3.20.76 \(\int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)} \, dx\) [1976]

3.20.76.1 Optimal result

Integrand size = 24, antiderivative size = 113 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)} \, dx=\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3}+\frac {112 \sqrt {1-2 x}}{9 (2+3 x)^2}+\frac {1073 \sqrt {1-2 x}}{9 (2+3 x)}+\frac {74020 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9 \sqrt {21}}-242 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

output

7/9*(1-2*x)^(3/2)/(2+3*x)^3+74020/189*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))* 
21^(1/2)-242*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+112/9*(1-2*x)^( 
1/2)/(2+3*x)^2+1073/9*(1-2*x)^(1/2)/(2+3*x)
 

3.20.76.2 Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.73 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)} \, dx=\frac {\sqrt {1-2 x} \left (4523+13198 x+9657 x^2\right )}{9 (2+3 x)^3}+\frac {74020 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9 \sqrt {21}}-242 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

input

Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^4*(3 + 5*x)),x]
 

output

(Sqrt[1 - 2*x]*(4523 + 13198*x + 9657*x^2))/(9*(2 + 3*x)^3) + (74020*ArcTa 
nh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(9*Sqrt[21]) - 242*Sqrt[55]*ArcTanh[Sqrt[5/11 
]*Sqrt[1 - 2*x]]
 

3.20.76.3 Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {109, 27, 166, 27, 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.

\(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^4 (5 x+3)} \, dx\)

\(\Big \downarrow \) 109

\(\displaystyle \frac {1}{9} \int \frac {3 (54-31 x) \sqrt {1-2 x}}{(3 x+2)^3 (5 x+3)}dx+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \int \frac {(54-31 x) \sqrt {1-2 x}}{(3 x+2)^3 (5 x+3)}dx+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {1}{3} \left (\frac {112 \sqrt {1-2 x}}{3 (3 x+2)^2}-\frac {1}{6} \int -\frac {2 (1425-1618 x)}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \int \frac {1425-1618 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {112 \sqrt {1-2 x}}{3 (3 x+2)^2}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {1}{7} \int \frac {35 (1752-1073 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {1073 \sqrt {1-2 x}}{3 x+2}\right )+\frac {112 \sqrt {1-2 x}}{3 (3 x+2)^2}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (5 \int \frac {1752-1073 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {1073 \sqrt {1-2 x}}{3 x+2}\right )+\frac {112 \sqrt {1-2 x}}{3 (3 x+2)^2}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (5 \left (11979 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-7402 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {1073 \sqrt {1-2 x}}{3 x+2}\right )+\frac {112 \sqrt {1-2 x}}{3 (3 x+2)^2}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (5 \left (7402 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-11979 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {1073 \sqrt {1-2 x}}{3 x+2}\right )+\frac {112 \sqrt {1-2 x}}{3 (3 x+2)^2}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (5 \left (\frac {14804 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-2178 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {1073 \sqrt {1-2 x}}{3 x+2}\right )+\frac {112 \sqrt {1-2 x}}{3 (3 x+2)^2}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3}\)

input

Int[(1 - 2*x)^(5/2)/((2 + 3*x)^4*(3 + 5*x)),x]
 

output

(7*(1 - 2*x)^(3/2))/(9*(2 + 3*x)^3) + ((112*Sqrt[1 - 2*x])/(3*(2 + 3*x)^2) 
 + ((1073*Sqrt[1 - 2*x])/(2 + 3*x) + 5*((14804*ArcTanh[Sqrt[3/7]*Sqrt[1 - 
2*x]])/Sqrt[21] - 2178*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]))/3)/3
 

3.20.76.3.1 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 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])
 

3.20.76.4 Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.61

method	result	size
risch	\(-\frac {19314 x^{3}+16739 x^{2}-4152 x -4523}{9 \left (2+3 x \right )^{3} \sqrt {1-2 x}}+\frac {74020 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{189}-242 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}\)	\(69\)
derivativedivides	\(-242 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {54 \left (\frac {1073 \left (1-2 x \right )^{\frac {5}{2}}}{27}-\frac {45710 \left (1-2 x \right )^{\frac {3}{2}}}{243}+\frac {54145 \sqrt {1-2 x}}{243}\right )}{\left (-4-6 x \right )^{3}}+\frac {74020 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{189}\)	\(75\)
default	\(-242 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {54 \left (\frac {1073 \left (1-2 x \right )^{\frac {5}{2}}}{27}-\frac {45710 \left (1-2 x \right )^{\frac {3}{2}}}{243}+\frac {54145 \sqrt {1-2 x}}{243}\right )}{\left (-4-6 x \right )^{3}}+\frac {74020 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{189}\)	\(75\)
pseudoelliptic	\(\frac {74020 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{3} \sqrt {21}-45738 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{3} \sqrt {55}+21 \sqrt {1-2 x}\, \left (9657 x^{2}+13198 x +4523\right )}{189 \left (2+3 x \right )^{3}}\)	\(80\)
trager	\(\frac {\left (9657 x^{2}+13198 x +4523\right ) \sqrt {1-2 x}}{9 \left (2+3 x \right )^{3}}-121 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )-\frac {37010 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{189}\)	\(116\)

