3.20.54 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^8} \, dx\) [1954]

3.20.54.1 Optimal result

Integrand size = 24, antiderivative size = 168 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^8} \, dx=-\frac {(1-2 x)^{7/2}}{441 (2+3 x)^7}+\frac {23 (1-2 x)^{7/2}}{882 (2+3 x)^6}-\frac {467 (1-2 x)^{5/2}}{2646 (2+3 x)^5}+\frac {2335 (1-2 x)^{3/2}}{31752 (2+3 x)^4}-\frac {2335 \sqrt {1-2 x}}{95256 (2+3 x)^3}+\frac {2335 \sqrt {1-2 x}}{1333584 (2+3 x)^2}+\frac {2335 \sqrt {1-2 x}}{3111696 (2+3 x)}+\frac {2335 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1555848 \sqrt {21}} \]

output

-1/441*(1-2*x)^(7/2)/(2+3*x)^7+23/882*(1-2*x)^(7/2)/(2+3*x)^6-467/2646*(1- 
2*x)^(5/2)/(2+3*x)^5+2335/31752*(1-2*x)^(3/2)/(2+3*x)^4+2335/32672808*arct 
anh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-2335/95256*(1-2*x)^(1/2)/(2+3*x)^ 
3+2335/1333584*(1-2*x)^(1/2)/(2+3*x)^2+2335/3111696*(1-2*x)^(1/2)/(2+3*x)
 

3.20.54.2 Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.48 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^8} \, dx=\frac {\frac {21 \sqrt {1-2 x} \left (-1107536+1415408 x+1405308 x^2-23950566 x^3-24492348 x^4+8132805 x^5+1702215 x^6\right )}{2 (2+3 x)^7}+2335 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{32672808} \]

input

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^2)/(2 + 3*x)^8,x]
 

output

((21*Sqrt[1 - 2*x]*(-1107536 + 1415408*x + 1405308*x^2 - 23950566*x^3 - 24 
492348*x^4 + 8132805*x^5 + 1702215*x^6))/(2*(2 + 3*x)^7) + 2335*Sqrt[21]*A 
rcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/32672808
 

3.20.54.3 Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.18, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {100, 27, 87, 51, 51, 51, 52, 52, 73, 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.

\(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^2}{(3 x+2)^8} \, dx\)

