\(\int \frac {(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^2} \, dx\) [665]

Optimal result

Integrand size = 24, antiderivative size = 92 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^2} \, dx=-\frac {58}{75} \sqrt {1-2 x}-\frac {11 (1-2 x)^{3/2}}{5 (3+5 x)}-\frac {98}{3} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {836}{25} \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Output:

-58/75*(1-2*x)^(1/2)-11*(1-2*x)^(3/2)/(15+25*x)-98/9*21^(1/2)*arctanh(1/7* 
21^(1/2)*(1-2*x)^(1/2))+836/125*55^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/ 
2))
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^2} \, dx=-\frac {98}{3} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {1}{375} \left (\frac {5 \sqrt {1-2 x} (-339+40 x)}{3+5 x}+2508 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right ) \] Input:

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

Output:

(-98*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3 + ((5*Sqrt[1 - 2*x]*(-3 
39 + 40*x))/(3 + 5*x) + 2508*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/3 
75
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {109, 171, 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.

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

\(\Big \downarrow \) 109

\(\displaystyle -\frac {1}{5} \int \frac {\sqrt {1-2 x} (29 x+101)}{(3 x+2) (5 x+3)}dx-\frac {11 (1-2 x)^{3/2}}{5 (5 x+3)}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{5} \left (-\frac {2}{15} \int \frac {1863-1493 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {58}{15} \sqrt {1-2 x}\right )-\frac {11 (1-2 x)^{3/2}}{5 (5 x+3)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \left (-\frac {1}{15} \int \frac {1863-1493 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {58}{15} \sqrt {1-2 x}\right )-\frac {11 (1-2 x)^{3/2}}{5 (5 x+3)}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {1}{5} \left (\frac {1}{15} \left (8575 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-13794 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx\right )-\frac {58}{15} \sqrt {1-2 x}\right )-\frac {11 (1-2 x)^{3/2}}{5 (5 x+3)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{5} \left (\frac {1}{15} \left (13794 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-8575 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {58}{15} \sqrt {1-2 x}\right )-\frac {11 (1-2 x)^{3/2}}{5 (5 x+3)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{5} \left (\frac {1}{15} \left (2508 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )-2450 \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )-\frac {58}{15} \sqrt {1-2 x}\right )-\frac {11 (1-2 x)^{3/2}}{5 (5 x+3)}\)

Input:

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

Output:

(-11*(1 - 2*x)^(3/2))/(5*(3 + 5*x)) + ((-58*Sqrt[1 - 2*x])/15 + (-2450*Sqr 
t[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + 2508*Sqrt[11/5]*ArcTanh[Sqrt[5/1 
1]*Sqrt[1 - 2*x]])/15)/5
 

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 171

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && 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])
 

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.68

method	result	size
derivativedivides	\(\frac {8 \sqrt {1-2 x}}{75}+\frac {242 \sqrt {1-2 x}}{125 \left (-\frac {6}{5}-2 x \right )}+\frac {836 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{125}-\frac {98 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{9}\)	\(63\)
default	\(\frac {8 \sqrt {1-2 x}}{75}+\frac {242 \sqrt {1-2 x}}{125 \left (-\frac {6}{5}-2 x \right )}+\frac {836 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{125}-\frac {98 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{9}\)	\(63\)
risch	\(-\frac {80 x^{2}-718 x +339}{75 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {98 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{9}+\frac {836 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{125}\)	\(64\)
pseudoelliptic	\(\frac {-12250 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (3+5 x \right ) \sqrt {21}+7524 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right ) \sqrt {55}+15 \sqrt {1-2 x}\, \left (-339+40 x \right )}{3375+5625 x}\)	\(70\)
trager	\(\frac {\sqrt {1-2 x}\, \left (-339+40 x \right )}{225+375 x}+\frac {418 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{125}-\frac {49 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{9}\)	\(111\)

Input:

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

Output:

