3.20.87 \(\int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^2} \, dx\) [1987]

3.20.87.1 Optimal result

Integrand size = 24, antiderivative size = 131 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^2} \, dx=-\frac {6763 \sqrt {1-2 x}}{18 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)}+\frac {343 \sqrt {1-2 x}}{9 (2+3 x) (3+5 x)}-\frac {6665}{3} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+2288 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

output

7/6*(1-2*x)^(3/2)/(2+3*x)^2/(3+5*x)+2288/5*arctanh(1/11*55^(1/2)*(1-2*x)^( 
1/2))*55^(1/2)-6665/9*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-6763/18 
*(1-2*x)^(1/2)/(3+5*x)+343/9*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)
 

3.20.87.2 Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^2} \, dx=-\frac {\sqrt {1-2 x} \left (8553+26380 x+20289 x^2\right )}{6 (2+3 x)^2 (3+5 x)}-\frac {6665}{3} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+2288 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

input

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

output

-1/6*(Sqrt[1 - 2*x]*(8553 + 26380*x + 20289*x^2))/((2 + 3*x)^2*(3 + 5*x)) 
- (6665*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3 + 2288*Sqrt[11/5]*Ar 
cTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
 

3.20.87.3 Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {109, 166, 25, 168, 27, 174, 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.

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

\(\Big \downarrow \) 109

\(\displaystyle \frac {1}{6} \int \frac {(164-97 x) \sqrt {1-2 x}}{(3 x+2)^2 (5 x+3)^2}dx+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {1}{6} \left (\frac {686 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)}-\frac {1}{3} \int -\frac {8821-10096 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx\right )+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{6} \left (\frac {1}{3} \int \frac {8821-10096 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx+\frac {686 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (-\frac {1}{11} \int \frac {33 (11043-6763 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {6763 \sqrt {1-2 x}}{5 x+3}\right )+\frac {686 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (-3 \int \frac {11043-6763 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {6763 \sqrt {1-2 x}}{5 x+3}\right )+\frac {686 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (-3 \left (75504 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-46655 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {6763 \sqrt {1-2 x}}{5 x+3}\right )+\frac {686 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (-3 \left (46655 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-75504 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {6763 \sqrt {1-2 x}}{5 x+3}\right )+\frac {686 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (-3 \left (13330 \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-13728 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {6763 \sqrt {1-2 x}}{5 x+3}\right )+\frac {686 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)}\)

input

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

output

(7*(1 - 2*x)^(3/2))/(6*(2 + 3*x)^2*(3 + 5*x)) + ((686*Sqrt[1 - 2*x])/(3*(2 
 + 3*x)*(3 + 5*x)) + ((-6763*Sqrt[1 - 2*x])/(3 + 5*x) - 3*(13330*Sqrt[7/3] 
*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 13728*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sq 
rt[1 - 2*x]]))/3)/6
 

3.20.87.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 109

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) 
+ c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) 
 + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, 
d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 168

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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 174

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

3.20.87.4 Maple [A] (verified)

Time = 1.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.58

method	result	size
risch	\(\frac {40578 x^{3}+32471 x^{2}-9274 x -8553}{6 \left (3+5 x \right ) \sqrt {1-2 x}\, \left (2+3 x \right )^{2}}-\frac {6665 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9}+\frac {2288 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{5}\)	\(76\)
derivativedivides	\(\frac {242 \sqrt {1-2 x}}{5 \left (-\frac {6}{5}-2 x \right )}+\frac {2288 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{5}+\frac {917 \left (1-2 x \right )^{\frac {3}{2}}-\frac {6517 \sqrt {1-2 x}}{3}}{\left (-4-6 x \right )^{2}}-\frac {6665 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9}\)	\(82\)
default	\(\frac {242 \sqrt {1-2 x}}{5 \left (-\frac {6}{5}-2 x \right )}+\frac {2288 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{5}+\frac {917 \left (1-2 x \right )^{\frac {3}{2}}-\frac {6517 \sqrt {1-2 x}}{3}}{\left (-4-6 x \right )^{2}}-\frac {6665 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9}\)	\(82\)
pseudoelliptic	\(\frac {-66650 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{2} \left (3+5 x \right ) \sqrt {21}+41184 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{2} \left (3+5 x \right ) \sqrt {55}-15 \sqrt {1-2 x}\, \left (20289 x^{2}+26380 x +8553\right )}{90 \left (2+3 x \right )^{2} \left (3+5 x \right )}\)	\(97\)
trager	\(-\frac {\left (20289 x^{2}+26380 x +8553\right ) \sqrt {1-2 x}}{6 \left (2+3 x \right )^{2} \left (3+5 x \right )}+\frac {6665 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{18}+\frac {1144 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{5}\)	\(123\)

input

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

output

1/6*(40578*x^3+32471*x^2-9274*x-8553)/(3+5*x)/(1-2*x)^(1/2)/(2+3*x)^2-6665 
/9*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+2288/5*arctanh(1/11*55^(1/ 
2)*(1-2*x)^(1/2))*55^(1/2)
 

