 
*2 - 24500*sqrt(15)*log(3*sqrt(2*x + 3) + sqrt(15))*x - 4900*sqrt(15)*log( 
3*sqrt(2*x + 3) + sqrt(15)) - 42714*log(sqrt(2*x + 3) - 1)*x**4 - 142380*l 
og(sqrt(2*x + 3) - 1)*x**3 - 175602*log(sqrt(2*x + 3) - 1)*x**2 - 94920*lo 
g(sqrt(2*x + 3) - 1)*x - 18984*log(sqrt(2*x + 3) - 1) + 42714*log(sqrt(2*x 
 + 3) + 1)*x**4 + 142380*log(sqrt(2*x + 3) + 1)*x**3 + 175602*log(sqrt(2*x 
 + 3) + 1)*x**2 + 94920*log(sqrt(2*x + 3) + 1)*x + 18984*log(sqrt(2*x + 3) 
 + 1))/(6*(9*x**4 + 30*x**3 + 37*x**2 + 20*x + 4))