3.20.66 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^7} \, dx\) [1966]

3.20.66.1 Optimal result

Integrand size = 24, antiderivative size = 162 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^7} \, dx=-\frac {121 (1-2 x)^{3/2}}{4536 (2+3 x)^4}+\frac {33275 (1-2 x)^{3/2}}{95256 (2+3 x)^3}-\frac {559625 \sqrt {1-2 x}}{190512 (2+3 x)^2}+\frac {559625 \sqrt {1-2 x}}{1333584 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac {11 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^5}+\frac {559625 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{666792 \sqrt {21}} \]

output

-121/4536*(1-2*x)^(3/2)/(2+3*x)^4+33275/95256*(1-2*x)^(3/2)/(2+3*x)^3-1/18 
*(1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^6+11/27*(1-2*x)^(3/2)*(3+5*x)^3/(2+3*x)^5 
+559625/14002632*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-559625/19051 
2*(1-2*x)^(1/2)/(2+3*x)^2+559625/1333584*(1-2*x)^(1/2)/(2+3*x)
 

3.20.66.2 Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.46 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^7} \, dx=\frac {-\frac {21 \sqrt {1-2 x} \left (13847024+102558856 x+352611738 x^2+687940758 x^3+720187425 x^4+308539125 x^5\right )}{2 (2+3 x)^6}+559625 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{14002632} \]

input

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(2 + 3*x)^7,x]
 

output

((-21*Sqrt[1 - 2*x]*(13847024 + 102558856*x + 352611738*x^2 + 687940758*x^ 
3 + 720187425*x^4 + 308539125*x^5))/(2*(2 + 3*x)^6) + 559625*Sqrt[21]*ArcT 
anh[Sqrt[3/7]*Sqrt[1 - 2*x]])/14002632
 

3.20.66.3 Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.15, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {108, 27, 166, 27, 100, 27, 87, 51, 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.

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

\(\Big \downarrow \) 108

\(\displaystyle \frac {1}{18} \int -\frac {55 (1-2 x)^{3/2} x (5 x+3)^2}{(3 x+2)^6}dx-\frac {(1-2 x)^{5/2} (5 x+3)^3}{18 (3 x+2)^6}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {55}{18} \int \frac {(1-2 x)^{3/2} x (5 x+3)^2}{(3 x+2)^6}dx-\frac {(1-2 x)^{5/2} (5 x+3)^3}{18 (3 x+2)^6}\)

\(\Big \downarrow \) 166

\(\displaystyle -\frac {55}{18} \left (-\frac {1}{15} \int \frac {33 \sqrt {1-2 x} (5 x+3)^2}{(3 x+2)^5}dx-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{18 (3 x+2)^6}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {55}{18} \left (-\frac {11}{5} \int \frac {\sqrt {1-2 x} (5 x+3)^2}{(3 x+2)^5}dx-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{18 (3 x+2)^6}\)

\(\Big \downarrow \) 100

\(\displaystyle -\frac {55}{18} \left (-\frac {11}{5} \left (\frac {1}{252} \int \frac {75 \sqrt {1-2 x} (28 x+15)}{(3 x+2)^4}dx-\frac {(1-2 x)^{3/2}}{252 (3 x+2)^4}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{18 (3 x+2)^6}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {55}{18} \left (-\frac {11}{5} \left (\frac {25}{84} \int \frac {\sqrt {1-2 x} (28 x+15)}{(3 x+2)^4}dx-\frac {(1-2 x)^{3/2}}{252 (3 x+2)^4}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{18 (3 x+2)^6}\)

\(\Big \downarrow \) 87

\(\displaystyle -\frac {55}{18} \left (-\frac {11}{5} \left (\frac {25}{84} \left (\frac {185}{21} \int \frac {\sqrt {1-2 x}}{(3 x+2)^3}dx+\frac {11 (1-2 x)^{3/2}}{63 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2}}{252 (3 x+2)^4}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{18 (3 x+2)^6}\)

\(\Big \downarrow \) 51

\(\displaystyle -\frac {55}{18} \left (-\frac {11}{5} \left (\frac {25}{84} \left (\frac {185}{21} \left (-\frac {1}{6} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x}}{6 (3 x+2)^2}\right )+\frac {11 (1-2 x)^{3/2}}{63 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2}}{252 (3 x+2)^4}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{18 (3 x+2)^6}\)

