\(\int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx\) [559]

Optimal result

Integrand size = 24, antiderivative size = 153 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx=-\frac {1045 \sqrt {1-2 x}}{14 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^2}+\frac {139 \sqrt {1-2 x}}{14 (2+3 x) (3+5 x)^2}+\frac {34655 \sqrt {1-2 x}}{77 (3+5 x)}+\frac {43467}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {66325}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Output:

-1045/14*(1-2*x)^(1/2)/(3+5*x)^2+1/2*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^2+139 
/14*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^2+34655*(1-2*x)^(1/2)/(231+385*x)+43467/ 
49*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))-66325/121*55^(1/2)*arctanh 
(1/11*55^(1/2)*(1-2*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx=\frac {\sqrt {1-2 x} \left (788875+3748007 x+5926515 x^2+3118950 x^3\right )}{154 \left (6+19 x+15 x^2\right )^2}+\frac {43467}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {66325}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Input:

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

Output:

(Sqrt[1 - 2*x]*(788875 + 3748007*x + 5926515*x^2 + 3118950*x^3))/(154*(6 + 
 19*x + 15*x^2)^2) + (43467*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 
- (66325*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {110, 25, 168, 168, 27, 168, 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.

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

\(\Big \downarrow \) 110

\(\displaystyle \frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}-\frac {1}{2} \int -\frac {23-35 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^3}dx\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \int \frac {23-35 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^3}dx+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{2} \left (\frac {1}{7} \int \frac {2513-3475 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^3}dx+\frac {139 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{2} \left (\frac {1}{7} \left (-\frac {1}{22} \int \frac {22 (8219-9405 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {1045 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {139 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {1}{7} \left (-\int \frac {8219-9405 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {1045 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {139 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{2} \left (\frac {1}{7} \left (\frac {1}{11} \int \frac {339517-207930 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {69310 \sqrt {1-2 x}}{11 (5 x+3)}-\frac {1045 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {139 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {1}{2} \left (\frac {1}{7} \left (\frac {1}{11} \left (2321375 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-1434411 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {69310 \sqrt {1-2 x}}{11 (5 x+3)}-\frac {1045 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {139 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{2} \left (\frac {1}{7} \left (\frac {1}{11} \left (1434411 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-2321375 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {69310 \sqrt {1-2 x}}{11 (5 x+3)}-\frac {1045 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {139 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (\frac {1}{7} \left (\frac {1}{11} \left (956274 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-928550 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {69310 \sqrt {1-2 x}}{11 (5 x+3)}-\frac {1045 \sqrt {1-2 x}}{(5 x+3)^2}\right )+\frac {139 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}\)

Input:

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

Output:

Sqrt[1 - 2*x]/(2*(2 + 3*x)^2*(3 + 5*x)^2) + ((139*Sqrt[1 - 2*x])/(7*(2 + 3 
*x)*(3 + 5*x)^2) + ((-1045*Sqrt[1 - 2*x])/(3 + 5*x)^2 + (69310*Sqrt[1 - 2* 
x])/(11*(3 + 5*x)) + (956274*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 
928550*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11)/7)/2
 

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 110

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 + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f))   Int[(a + b*x)^(m + 1) 
*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + 
p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt 
Q[n, 0] && (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])
 