8/75*(1-2*x)^(1/2)+242/125*(1-2*x)^(1/2)/(-6/5-2*x)+836/125*55^(1/2)*arcta 
nh(1/11*55^(1/2)*(1-2*x)^(1/2))-98/9*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x) 
^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.03 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^2} \, dx=\frac {1254 \, \sqrt {\frac {11}{5}} {\left (5 \, x + 3\right )} \log \left (\frac {5 \, x - 5 \, \sqrt {\frac {11}{5}} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 1225 \, \sqrt {\frac {7}{3}} {\left (5 \, x + 3\right )} \log \left (\frac {3 \, x + 3 \, \sqrt {\frac {7}{3}} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + {\left (40 \, x - 339\right )} \sqrt {-2 \, x + 1}}{75 \, {\left (5 \, x + 3\right )}} \] Input:

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

Output:

1/75*(1254*sqrt(11/5)*(5*x + 3)*log((5*x - 5*sqrt(11/5)*sqrt(-2*x + 1) - 8 
)/(5*x + 3)) + 1225*sqrt(7/3)*(5*x + 3)*log((3*x + 3*sqrt(7/3)*sqrt(-2*x + 
 1) - 5)/(3*x + 2)) + (40*x - 339)*sqrt(-2*x + 1))/(5*x + 3)
 

Sympy [A] (verification not implemented)

Time = 15.01 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.35 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^2} \, dx=\frac {8 \sqrt {1 - 2 x}}{75} + \frac {49 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{9} - \frac {429 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{125} - \frac {5324 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{25} \] Input:

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

Output:

8*sqrt(1 - 2*x)/75 + 49*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sq 
rt(1 - 2*x) + sqrt(21)/3))/9 - 429*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/ 
5) - log(sqrt(1 - 2*x) + sqrt(55)/5))/125 - 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)*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)))/ 
25
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.07 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^2} \, dx=-\frac {418}{125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {49}{9} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {8}{75} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{25 \, {\left (5 \, x + 3\right )}} \] Input:

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

Output:

-418/125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2 
*x + 1))) + 49/9*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3 
*sqrt(-2*x + 1))) + 8/75*sqrt(-2*x + 1) - 121/25*sqrt(-2*x + 1)/(5*x + 3)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.13 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^2} \, dx=-\frac {418}{125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {49}{9} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {8}{75} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{25 \, {\left (5 \, x + 3\right )}} \] Input:

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

Output:

-418/125*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 
 5*sqrt(-2*x + 1))) + 49/9*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x 
+ 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 8/75*sqrt(-2*x + 1) - 121/25*sqrt(- 
2*x + 1)/(5*x + 3)
 

Mupad [B] (verification not implemented)

Time = 1.13 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^2} \, dx=\frac {8\,\sqrt {1-2\,x}}{75}-\frac {242\,\sqrt {1-2\,x}}{125\,\left (2\,x+\frac {6}{5}\right )}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,98{}\mathrm {i}}{9}-\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,836{}\mathrm {i}}{125} \] Input:

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

Output:

(21^(1/2)*atan((21^(1/2)*(1 - 2*x)^(1/2)*1i)/7)*98i)/9 - (55^(1/2)*atan((5 
5^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*836i)/125 - (242*(1 - 2*x)^(1/2))/(125*(2* 
x + 6/5)) + (8*(1 - 2*x)^(1/2))/75
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.80 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^2} \, dx=\frac {600 \sqrt {-2 x +1}\, x -5085 \sqrt {-2 x +1}-18810 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x -11286 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right )+18810 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x +11286 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right )+30625 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x +18375 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )-30625 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x -18375 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{5625 x +3375} \] Input:

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

Output:

(600*sqrt( - 2*x + 1)*x - 5085*sqrt( - 2*x + 1) - 18810*sqrt(55)*log(5*sqr 
t( - 2*x + 1) - sqrt(55))*x - 11286*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt 
(55)) + 18810*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x + 11286*sqrt(5 
5)*log(5*sqrt( - 2*x + 1) + sqrt(55)) + 30625*sqrt(21)*log(3*sqrt( - 2*x + 
 1) - sqrt(21))*x + 18375*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21)) - 30 
625*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x - 18375*sqrt(21)*log(3*s 
qrt( - 2*x + 1) + sqrt(21)))/(1125*(5*x + 3))