3.20.82 \(\int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^2} \, dx\) [1982]

3.20.82.1 Optimal result
3.20.82.2 Mathematica [A] (verified)
3.20.82.3 Rubi [A] (verified)
3.20.82.4 Maple [A] (verified)
3.20.82.5 Fricas [A] (verification not implemented)
3.20.82.6 Sympy [A] (verification not implemented)
3.20.82.7 Maxima [A] (verification not implemented)
3.20.82.8 Giac [A] (verification not implemented)
3.20.82.9 Mupad [B] (verification not implemented)

3.20.82.1 Optimal result

Integrand size = 24, antiderivative size = 102 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^2} \, dx=\frac {1342 \sqrt {1-2 x}}{3125}+\frac {122 (1-2 x)^{3/2}}{1875}+\frac {122 (1-2 x)^{5/2}}{6875}-\frac {9}{175} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{275 (3+5 x)}-\frac {1342 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \]

output
122/1875*(1-2*x)^(3/2)+122/6875*(1-2*x)^(5/2)-9/175*(1-2*x)^(7/2)-1/275*(1 
-2*x)^(7/2)/(3+5*x)-1342/15625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/ 
2)+1342/3125*(1-2*x)^(1/2)
 
3.20.82.2 Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^2} \, dx=\frac {\frac {5 \sqrt {1-2 x} \left (90486+173795 x-75130 x^2-96300 x^3+135000 x^4\right )}{3+5 x}-28182 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{328125} \]

input
Integrate[((1 - 2*x)^(5/2)*(2 + 3*x)^2)/(3 + 5*x)^2,x]
 
output
((5*Sqrt[1 - 2*x]*(90486 + 173795*x - 75130*x^2 - 96300*x^3 + 135000*x^4)) 
/(3 + 5*x) - 28182*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/328125
 
3.20.82.3 Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {100, 90, 60, 60, 60, 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.

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

\(\Big \downarrow \) 100

\(\displaystyle \frac {1}{275} \int \frac {(1-2 x)^{5/2} (495 x+358)}{5 x+3}dx-\frac {(1-2 x)^{7/2}}{275 (5 x+3)}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {1}{275} \left (61 \int \frac {(1-2 x)^{5/2}}{5 x+3}dx-\frac {99}{7} (1-2 x)^{7/2}\right )-\frac {(1-2 x)^{7/2}}{275 (5 x+3)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{275} \left (61 \left (\frac {11}{5} \int \frac {(1-2 x)^{3/2}}{5 x+3}dx+\frac {2}{25} (1-2 x)^{5/2}\right )-\frac {99}{7} (1-2 x)^{7/2}\right )-\frac {(1-2 x)^{7/2}}{275 (5 x+3)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{275} \left (61 \left (\frac {11}{5} \left (\frac {11}{5} \int \frac {\sqrt {1-2 x}}{5 x+3}dx+\frac {2}{15} (1-2 x)^{3/2}\right )+\frac {2}{25} (1-2 x)^{5/2}\right )-\frac {99}{7} (1-2 x)^{7/2}\right )-\frac {(1-2 x)^{7/2}}{275 (5 x+3)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{275} \left (61 \left (\frac {11}{5} \left (\frac {11}{5} \left (\frac {11}{5} \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx+\frac {2}{5} \sqrt {1-2 x}\right )+\frac {2}{15} (1-2 x)^{3/2}\right )+\frac {2}{25} (1-2 x)^{5/2}\right )-\frac {99}{7} (1-2 x)^{7/2}\right )-\frac {(1-2 x)^{7/2}}{275 (5 x+3)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{275} \left (61 \left (\frac {11}{5} \left (\frac {11}{5} \left (\frac {2}{5} \sqrt {1-2 x}-\frac {11}{5} \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {2}{15} (1-2 x)^{3/2}\right )+\frac {2}{25} (1-2 x)^{5/2}\right )-\frac {99}{7} (1-2 x)^{7/2}\right )-\frac {(1-2 x)^{7/2}}{275 (5 x+3)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{275} \left (61 \left (\frac {11}{5} \left (\frac {11}{5} \left (\frac {2}{5} \sqrt {1-2 x}-\frac {2}{5} \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {2}{15} (1-2 x)^{3/2}\right )+\frac {2}{25} (1-2 x)^{5/2}\right )-\frac {99}{7} (1-2 x)^{7/2}\right )-\frac {(1-2 x)^{7/2}}{275 (5 x+3)}\)