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.52

method	result	size
risch	\(-\frac {\left (-1+2 x \right ) \left (3118950 x^{3}+5926515 x^{2}+3748007 x +788875\right )}{154 \left (15 x^{2}+19 x +6\right )^{2} \sqrt {1-2 x}}+\frac {43467 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{49}-\frac {66325 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{121}\)	\(79\)
derivativedivides	\(\frac {-\frac {24875 \left (1-2 x \right )^{\frac {3}{2}}}{11}+4925 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-\frac {66325 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{121}-\frac {972 \left (\frac {209 \left (1-2 x \right )^{\frac {3}{2}}}{252}-\frac {211 \sqrt {1-2 x}}{108}\right )}{\left (-4-6 x \right )^{2}}+\frac {43467 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{49}\)	\(94\)
default	\(\frac {-\frac {24875 \left (1-2 x \right )^{\frac {3}{2}}}{11}+4925 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-\frac {66325 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{121}-\frac {972 \left (\frac {209 \left (1-2 x \right )^{\frac {3}{2}}}{252}-\frac {211 \sqrt {1-2 x}}{108}\right )}{\left (-4-6 x \right )^{2}}+\frac {43467 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{49}\)	\(94\)
pseudoelliptic	\(\frac {10519014 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (15 x^{2}+19 x +6\right )^{2} \sqrt {21}-6499850 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (15 x^{2}+19 x +6\right )^{2} \sqrt {55}+77 \sqrt {1-2 x}\, \left (3118950 x^{3}+5926515 x^{2}+3748007 x +788875\right )}{11858 \left (2+3 x \right )^{2} \left (3+5 x \right )^{2}}\)	\(102\)
trager	\(\frac {\left (3118950 x^{3}+5926515 x^{2}+3748007 x +788875\right ) \sqrt {1-2 x}}{154 \left (15 x^{2}+19 x +6\right )^{2}}-\frac {175 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7900255\right ) \ln \left (-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7900255\right ) x -8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7900255\right )-20845 \sqrt {1-2 x}}{3+5 x}\right )}{242}+\frac {43467 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{98}\)	\(127\)

Input:

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

Output:

-1/154*(-1+2*x)*(3118950*x^3+5926515*x^2+3748007*x+788875)/(15*x^2+19*x+6) 
^2/(1-2*x)^(1/2)+43467/49*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))-663 
25/121*55^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx=\frac {464275 \, \sqrt {\frac {5}{11}} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac {5 \, x + 11 \, \sqrt {\frac {5}{11}} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 478137 \, \sqrt {\frac {3}{7}} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac {3 \, x - 7 \, \sqrt {\frac {3}{7}} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + {\left (3118950 \, x^{3} + 5926515 \, x^{2} + 3748007 \, x + 788875\right )} \sqrt {-2 \, x + 1}}{154 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \] Input:

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

Output:

1/154*(464275*sqrt(5/11)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log((5 
*x + 11*sqrt(5/11)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 478137*sqrt(3/7)*(225* 
x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log((3*x - 7*sqrt(3/7)*sqrt(-2*x + 1 
) - 5)/(3*x + 2)) + (3118950*x^3 + 5926515*x^2 + 3748007*x + 788875)*sqrt( 
-2*x + 1))/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)
 

Sympy [A] (verification not implemented)

Time = 82.78 (sec) , antiderivative size = 654, normalized size of antiderivative = 4.27 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-3060*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt 
(21)/3))/7 + 3060*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 
 2*x) + sqrt(55)/5))/11 + 3708*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))) - 504*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))) + 10100*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))) + 2200*Piecewise 
((sqrt(55)*(3*log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/16 - 3*log(sqrt(55)*sqrt( 
1 - 2*x)/11 + 1)/16 + 3/(16*(sqrt(55)*sqrt(1 - 2*x)/11 + 1)) + 1/(16*(sqrt 
(55)*sqrt(1 - 2*x)/11 + 1)**2) + 3/(16*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)) - 
1/(16*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)**2))/6655, (sqrt(1 - 2*x) > -sqrt(55 
)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx=\frac {66325}{242} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {43467}{98} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (1559475 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 10604940 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 24027469 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 18137504 \, \sqrt {-2 \, x + 1}\right )}}{77 \, {\left (225 \, {\left (2 \, x - 1\right )}^{4} + 2040 \, {\left (2 \, x - 1\right )}^{3} + 6934 \, {\left (2 \, x - 1\right )}^{2} + 20944 \, x - 4543\right )}} \] Input:

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

Output:

66325/242*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(- 
2*x + 1))) - 43467/98*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21 
) + 3*sqrt(-2*x + 1))) - 2/77*(1559475*(-2*x + 1)^(7/2) - 10604940*(-2*x + 
 1)^(5/2) + 24027469*(-2*x + 1)^(3/2) - 18137504*sqrt(-2*x + 1))/(225*(2*x 
 - 1)^4 + 2040*(2*x - 1)^3 + 6934*(2*x - 1)^2 + 20944*x - 4543)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx=\frac {66325}{242} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {43467}{98} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2 \, {\left (1559475 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 10604940 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 24027469 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 18137504 \, \sqrt {-2 \, x + 1}\right )}}{77 \, {\left (15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \] Input:

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

Output:

