3.22.97 \(\int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^3} \, dx\) [2197]

3.22.97.1 Optimal result

Integrand size = 24, antiderivative size = 125 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^3} \, dx=-\frac {5969}{27951 (1-2 x)^{3/2}}-\frac {65167}{717409 \sqrt {1-2 x}}-\frac {5}{22 (1-2 x)^{3/2} (3+5 x)^2}+\frac {295}{242 (1-2 x)^{3/2} (3+5 x)}+\frac {162}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {47075 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641} \]

output

-5969/27951/(1-2*x)^(3/2)-5/22/(1-2*x)^(3/2)/(3+5*x)^2+295/242/(1-2*x)^(3/ 
2)/(3+5*x)+162/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-47075/1610 
51*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-65167/717409/(1-2*x)^(1/2 
)
 

3.22.97.2 Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^3} \, dx=\frac {162}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {\frac {11 \left (2971158-6032979 x-9295580 x^2+19550100 x^3\right )}{(1-2 x)^{3/2} (3+5 x)^2}-13840050 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{47348994} \]

input

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

output

(162*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + ((11*(2971158 - 6032 
979*x - 9295580*x^2 + 19550100*x^3))/((1 - 2*x)^(3/2)*(3 + 5*x)^2) - 13840 
050*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/47348994
 

3.22.97.3 Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.13, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {114, 25, 168, 169, 27, 169, 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.

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

\(\Big \downarrow \) 114

\(\displaystyle -\frac {1}{22} \int -\frac {105 x+4}{(1-2 x)^{5/2} (3 x+2) (5 x+3)^2}dx-\frac {5}{22 (1-2 x)^{3/2} (5 x+3)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{22} \int \frac {105 x+4}{(1-2 x)^{5/2} (3 x+2) (5 x+3)^2}dx-\frac {5}{22 (1-2 x)^{3/2} (5 x+3)^2}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{22} \left (\frac {295}{11 (1-2 x)^{3/2} (5 x+3)}-\frac {1}{11} \int \frac {4425 x+772}{(1-2 x)^{5/2} (3 x+2) (5 x+3)}dx\right )-\frac {5}{22 (1-2 x)^{3/2} (5 x+3)^2}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {2}{231} \int \frac {3 (12184-89535 x)}{2 (1-2 x)^{3/2} (3 x+2) (5 x+3)}dx-\frac {11938}{231 (1-2 x)^{3/2}}\right )+\frac {295}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {5}{22 (1-2 x)^{3/2} (5 x+3)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{77} \int \frac {12184-89535 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx-\frac {11938}{231 (1-2 x)^{3/2}}\right )+\frac {295}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {5}{22 (1-2 x)^{3/2} (5 x+3)^2}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{77} \left (-\frac {2}{77} \int -\frac {1720172-977505 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {130334}{77 \sqrt {1-2 x}}\right )-\frac {11938}{231 (1-2 x)^{3/2}}\right )+\frac {295}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {5}{22 (1-2 x)^{3/2} (5 x+3)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{77} \left (\frac {1}{77} \int \frac {1720172-977505 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {130334}{77 \sqrt {1-2 x}}\right )-\frac {11938}{231 (1-2 x)^{3/2}}\right )+\frac {295}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {5}{22 (1-2 x)^{3/2} (5 x+3)^2}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{77} \left (\frac {1}{77} \left (11533375 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-7115526 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {130334}{77 \sqrt {1-2 x}}\right )-\frac {11938}{231 (1-2 x)^{3/2}}\right )+\frac {295}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {5}{22 (1-2 x)^{3/2} (5 x+3)^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{77} \left (\frac {1}{77} \left (7115526 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-11533375 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {130334}{77 \sqrt {1-2 x}}\right )-\frac {11938}{231 (1-2 x)^{3/2}}\right )+\frac {295}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {5}{22 (1-2 x)^{3/2} (5 x+3)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{77} \left (\frac {1}{77} \left (4743684 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-4613350 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {130334}{77 \sqrt {1-2 x}}\right )-\frac {11938}{231 (1-2 x)^{3/2}}\right )+\frac {295}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {5}{22 (1-2 x)^{3/2} (5 x+3)^2}\)

input

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

output

-5/(22*(1 - 2*x)^(3/2)*(3 + 5*x)^2) + (295/(11*(1 - 2*x)^(3/2)*(3 + 5*x)) 
+ (-11938/(231*(1 - 2*x)^(3/2)) + (-130334/(77*Sqrt[1 - 2*x]) + (4743684*S 
qrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 4613350*Sqrt[5/11]*ArcTanh[Sqr 
t[5/11]*Sqrt[1 - 2*x]])/77)/77)/11)/22
 