\(\Big \downarrow \) 100

\(\displaystyle \frac {1}{441} \int \frac {7 (1-2 x)^{5/2} (525 x+281)}{(3 x+2)^7}dx-\frac {(1-2 x)^{7/2}}{441 (3 x+2)^7}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{63} \int \frac {(1-2 x)^{5/2} (525 x+281)}{(3 x+2)^7}dx-\frac {(1-2 x)^{7/2}}{441 (3 x+2)^7}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {1}{63} \left (\frac {2335}{14} \int \frac {(1-2 x)^{5/2}}{(3 x+2)^6}dx+\frac {23 (1-2 x)^{7/2}}{14 (3 x+2)^6}\right )-\frac {(1-2 x)^{7/2}}{441 (3 x+2)^7}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {1}{63} \left (\frac {2335}{14} \left (-\frac {1}{3} \int \frac {(1-2 x)^{3/2}}{(3 x+2)^5}dx-\frac {(1-2 x)^{5/2}}{15 (3 x+2)^5}\right )+\frac {23 (1-2 x)^{7/2}}{14 (3 x+2)^6}\right )-\frac {(1-2 x)^{7/2}}{441 (3 x+2)^7}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {1}{63} \left (\frac {2335}{14} \left (\frac {1}{3} \left (\frac {1}{4} \int \frac {\sqrt {1-2 x}}{(3 x+2)^4}dx+\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2}}{15 (3 x+2)^5}\right )+\frac {23 (1-2 x)^{7/2}}{14 (3 x+2)^6}\right )-\frac {(1-2 x)^{7/2}}{441 (3 x+2)^7}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {1}{63} \left (\frac {2335}{14} \left (\frac {1}{3} \left (\frac {1}{4} \left (-\frac {1}{9} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3}dx-\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2}}{15 (3 x+2)^5}\right )+\frac {23 (1-2 x)^{7/2}}{14 (3 x+2)^6}\right )-\frac {(1-2 x)^{7/2}}{441 (3 x+2)^7}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {1}{63} \left (\frac {2335}{14} \left (\frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{9} \left (\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}-\frac {3}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx\right )-\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2}}{15 (3 x+2)^5}\right )+\frac {23 (1-2 x)^{7/2}}{14 (3 x+2)^6}\right )-\frac {(1-2 x)^{7/2}}{441 (3 x+2)^7}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {1}{63} \left (\frac {2335}{14} \left (\frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{9} \left (\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}-\frac {3}{14} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )\right )-\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2}}{15 (3 x+2)^5}\right )+\frac {23 (1-2 x)^{7/2}}{14 (3 x+2)^6}\right )-\frac {(1-2 x)^{7/2}}{441 (3 x+2)^7}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{63} \left (\frac {2335}{14} \left (\frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{9} \left (\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}-\frac {3}{14} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )\right )-\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2}}{15 (3 x+2)^5}\right )+\frac {23 (1-2 x)^{7/2}}{14 (3 x+2)^6}\right )-\frac {(1-2 x)^{7/2}}{441 (3 x+2)^7}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{63} \left (\frac {2335}{14} \left (\frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{9} \left (\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}-\frac {3}{14} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )\right )-\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2}}{15 (3 x+2)^5}\right )+\frac {23 (1-2 x)^{7/2}}{14 (3 x+2)^6}\right )-\frac {(1-2 x)^{7/2}}{441 (3 x+2)^7}\)

input

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^2)/(2 + 3*x)^8,x]
 

output

-1/441*(1 - 2*x)^(7/2)/(2 + 3*x)^7 + ((23*(1 - 2*x)^(7/2))/(14*(2 + 3*x)^6 
) + (2335*(-1/15*(1 - 2*x)^(5/2)/(2 + 3*x)^5 + ((1 - 2*x)^(3/2)/(12*(2 + 3 
*x)^4) + (-1/9*Sqrt[1 - 2*x]/(2 + 3*x)^3 + (Sqrt[1 - 2*x]/(14*(2 + 3*x)^2) 
 - (3*(-1/7*Sqrt[1 - 2*x]/(2 + 3*x) - (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]) 
/(7*Sqrt[21])))/14)/9)/4)/3))/14)/63
 

3.20.54.3.1 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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

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])
 