66325/242*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) 
+ 5*sqrt(-2*x + 1))) - 43467/98*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt( 
-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/77*(1559475*(2*x - 1)^3*sqrt 
(-2*x + 1) + 10604940*(2*x - 1)^2*sqrt(-2*x + 1) - 24027469*(-2*x + 1)^(3/ 
2) + 18137504*sqrt(-2*x + 1))/(15*(2*x - 1)^2 + 136*x + 9)^2
 

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx=\frac {43467\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{49}-\frac {66325\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121}+\frac {\frac {471104\,\sqrt {1-2\,x}}{225}-\frac {48054938\,{\left (1-2\,x\right )}^{3/2}}{17325}+\frac {1413992\,{\left (1-2\,x\right )}^{5/2}}{1155}-\frac {13862\,{\left (1-2\,x\right )}^{7/2}}{77}}{\frac {20944\,x}{225}+\frac {6934\,{\left (2\,x-1\right )}^2}{225}+\frac {136\,{\left (2\,x-1\right )}^3}{15}+{\left (2\,x-1\right )}^4-\frac {4543}{225}} \] Input:

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

Output:

(43467*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/49 - (66325*55^(1/2)* 
atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/121 + ((471104*(1 - 2*x)^(1/2))/225 
- (48054938*(1 - 2*x)^(3/2))/17325 + (1413992*(1 - 2*x)^(5/2))/1155 - (138 
62*(1 - 2*x)^(7/2))/77)/((20944*x)/225 + (6934*(2*x - 1)^2)/225 + (136*(2* 
x - 1)^3)/15 + (2*x - 1)^4 - 4543/225)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 443, normalized size of antiderivative = 2.90 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx=\frac {240159150 \sqrt {-2 x +1}\, x^{3}[phone] \sqrt {-2 x +1}\, x^{2}[phone] \sqrt {-2 x +1}\, x [phone] \sqrt {-2 x +1}[phone] \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{4}[phone] \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{3}[phone] \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{2}[phone] \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x [phone] \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right )-731233125 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{4}-1852457250 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{3}-1758209425 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{2}-740982900 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x -116997300 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right )-1183389075 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{4}-2997918990 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{3}-2845393287 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{2}-1199167596 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x -189342252 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )[phone] \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{4}[phone] \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{3}[phone] \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{2}[phone] \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x [phone] \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{2668050 x^{4}+6759060 x^{3}+6415178 x^{2}+2703624 x +426888} \] Input:

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

Output:

(240159150*sqrt( - 2*x + 1)*x**3 + 456341655*sqrt( - 2*x + 1)*x**2 + 28859 
6539*sqrt( - 2*x + 1)*x + 60743375*sqrt( - 2*x + 1) + 731233125*sqrt(55)*l 
og(5*sqrt( - 2*x + 1) - sqrt(55))*x**4 + 1852457250*sqrt(55)*log(5*sqrt( - 
 2*x + 1) - sqrt(55))*x**3 + 1758209425*sqrt(55)*log(5*sqrt( - 2*x + 1) - 
sqrt(55))*x**2 + 740982900*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x + 
 116997300*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55)) - 731233125*sqrt(55 
)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x**4 - 1852457250*sqrt(55)*log(5*sqrt 
( - 2*x + 1) + sqrt(55))*x**3 - 1758209425*sqrt(55)*log(5*sqrt( - 2*x + 1) 
 + sqrt(55))*x**2 - 740982900*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))* 
x - 116997300*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55)) - 1183389075*sqr 
t(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**4 - 2997918990*sqrt(21)*log(3* 
sqrt( - 2*x + 1) - sqrt(21))*x**3 - 2845393287*sqrt(21)*log(3*sqrt( - 2*x 
+ 1) - sqrt(21))*x**2 - 1199167596*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt( 
21))*x - 189342252*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21)) + 118338907 
5*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x**4 + 2997918990*sqrt(21)*l 
og(3*sqrt( - 2*x + 1) + sqrt(21))*x**3 + 2845393287*sqrt(21)*log(3*sqrt( - 
 2*x + 1) + sqrt(21))*x**2 + 1199167596*sqrt(21)*log(3*sqrt( - 2*x + 1) + 
sqrt(21))*x + 189342252*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21)))/(1185 
8*(225*x**4 + 570*x**3 + 541*x**2 + 228*x + 36))