3.22.97.3.1 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 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(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] + Simp[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*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 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 169

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] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

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])
 

3.22.97.4 Maple [A] (verified)

Time = 1.16 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.67

method	result	size
derivativedivides	\(\frac {-\frac {3125 \left (1-2 x \right )^{\frac {3}{2}}}{1331}+\frac {6625 \sqrt {1-2 x}}{1331}}{\left (-6-10 x \right )^{2}}-\frac {47075 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{161051}+\frac {162 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}+\frac {16}{27951 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {2208}{717409 \sqrt {1-2 x}}\)	\(84\)
default	\(\frac {-\frac {3125 \left (1-2 x \right )^{\frac {3}{2}}}{1331}+\frac {6625 \sqrt {1-2 x}}{1331}}{\left (-6-10 x \right )^{2}}-\frac {47075 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{161051}+\frac {162 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}+\frac {16}{27951 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {2208}{717409 \sqrt {1-2 x}}\)	\(84\)
pseudoelliptic	\(\frac {\frac {495193}{717409}-\frac {162 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (-1+2 x \right ) \left (3+5 x \right )^{2} \sqrt {21}}{343}+\frac {47075 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (-1+2 x \right ) \left (3+5 x \right )^{2} \sqrt {55}}{161051}+\frac {3258350 x^{3}}{717409}-\frac {663970 x^{2}}{307461}-\frac {2010993 x}{1434818}}{\left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{2}}\)	\(106\)
trager	\(\frac {\left (19550100 x^{3}-9295580 x^{2}-6032979 x +2971158\right ) \sqrt {1-2 x}}{4304454 \left (10 x^{2}+x -3\right )^{2}}-\frac {81 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{343}+\frac {175 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3979855\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3979855\right ) x +14795 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3979855\right )}{3+5 x}\right )}{322102}\)	\(124\)

input

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

output

31250/14641*(-11/10*(1-2*x)^(3/2)+583/250*(1-2*x)^(1/2))/(-6-10*x)^2-47075 
/161051*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+162/343*arctanh(1/7* 
21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+16/27951/(1-2*x)^(3/2)+2208/717409/(1-2*x 
)^(1/2)
 

3.22.97.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.30 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^3} \, dx=\frac {48440175 \, \sqrt {11} \sqrt {5} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 78270786 \, \sqrt {7} \sqrt {3} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (19550100 \, x^{3} - 9295580 \, x^{2} - 6032979 \, x + 2971158\right )} \sqrt {-2 \, x + 1}}{331442958 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]

input

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

output

1/331442958*(48440175*sqrt(11)*sqrt(5)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 
9)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 78270786*s 
qrt(7)*sqrt(3)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(-(sqrt(7)*sqrt(3) 
*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(19550100*x^3 - 9295580*x^2 - 6 
032979*x + 2971158)*sqrt(-2*x + 1))/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)
 

3.22.97.6 Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 9.43 (sec) , antiderivative size = 1352, normalized size of antiderivative = 10.82 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^3} \, dx=\text {Too large to display} \]

