3.20.16 \(\int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^2} \, dx\) [1916]

3.20.16.1 Optimal result

Integrand size = 24, antiderivative size = 127 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^2} \, dx=-\frac {335 \sqrt {1-2 x}}{2 (3+5 x)}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)}+\frac {50 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)}-2311 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+204 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

output

-2311/7*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+204*arctanh(1/11*55^( 
1/2)*(1-2*x)^(1/2))*55^(1/2)-335/2*(1-2*x)^(1/2)/(3+5*x)+7/6*(1-2*x)^(1/2) 
/(2+3*x)^2/(3+5*x)+50/3*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)
 

3.20.16.2 Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^2} \, dx=-\frac {\sqrt {1-2 x} \left (1271+3920 x+3015 x^2\right )}{2 (2+3 x)^2 (3+5 x)}-2311 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+204 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

input

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

output

-1/2*(Sqrt[1 - 2*x]*(1271 + 3920*x + 3015*x^2))/((2 + 3*x)^2*(3 + 5*x)) - 
2311*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + 204*Sqrt[55]*ArcTanh[Sqr 
t[5/11]*Sqrt[1 - 2*x]]
 

3.20.16.3 Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {109, 168, 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.

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

\(\Big \downarrow \) 109

\(\displaystyle \frac {1}{6} \int \frac {122-167 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^2}dx+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{6} \left (\frac {1}{7} \int \frac {21 (437-500 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx+\frac {100 \sqrt {1-2 x}}{(3 x+2) (5 x+3)}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (3 \int \frac {437-500 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx+\frac {100 \sqrt {1-2 x}}{(3 x+2) (5 x+3)}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{6} \left (3 \left (-\frac {1}{11} \int \frac {33 (547-335 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {335 \sqrt {1-2 x}}{5 x+3}\right )+\frac {100 \sqrt {1-2 x}}{(3 x+2) (5 x+3)}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (3 \left (-3 \int \frac {547-335 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {335 \sqrt {1-2 x}}{5 x+3}\right )+\frac {100 \sqrt {1-2 x}}{(3 x+2) (5 x+3)}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {1}{6} \left (3 \left (-3 \left (3740 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-2311 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {335 \sqrt {1-2 x}}{5 x+3}\right )+\frac {100 \sqrt {1-2 x}}{(3 x+2) (5 x+3)}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{6} \left (3 \left (-3 \left (2311 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-3740 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {335 \sqrt {1-2 x}}{5 x+3}\right )+\frac {100 \sqrt {1-2 x}}{(3 x+2) (5 x+3)}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{6} \left (3 \left (-3 \left (\frac {4622 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-136 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {335 \sqrt {1-2 x}}{5 x+3}\right )+\frac {100 \sqrt {1-2 x}}{(3 x+2) (5 x+3)}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)}\)

input

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

output

(7*Sqrt[1 - 2*x])/(6*(2 + 3*x)^2*(3 + 5*x)) + ((100*Sqrt[1 - 2*x])/((2 + 3 
*x)*(3 + 5*x)) + 3*((-335*Sqrt[1 - 2*x])/(3 + 5*x) - 3*((4622*ArcTanh[Sqrt 
[3/7]*Sqrt[1 - 2*x]])/Sqrt[21] - 136*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 
2*x]])))/6
 

3.20.16.3.1 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 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])
 

3.20.16.4 Maple [A] (verified)

Time = 1.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.60

method	result	size
risch	\(\frac {6030 x^{3}+4825 x^{2}-1378 x -1271}{2 \left (3+5 x \right ) \sqrt {1-2 x}\, \left (2+3 x \right )^{2}}-\frac {2311 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}+204 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}\)	\(76\)
derivativedivides	\(\frac {22 \sqrt {1-2 x}}{-\frac {6}{5}-2 x}+204 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}+\frac {405 \left (1-2 x \right )^{\frac {3}{2}}-959 \sqrt {1-2 x}}{\left (-4-6 x \right )^{2}}-\frac {2311 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\)	\(82\)
default	\(\frac {22 \sqrt {1-2 x}}{-\frac {6}{5}-2 x}+204 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}+\frac {405 \left (1-2 x \right )^{\frac {3}{2}}-959 \sqrt {1-2 x}}{\left (-4-6 x \right )^{2}}-\frac {2311 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\)	\(82\)
pseudoelliptic	\(\frac {-4622 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{2} \left (3+5 x \right ) \sqrt {21}+2856 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{2} \left (3+5 x \right ) \sqrt {55}-7 \sqrt {1-2 x}\, \left (3015 x^{2}+3920 x +1271\right )}{14 \left (2+3 x \right )^{2} \left (3+5 x \right )}\)	\(97\)
trager	\(-\frac {\left (3015 x^{2}+3920 x +1271\right ) \sqrt {1-2 x}}{2 \left (2+3 x \right )^{2} \left (3+5 x \right )}-\frac {2311 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{14}-102 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )\)	\(123\)