\(\Big \downarrow \) 52

\(\displaystyle -\frac {55}{18} \left (-\frac {11}{5} \left (\frac {25}{84} \left (\frac {185}{21} \left (\frac {1}{6} \left (\frac {\sqrt {1-2 x}}{7 (3 x+2)}-\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {\sqrt {1-2 x}}{6 (3 x+2)^2}\right )+\frac {11 (1-2 x)^{3/2}}{63 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2}}{252 (3 x+2)^4}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{18 (3 x+2)^6}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {55}{18} \left (-\frac {11}{5} \left (\frac {25}{84} \left (\frac {185}{21} \left (\frac {1}{6} \left (\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}+\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{6 (3 x+2)^2}\right )+\frac {11 (1-2 x)^{3/2}}{63 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2}}{252 (3 x+2)^4}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{18 (3 x+2)^6}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {55}{18} \left (-\frac {11}{5} \left (\frac {25}{84} \left (\frac {185}{21} \left (\frac {1}{6} \left (\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}+\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{6 (3 x+2)^2}\right )+\frac {11 (1-2 x)^{3/2}}{63 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2}}{252 (3 x+2)^4}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{18 (3 x+2)^6}\)

input

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(2 + 3*x)^7,x]
 

output

-1/18*((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(2 + 3*x)^6 - (55*((-2*(1 - 2*x)^(3/2) 
*(3 + 5*x)^3)/(15*(2 + 3*x)^5) - (11*(-1/252*(1 - 2*x)^(3/2)/(2 + 3*x)^4 + 
 (25*((11*(1 - 2*x)^(3/2))/(63*(2 + 3*x)^3) + (185*(-1/6*Sqrt[1 - 2*x]/(2 
+ 3*x)^2 + (Sqrt[1 - 2*x]/(7*(2 + 3*x)) + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2* 
x]])/(7*Sqrt[21]))/6))/21))/84))/5))/18
 

3.20.66.3.1 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 108

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

rule 166

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - 
a*f)*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* 
c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h 
)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, 
e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
 

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])
 

3.20.66.4 Maple [A] (verified)

Time = 3.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.41

method	result	size
risch	\(\frac {617078250 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}-147494026 x^{2}-74864808 x -13847024}{1333584 \left (2+3 x \right )^{6} \sqrt {1-2 x}}+\frac {559625 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{14002632}\)	\(66\)
pseudoelliptic	\(\frac {1119250 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{6} \sqrt {21}-21 \sqrt {1-2 x}\, \left (308539125 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right )}{28005264 \left (2+3 x \right )^{6}}\)	\(70\)
derivativedivides	\(-\frac {11664 \left (-\frac {3809125 \left (1-2 x \right )^{\frac {11}{2}}}{96018048}+\frac {47350325 \left (1-2 x \right )^{\frac {9}{2}}}{123451776}-\frac {4383467 \left (1-2 x \right )^{\frac {7}{2}}}{2939328}+\frac {1231175 \left (1-2 x \right )^{\frac {5}{2}}}{419904}-\frac {66595375 \left (1-2 x \right )^{\frac {3}{2}}}{22674816}+\frac {27421625 \sqrt {1-2 x}}{22674816}\right )}{\left (-4-6 x \right )^{6}}+\frac {559625 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{14002632}\)	\(84\)
default	\(-\frac {11664 \left (-\frac {3809125 \left (1-2 x \right )^{\frac {11}{2}}}{96018048}+\frac {47350325 \left (1-2 x \right )^{\frac {9}{2}}}{123451776}-\frac {4383467 \left (1-2 x \right )^{\frac {7}{2}}}{2939328}+\frac {1231175 \left (1-2 x \right )^{\frac {5}{2}}}{419904}-\frac {66595375 \left (1-2 x \right )^{\frac {3}{2}}}{22674816}+\frac {27421625 \sqrt {1-2 x}}{22674816}\right )}{\left (-4-6 x \right )^{6}}+\frac {559625 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{14002632}\)	\(84\)
trager	\(-\frac {\left (308539125 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {1-2 x}}{1333584 \left (2+3 x \right )^{6}}+\frac {559625 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{28005264}\)	\(87\)

input

int((1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^7,x,method=_RETURNVERBOSE)
 

output

1/1333584*(617078250*x^6+1131835725*x^5+655694091*x^4+17282718*x^3-1474940 
26*x^2-74864808*x-13847024)/(2+3*x)^6/(1-2*x)^(1/2)+559625/14002632*arctan 
h(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
 

3.20.66.5 Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.80 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^7} \, dx=\frac {559625 \, \sqrt {21} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (308539125 \, x^{5} + 720187425 \, x^{4} + 687940758 \, x^{3} + 352611738 \, x^{2} + 102558856 \, x + 13847024\right )} \sqrt {-2 \, x + 1}}{28005264 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

input

integrate((1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^7,x, algorithm="fricas")
 

output

1/28005264*(559625*sqrt(21)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 21 
60*x^2 + 576*x + 64)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 
21*(308539125*x^5 + 720187425*x^4 + 687940758*x^3 + 352611738*x^2 + 102558 
856*x + 13847024)*sqrt(-2*x + 1))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^ 
3 + 2160*x^2 + 576*x + 64)
 