input

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

output

-96880350000*sqrt(55)*I*(x - 1/2)**(11/2)*atan(sqrt(110)*sqrt(x - 1/2)/11) 
/(331442958000*(x - 1/2)**(11/2) + 1093761761400*(x - 1/2)**(9/2) + 120313 
7937540*(x - 1/2)**(7/2) + 441150577098*(x - 1/2)**(5/2)) + 156541572000*s 
qrt(21)*I*(x - 1/2)**(11/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(331442958000*( 
x - 1/2)**(11/2) + 1093761761400*(x - 1/2)**(9/2) + 1203137937540*(x - 1/2 
)**(7/2) + 441150577098*(x - 1/2)**(5/2)) - 78270786000*sqrt(21)*I*pi*(x - 
 1/2)**(11/2)/(331442958000*(x - 1/2)**(11/2) + 1093761761400*(x - 1/2)**( 
9/2) + 1203137937540*(x - 1/2)**(7/2) + 441150577098*(x - 1/2)**(5/2)) + 4 
8440175000*sqrt(55)*I*pi*(x - 1/2)**(11/2)/(331442958000*(x - 1/2)**(11/2) 
 + 1093761761400*(x - 1/2)**(9/2) + 1203137937540*(x - 1/2)**(7/2) + 44115 
0577098*(x - 1/2)**(5/2)) - 319705155000*sqrt(55)*I*(x - 1/2)**(9/2)*atan( 
sqrt(110)*sqrt(x - 1/2)/11)/(331442958000*(x - 1/2)**(11/2) + 109376176140 
0*(x - 1/2)**(9/2) + 1203137937540*(x - 1/2)**(7/2) + 441150577098*(x - 1/ 
2)**(5/2)) + 516587187600*sqrt(21)*I*(x - 1/2)**(9/2)*atan(sqrt(42)*sqrt(x 
 - 1/2)/7)/(331442958000*(x - 1/2)**(11/2) + 1093761761400*(x - 1/2)**(9/2 
) + 1203137937540*(x - 1/2)**(7/2) + 441150577098*(x - 1/2)**(5/2)) - 2582 
93593800*sqrt(21)*I*pi*(x - 1/2)**(9/2)/(331442958000*(x - 1/2)**(11/2) + 
1093761761400*(x - 1/2)**(9/2) + 1203137937540*(x - 1/2)**(7/2) + 44115057 
7098*(x - 1/2)**(5/2)) + 159852577500*sqrt(55)*I*pi*(x - 1/2)**(9/2)/(3314 
42958000*(x - 1/2)**(11/2) + 1093761761400*(x - 1/2)**(9/2) + 120313793...
 

3.22.97.7 Maxima [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^3} \, dx=\frac {47075}{322102} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {81}{343} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4887525 \, {\left (2 \, x - 1\right )}^{3} + 10014785 \, {\left (2 \, x - 1\right )}^{2} - 1331968 \, x + 815056}{2152227 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 121 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]

input

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

output

47075/322102*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqr 
t(-2*x + 1))) - 81/343*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(2 
1) + 3*sqrt(-2*x + 1))) + 1/2152227*(4887525*(2*x - 1)^3 + 10014785*(2*x - 
 1)^2 - 1331968*x + 815056)/(25*(-2*x + 1)^(7/2) - 110*(-2*x + 1)^(5/2) + 
121*(-2*x + 1)^(3/2))
 

3.22.97.8 Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^3} \, dx=\frac {47075}{322102} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {81}{343} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {16 \, {\left (828 \, x - 491\right )}}{2152227 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {125 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 53 \, \sqrt {-2 \, x + 1}\right )}}{5324 \, {\left (5 \, x + 3\right )}^{2}} \]

input

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

output

47075/322102*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(5 
5) + 5*sqrt(-2*x + 1))) - 81/343*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt 
(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 16/2152227*(828*x - 491)/((2* 
x - 1)*sqrt(-2*x + 1)) - 125/5324*(25*(-2*x + 1)^(3/2) - 53*sqrt(-2*x + 1) 
)/(5*x + 3)^2
 

3.22.97.9 Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.71 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^3} \, dx=\frac {162\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {47075\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{161051}+\frac {\frac {182087\,{\left (2\,x-1\right )}^2}{978285}-\frac {11008\,x}{444675}+\frac {65167\,{\left (2\,x-1\right )}^3}{717409}+\frac {6736}{444675}}{\frac {121\,{\left (1-2\,x\right )}^{3/2}}{25}-\frac {22\,{\left (1-2\,x\right )}^{5/2}}{5}+{\left (1-2\,x\right )}^{7/2}} \]

input

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

output

(162*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/343 - (47075*55^(1/2)*a 
tanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/161051 + ((182087*(2*x - 1)^2)/978285 
 - (11008*x)/444675 + (65167*(2*x - 1)^3)/717409 + 6736/444675)/((121*(1 - 
 2*x)^(3/2))/25 - (22*(1 - 2*x)^(5/2))/5 + (1 - 2*x)^(7/2))