input

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

output

1/2*(6030*x^3+4825*x^2-1378*x-1271)/(3+5*x)/(1-2*x)^(1/2)/(2+3*x)^2-2311/7 
*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+204*arctanh(1/11*55^(1/2)*(1 
-2*x)^(1/2))*55^(1/2)
 

3.20.16.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.07 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^2} \, dx=\frac {2311 \, \sqrt {7} \sqrt {3} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) + 1428 \, \sqrt {55} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac {5 \, x - \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 7 \, {\left (3015 \, x^{2} + 3920 \, x + 1271\right )} \sqrt {-2 \, x + 1}}{14 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \]

input

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

output

1/14*(2311*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)) + 1428*sqrt(55)*(45*x^3 + 87*x^2 
+ 56*x + 12)*log((5*x - sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - 7*(3015* 
x^2 + 3920*x + 1271)*sqrt(-2*x + 1))/(45*x^3 + 87*x^2 + 56*x + 12)
 

3.20.16.6 Sympy [A] (verification not implemented)

Time = 52.86 (sec) , antiderivative size = 486, normalized size of antiderivative = 3.83 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^2} \, dx=\frac {1133 \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} - 103 \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 ) - 1848 \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 ) + 392 \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 ) - 2420 \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)**(3/2)/(2+3*x)**3/(3+5*x)**2,x)
 

output

1133*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt( 
21)/3))/7 - 103*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2 
*x) + sqrt(55)/5)) - 1848*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))) + 392*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)*s 
qrt(1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2 
*x) < sqrt(21)/3))) - 2420*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)*sqrt(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)))
 

3.20.16.7 Maxima [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.01 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^2} \, dx=-102 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2311}{14} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {3015 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 13870 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 15939 \, \sqrt {-2 \, x + 1}}{45 \, {\left (2 \, x - 1\right )}^{3} + 309 \, {\left (2 \, x - 1\right )}^{2} + 1414 \, x - 168} \]

input

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

output

-102*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 
 1))) + 2311/14*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3* 
sqrt(-2*x + 1))) - (3015*(-2*x + 1)^(5/2) - 13870*(-2*x + 1)^(3/2) + 15939 
*sqrt(-2*x + 1))/(45*(2*x - 1)^3 + 309*(2*x - 1)^2 + 1414*x - 168)
 

3.20.16.8 Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.97 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^2} \, dx=-102 \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2311}{14} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {55 \, \sqrt {-2 \, x + 1}}{5 \, x + 3} + \frac {405 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 959 \, \sqrt {-2 \, x + 1}}{4 \, {\left (3 \, x + 2\right )}^{2}} \]

input

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

output

-102*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*s 
qrt(-2*x + 1))) + 2311/14*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 
 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 55*sqrt(-2*x + 1)/(5*x + 3) + 1/4*(4 
05*(-2*x + 1)^(3/2) - 959*sqrt(-2*x + 1))/(3*x + 2)^2
 

3.20.16.9 Mupad [B] (verification not implemented)

Time = 1.54 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^2} \, dx=204\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )-\frac {2311\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{7}-\frac {\frac {1771\,\sqrt {1-2\,x}}{5}-\frac {2774\,{\left (1-2\,x\right )}^{3/2}}{9}+67\,{\left (1-2\,x\right )}^{5/2}}{\frac {1414\,x}{45}+\frac {103\,{\left (2\,x-1\right )}^2}{15}+{\left (2\,x-1\right )}^3-\frac {56}{15}} \]

input

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

output

204*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11) - (2311*21^(1/2)*atanh(( 
21^(1/2)*(1 - 2*x)^(1/2))/7))/7 - ((1771*(1 - 2*x)^(1/2))/5 - (2774*(1 - 2 
*x)^(3/2))/9 + 67*(1 - 2*x)^(5/2))/((1414*x)/45 + (103*(2*x - 1)^2)/15 + ( 
2*x - 1)^3 - 56/15)