3.20.66.6 Sympy [F(-1)]

Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^7} \, dx=\text {Timed out} \]

input

integrate((1-2*x)**(5/2)*(3+5*x)**3/(2+3*x)**7,x)
 

output

Timed out
 

3.20.66.7 Maxima [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^7} \, dx=-\frac {559625}{28005264} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {308539125 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 2983070475 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 11598653682 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 22803823350 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 22842213625 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 9405617375 \, \sqrt {-2 \, x + 1}}{666792 \, {\left (729 \, {\left (2 \, x - 1\right )}^{6} + 10206 \, {\left (2 \, x - 1\right )}^{5} + 59535 \, {\left (2 \, x - 1\right )}^{4} + 185220 \, {\left (2 \, x - 1\right )}^{3} + 324135 \, {\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \]

input

integrate((1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^7,x, algorithm="maxima")
 

output

-559625/28005264*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3 
*sqrt(-2*x + 1))) + 1/666792*(308539125*(-2*x + 1)^(11/2) - 2983070475*(-2 
*x + 1)^(9/2) + 11598653682*(-2*x + 1)^(7/2) - 22803823350*(-2*x + 1)^(5/2 
) + 22842213625*(-2*x + 1)^(3/2) - 9405617375*sqrt(-2*x + 1))/(729*(2*x - 
1)^6 + 10206*(2*x - 1)^5 + 59535*(2*x - 1)^4 + 185220*(2*x - 1)^3 + 324135 
*(2*x - 1)^2 + 605052*x - 184877)
 

3.20.66.8 Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.81 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^7} \, dx=-\frac {559625}{28005264} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {308539125 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + 2983070475 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 11598653682 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 22803823350 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 22842213625 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 9405617375 \, \sqrt {-2 \, x + 1}}{42674688 \, {\left (3 \, x + 2\right )}^{6}} \]

input

integrate((1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^7,x, algorithm="giac")
 

output

-559625/28005264*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqr 
t(21) + 3*sqrt(-2*x + 1))) - 1/42674688*(308539125*(2*x - 1)^5*sqrt(-2*x + 
 1) + 2983070475*(2*x - 1)^4*sqrt(-2*x + 1) + 11598653682*(2*x - 1)^3*sqrt 
(-2*x + 1) + 22803823350*(2*x - 1)^2*sqrt(-2*x + 1) - 22842213625*(-2*x + 
1)^(3/2) + 9405617375*sqrt(-2*x + 1))/(3*x + 2)^6
 

3.20.66.9 Mupad [B] (verification not implemented)

Time = 1.43 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.78 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^7} \, dx=\frac {559625\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{14002632}-\frac {\frac {27421625\,\sqrt {1-2\,x}}{1417176}-\frac {66595375\,{\left (1-2\,x\right )}^{3/2}}{1417176}+\frac {1231175\,{\left (1-2\,x\right )}^{5/2}}{26244}-\frac {4383467\,{\left (1-2\,x\right )}^{7/2}}{183708}+\frac {47350325\,{\left (1-2\,x\right )}^{9/2}}{7715736}-\frac {3809125\,{\left (1-2\,x\right )}^{11/2}}{6001128}}{\frac {67228\,x}{81}+\frac {12005\,{\left (2\,x-1\right )}^2}{27}+\frac {6860\,{\left (2\,x-1\right )}^3}{27}+\frac {245\,{\left (2\,x-1\right )}^4}{3}+14\,{\left (2\,x-1\right )}^5+{\left (2\,x-1\right )}^6-\frac {184877}{729}} \]

input

int(((1 - 2*x)^(5/2)*(5*x + 3)^3)/(3*x + 2)^7,x)
 

output

(559625*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/14002632 - ((2742162 
5*(1 - 2*x)^(1/2))/1417176 - (66595375*(1 - 2*x)^(3/2))/1417176 + (1231175 
*(1 - 2*x)^(5/2))/26244 - (4383467*(1 - 2*x)^(7/2))/183708 + (47350325*(1 
- 2*x)^(9/2))/7715736 - (3809125*(1 - 2*x)^(11/2))/6001128)/((67228*x)/81 
+ (12005*(2*x - 1)^2)/27 + (6860*(2*x - 1)^3)/27 + (245*(2*x - 1)^4)/3 + 1 
4*(2*x - 1)^5 + (2*x - 1)^6 - 184877/729)