3.20.87.5 Fricas [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.08 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^2} \, dx=\frac {20592 \, \sqrt {11} \sqrt {5} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 33325 \, \sqrt {7} \sqrt {3} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 15 \, {\left (20289 \, x^{2} + 26380 \, x + 8553\right )} \sqrt {-2 \, x + 1}}{90 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \]

input

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

output

1/90*(20592*sqrt(11)*sqrt(5)*(45*x^3 + 87*x^2 + 56*x + 12)*log(-(sqrt(11)* 
sqrt(5)*sqrt(-2*x + 1) - 5*x + 8)/(5*x + 3)) + 33325*sqrt(7)*sqrt(3)*(45*x 
^3 + 87*x^2 + 56*x + 12)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3 
*x + 2)) - 15*(20289*x^2 + 26380*x + 8553)*sqrt(-2*x + 1))/(45*x^3 + 87*x^ 
2 + 56*x + 12)
 

3.20.87.6 Sympy [A] (verification not implemented)

Time = 49.32 (sec) , antiderivative size = 488, normalized size of antiderivative = 3.73 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^2} \, dx=363 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right ) - 231 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right ) - \frac {12544 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{3} + \frac {2744 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{3} - 5324 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right ) \]

input

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

output

363*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(2 
1)/3)) - 231*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) 
 + sqrt(55)/5)) - 12544*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 
 - 1)/4 + log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(1 - 2* 
x)/7 + 1)) - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (sqrt(1 - 2*x) > - 
sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/3 + 2744*Piecewise((sqrt(21)* 
(3*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(1 - 2*x)/7 + 
 1)/16 + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(1 - 
2*x)/7 + 1)**2) + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) - 1/(16*(sqrt(21)* 
sqrt(1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 
2*x) < sqrt(21)/3)))/3 - 5324*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(1 - 
2*x)/11 - 1)/4 + log(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/4 - 1/(4*(sqrt(55)*sqr 
t(1 - 2*x)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)))/605, (sqrt(1 
- 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))
 

3.20.87.7 Maxima [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^2} \, dx=-\frac {1144}{5} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {6665}{18} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {20289 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 93338 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 107261 \, \sqrt {-2 \, x + 1}}{3 \, {\left (45 \, {\left (2 \, x - 1\right )}^{3} + 309 \, {\left (2 \, x - 1\right )}^{2} + 1414 \, x - 168\right )}} \]

input

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

output

-1144/5*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2* 
x + 1))) + 6665/18*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 
 3*sqrt(-2*x + 1))) - 1/3*(20289*(-2*x + 1)^(5/2) - 93338*(-2*x + 1)^(3/2) 
 + 107261*sqrt(-2*x + 1))/(45*(2*x - 1)^3 + 309*(2*x - 1)^2 + 1414*x - 168 
)
 

3.20.87.8 Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.94 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^2} \, dx=-\frac {1144}{5} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {6665}{18} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {121 \, \sqrt {-2 \, x + 1}}{5 \, x + 3} + \frac {7 \, {\left (393 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 931 \, \sqrt {-2 \, x + 1}\right )}}{12 \, {\left (3 \, x + 2\right )}^{2}} \]

input

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

output

-1144/5*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 
5*sqrt(-2*x + 1))) + 6665/18*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2* 
x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 121*sqrt(-2*x + 1)/(5*x + 3) + 7/ 
12*(393*(-2*x + 1)^(3/2) - 931*sqrt(-2*x + 1))/(3*x + 2)^2
 

3.20.87.9 Mupad [B] (verification not implemented)

Time = 1.48 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^2} \, dx=\frac {2288\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{5}-\frac {6665\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{9}-\frac {\frac {107261\,\sqrt {1-2\,x}}{135}-\frac {93338\,{\left (1-2\,x\right )}^{3/2}}{135}+\frac {6763\,{\left (1-2\,x\right )}^{5/2}}{45}}{\frac {1414\,x}{45}+\frac {103\,{\left (2\,x-1\right )}^2}{15}+{\left (2\,x-1\right )}^3-\frac {56}{15}} \]

input

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

output

(2288*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/5 - (6665*21^(1/2)*at 
anh((21^(1/2)*(1 - 2*x)^(1/2))/7))/9 - ((107261*(1 - 2*x)^(1/2))/135 - (93 
338*(1 - 2*x)^(3/2))/135 + (6763*(1 - 2*x)^(5/2))/45)/((1414*x)/45 + (103* 
(2*x - 1)^2)/15 + (2*x - 1)^3 - 56/15)