3.20.54.4 Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.42

method	result	size
risch	\(-\frac {3404430 x^{7}[phone] x^{6}-57117501 x^{5}-23408784 x^{4}[phone] x^{3}+1425508 x^{2}-3630480 x +1107536}{3111696 \left (2+3 x \right )^{7} \sqrt {1-2 x}}+\frac {2335 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{32672808}\)	\(71\)
pseudoelliptic	\(\frac {4670 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{7} \sqrt {21}+21 \sqrt {1-2 x}\, \left (1702215 x^{6}+8132805 x^{5}-24492348 x^{4}-23950566 x^{3}+1405308 x^{2}+1415408 x -1107536\right )}{65345616 \left (2+3 x \right )^{7}}\)	\(75\)
trager	\(\frac {\left (1702215 x^{6}+8132805 x^{5}-24492348 x^{4}-23950566 x^{3}+1405308 x^{2}+1415408 x -1107536\right ) \sqrt {1-2 x}}{3111696 \left (2+3 x \right )^{7}}+\frac {2335 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{65345616}\)	\(92\)
derivativedivides	\(-\frac {139968 \left (\frac {2335 \left (1-2 x \right )^{\frac {13}{2}}}{298722816}-\frac {11675 \left (1-2 x \right )^{\frac {11}{2}}}{96018048}+\frac {6721 \left (1-2 x \right )^{\frac {9}{2}}}{164602368}+\frac {571 \left (1-2 x \right )^{\frac {7}{2}}}{321489}-\frac {132161 \left (1-2 x \right )^{\frac {5}{2}}}{30233088}+\frac {81725 \left (1-2 x \right )^{\frac {3}{2}}}{22674816}-\frac {114415 \sqrt {1-2 x}}{90699264}\right )}{\left (-4-6 x \right )^{7}}+\frac {2335 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{32672808}\)	\(93\)
default	\(-\frac {139968 \left (\frac {2335 \left (1-2 x \right )^{\frac {13}{2}}}{298722816}-\frac {11675 \left (1-2 x \right )^{\frac {11}{2}}}{96018048}+\frac {6721 \left (1-2 x \right )^{\frac {9}{2}}}{164602368}+\frac {571 \left (1-2 x \right )^{\frac {7}{2}}}{321489}-\frac {132161 \left (1-2 x \right )^{\frac {5}{2}}}{30233088}+\frac {81725 \left (1-2 x \right )^{\frac {3}{2}}}{22674816}-\frac {114415 \sqrt {1-2 x}}{90699264}\right )}{\left (-4-6 x \right )^{7}}+\frac {2335 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{32672808}\)	\(93\)

input

int((1-2*x)^(5/2)*(3+5*x)^2/(2+3*x)^8,x,method=_RETURNVERBOSE)
 

output

-1/3111696*(3404430*x^7+14563395*x^6-57117501*x^5-23408784*x^4+26761182*x^ 
3+1425508*x^2-3630480*x+1107536)/(2+3*x)^7/(1-2*x)^(1/2)+2335/32672808*arc 
tanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
 

3.20.54.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.86 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^8} \, dx=\frac {2335 \, \sqrt {21} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (1702215 \, x^{6} + 8132805 \, x^{5} - 24492348 \, x^{4} - 23950566 \, x^{3} + 1405308 \, x^{2} + 1415408 \, x - 1107536\right )} \sqrt {-2 \, x + 1}}{65345616 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]

input

integrate((1-2*x)^(5/2)*(3+5*x)^2/(2+3*x)^8,x, algorithm="fricas")
 

output

1/65345616*(2335*sqrt(21)*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 
15120*x^3 + 6048*x^2 + 1344*x + 128)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 
5)/(3*x + 2)) + 21*(1702215*x^6 + 8132805*x^5 - 24492348*x^4 - 23950566*x^ 
3 + 1405308*x^2 + 1415408*x - 1107536)*sqrt(-2*x + 1))/(2187*x^7 + 10206*x 
^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)
 

3.20.54.6 Sympy [F(-1)]

Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^8} \, dx=\text {Timed out} \]

input

integrate((1-2*x)**(5/2)*(3+5*x)**2/(2+3*x)**8,x)
 

output

Timed out
 

3.20.54.7 Maxima [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.98 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^8} \, dx=-\frac {2335}{65345616} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1702215 \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - 26478900 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + 8891883 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 386781696 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 951955683 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 784886900 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 274710415 \, \sqrt {-2 \, x + 1}}{1555848 \, {\left (2187 \, {\left (2 \, x - 1\right )}^{7} + 35721 \, {\left (2 \, x - 1\right )}^{6} + 250047 \, {\left (2 \, x - 1\right )}^{5} + 972405 \, {\left (2 \, x - 1\right )}^{4} + 2268945 \, {\left (2 \, x - 1\right )}^{3} + 3176523 \, {\left (2 \, x - 1\right )}^{2} + 4941258 \, x - 1647086\right )}} \]

input

integrate((1-2*x)^(5/2)*(3+5*x)^2/(2+3*x)^8,x, algorithm="maxima")
 