input

int((1-2*x)^(5/2)/(2+3*x)^4/(3+5*x),x,method=_RETURNVERBOSE)
 

output

-1/9*(19314*x^3+16739*x^2-4152*x-4523)/(2+3*x)^3/(1-2*x)^(1/2)+74020/189*a 
rctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-242*arctanh(1/11*55^(1/2)*(1-2 
*x)^(1/2))*55^(1/2)
 

3.20.76.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.15 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)} \, dx=\frac {22869 \, \sqrt {55} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 37010 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (9657 \, x^{2} + 13198 \, x + 4523\right )} \sqrt {-2 \, x + 1}}{189 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

input

integrate((1-2*x)^(5/2)/(2+3*x)^4/(3+5*x),x, algorithm="fricas")
 

output

1/189*(22869*sqrt(55)*(27*x^3 + 54*x^2 + 36*x + 8)*log((5*x + sqrt(55)*sqr 
t(-2*x + 1) - 8)/(5*x + 3)) + 37010*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)* 
log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(9657*x^2 + 13198* 
x + 4523)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)
 

3.20.76.6 Sympy [A] (verification not implemented)

Time = 73.28 (sec) , antiderivative size = 583, normalized size of antiderivative = 5.16 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)} \, dx=- \frac {1331 \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 )}{7} + 121 \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 {28756 \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 )}{9} - \frac {11368 \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 )}{9} + \frac {5488 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {5 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{32} + \frac {5 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{32} - \frac {5}{32 \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 {1}{48 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{3}} - \frac {5}{32 \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 {1}{48 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{3}}\right )}{7203} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{9} \]

input

integrate((1-2*x)**(5/2)/(2+3*x)**4/(3+5*x),x)
 

output

-1331*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt 
(21)/3))/7 + 121*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 
2*x) + sqrt(55)/5)) + 28756*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)))/9 - 11368*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)))/9 + 5488*Piecewise((sqrt(21)*(-5*log(sqrt(21)*sq 
rt(1 - 2*x)/7 - 1)/32 + 5*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/32 - 5/(32*(sq 
rt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) - 
1/(48*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**3) - 5/(32*(sqrt(21)*sqrt(1 - 2*x)/7 
 - 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2) - 1/(48*(sqrt(21)*sqrt(1 
 - 2*x)/7 - 1)**3))/7203, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < 
 sqrt(21)/3)))/9
 

3.20.76.7 Maxima [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.13 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)} \, dx=121 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {37010}{189} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (9657 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 45710 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 54145 \, \sqrt {-2 \, x + 1}\right )}}{9 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]

input

integrate((1-2*x)^(5/2)/(2+3*x)^4/(3+5*x),x, algorithm="maxima")
 

output

121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 
1))) - 37010/189*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3 
*sqrt(-2*x + 1))) + 2/9*(9657*(-2*x + 1)^(5/2) - 45710*(-2*x + 1)^(3/2) + 
54145*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)
 

3.20.76.8 Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.09 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)} \, dx=121 \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {37010}{189} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {9657 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 45710 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 54145 \, \sqrt {-2 \, x + 1}}{36 \, {\left (3 \, x + 2\right )}^{3}} \]

input

integrate((1-2*x)^(5/2)/(2+3*x)^4/(3+5*x),x, algorithm="giac")
 

output

121*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sq 
rt(-2*x + 1))) - 37010/189*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x 
+ 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/36*(9657*(2*x - 1)^2*sqrt(-2*x + 
1) - 45710*(-2*x + 1)^(3/2) + 54145*sqrt(-2*x + 1))/(3*x + 2)^3
 

3.20.76.9 Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.79 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)} \, dx=\frac {74020\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{189}-242\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {108290\,\sqrt {1-2\,x}}{243}-\frac {91420\,{\left (1-2\,x\right )}^{3/2}}{243}+\frac {2146\,{\left (1-2\,x\right )}^{5/2}}{27}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}} \]

input

int((1 - 2*x)^(5/2)/((3*x + 2)^4*(5*x + 3)),x)
 

output

(74020*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/189 - 242*55^(1/2)*at 
anh((55^(1/2)*(1 - 2*x)^(1/2))/11) + ((108290*(1 - 2*x)^(1/2))/243 - (9142 
0*(1 - 2*x)^(3/2))/243 + (2146*(1 - 2*x)^(5/2))/27)/((98*x)/3 + 7*(2*x - 1 
)^2 + (2*x - 1)^3 - 98/27)