output

-2335/65345616*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*s 
qrt(-2*x + 1))) + 1/1555848*(1702215*(-2*x + 1)^(13/2) - 26478900*(-2*x + 
1)^(11/2) + 8891883*(-2*x + 1)^(9/2) + 386781696*(-2*x + 1)^(7/2) - 951955 
683*(-2*x + 1)^(5/2) + 784886900*(-2*x + 1)^(3/2) - 274710415*sqrt(-2*x + 
1))/(2187*(2*x - 1)^7 + 35721*(2*x - 1)^6 + 250047*(2*x - 1)^5 + 972405*(2 
*x - 1)^4 + 2268945*(2*x - 1)^3 + 3176523*(2*x - 1)^2 + 4941258*x - 164708 
6)
 

3.20.54.8 Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^8} \, dx=-\frac {2335}{65345616} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {1702215 \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + 26478900 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + 8891883 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - 386781696 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 951955683 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 784886900 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 274710415 \, \sqrt {-2 \, x + 1}}{199148544 \, {\left (3 \, x + 2\right )}^{7}} \]

input

integrate((1-2*x)^(5/2)*(3+5*x)^2/(2+3*x)^8,x, algorithm="giac")
 

output

-2335/65345616*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt( 
21) + 3*sqrt(-2*x + 1))) + 1/199148544*(1702215*(2*x - 1)^6*sqrt(-2*x + 1) 
 + 26478900*(2*x - 1)^5*sqrt(-2*x + 1) + 8891883*(2*x - 1)^4*sqrt(-2*x + 1 
) - 386781696*(2*x - 1)^3*sqrt(-2*x + 1) - 951955683*(2*x - 1)^2*sqrt(-2*x 
 + 1) + 784886900*(-2*x + 1)^(3/2) - 274710415*sqrt(-2*x + 1))/(3*x + 2)^7
 

3.20.54.9 Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^8} \, dx=\frac {\frac {81725\,{\left (1-2\,x\right )}^{3/2}}{354294}-\frac {114415\,\sqrt {1-2\,x}}{1417176}-\frac {132161\,{\left (1-2\,x\right )}^{5/2}}{472392}+\frac {36544\,{\left (1-2\,x\right )}^{7/2}}{321489}+\frac {6721\,{\left (1-2\,x\right )}^{9/2}}{2571912}-\frac {11675\,{\left (1-2\,x\right )}^{11/2}}{1500282}+\frac {2335\,{\left (1-2\,x\right )}^{13/2}}{4667544}}{\frac {1647086\,x}{729}+\frac {117649\,{\left (2\,x-1\right )}^2}{81}+\frac {84035\,{\left (2\,x-1\right )}^3}{81}+\frac {12005\,{\left (2\,x-1\right )}^4}{27}+\frac {343\,{\left (2\,x-1\right )}^5}{3}+\frac {49\,{\left (2\,x-1\right )}^6}{3}+{\left (2\,x-1\right )}^7-\frac {1647086}{2187}}+\frac {2335\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{32672808} \]

input

int(((1 - 2*x)^(5/2)*(5*x + 3)^2)/(3*x + 2)^8,x)
 

output

((81725*(1 - 2*x)^(3/2))/354294 - (114415*(1 - 2*x)^(1/2))/1417176 - (1321 
61*(1 - 2*x)^(5/2))/472392 + (36544*(1 - 2*x)^(7/2))/321489 + (6721*(1 - 2 
*x)^(9/2))/2571912 - (11675*(1 - 2*x)^(11/2))/1500282 + (2335*(1 - 2*x)^(1 
3/2))/4667544)/((1647086*x)/729 + (117649*(2*x - 1)^2)/81 + (84035*(2*x - 
1)^3)/81 + (12005*(2*x - 1)^4)/27 + (343*(2*x - 1)^5)/3 + (49*(2*x - 1)^6) 
/3 + (2*x - 1)^7 - 1647086/2187) + (2335*21^(1/2)*atanh((21^(1/2)*(1 - 2*x 
)^(1/2))/7))/32672808