input
Int[((1 - 2*x)^(5/2)*(2 + 3*x)^2)/(3 + 5*x)^2,x]
 
output
-1/275*(1 - 2*x)^(7/2)/(3 + 5*x) + ((-99*(1 - 2*x)^(7/2))/7 + 61*((2*(1 - 
2*x)^(5/2))/25 + (11*((2*(1 - 2*x)^(3/2))/15 + (11*((2*Sqrt[1 - 2*x])/5 - 
(2*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/5))/5))/5))/275
 

3.20.82.3.1 Defintions of rubi rules used

rule 60
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, 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 90
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 100
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

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])
 
3.20.82.4 Maple [A] (verified)

Time = 3.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.60

method result size
risch \(-\frac {270000 x^{5}-327600 x^{4}-53960 x^{3}+422720 x^{2}+7177 x -90486}{65625 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {1342 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}\) \(61\)
pseudoelliptic \(\frac {-28182 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right ) \sqrt {55}+5 \sqrt {1-2 x}\, \left (135000 x^{4}-96300 x^{3}-75130 x^{2}+173795 x +90486\right )}{984375+1640625 x}\) \(62\)
derivativedivides \(-\frac {9 \left (1-2 x \right )^{\frac {7}{2}}}{175}+\frac {12 \left (1-2 x \right )^{\frac {5}{2}}}{625}+\frac {128 \left (1-2 x \right )^{\frac {3}{2}}}{1875}+\frac {1364 \sqrt {1-2 x}}{3125}+\frac {242 \sqrt {1-2 x}}{15625 \left (-\frac {6}{5}-2 x \right )}-\frac {1342 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}\) \(72\)
default \(-\frac {9 \left (1-2 x \right )^{\frac {7}{2}}}{175}+\frac {12 \left (1-2 x \right )^{\frac {5}{2}}}{625}+\frac {128 \left (1-2 x \right )^{\frac {3}{2}}}{1875}+\frac {1364 \sqrt {1-2 x}}{3125}+\frac {242 \sqrt {1-2 x}}{15625 \left (-\frac {6}{5}-2 x \right )}-\frac {1342 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}\) \(72\)
trager \(\frac {\sqrt {1-2 x}\, \left (135000 x^{4}-96300 x^{3}-75130 x^{2}+173795 x +90486\right )}{196875+328125 x}+\frac {671 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{15625}\) \(82\)

input
int((1-2*x)^(5/2)*(2+3*x)^2/(3+5*x)^2,x,method=_RETURNVERBOSE)
 
output
-1/65625*(270000*x^5-327600*x^4-53960*x^3+422720*x^2+7177*x-90486)/(3+5*x) 
/(1-2*x)^(1/2)-1342/15625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
 
3.20.82.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.78 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^2} \, dx=\frac {14091 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 5 \, {\left (135000 \, x^{4} - 96300 \, x^{3} - 75130 \, x^{2} + 173795 \, x + 90486\right )} \sqrt {-2 \, x + 1}}{328125 \, {\left (5 \, x + 3\right )}} \]

input
integrate((1-2*x)^(5/2)*(2+3*x)^2/(3+5*x)^2,x, algorithm="fricas")
 
output
1/328125*(14091*sqrt(11)*sqrt(5)*(5*x + 3)*log((sqrt(11)*sqrt(5)*sqrt(-2*x 
 + 1) + 5*x - 8)/(5*x + 3)) + 5*(135000*x^4 - 96300*x^3 - 75130*x^2 + 1737 
95*x + 90486)*sqrt(-2*x + 1))/(5*x + 3)
 
3.20.82.6 Sympy [A] (verification not implemented)

Time = 30.69 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.05 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^2} \, dx=- \frac {9 \left (1 - 2 x\right )^{\frac {7}{2}}}{175} + \frac {12 \left (1 - 2 x\right )^{\frac {5}{2}}}{625} + \frac {128 \left (1 - 2 x\right )^{\frac {3}{2}}}{1875} + \frac {1364 \sqrt {1 - 2 x}}{3125} + \frac {132 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{3125} - \frac {5324 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{3125} \]

input
integrate((1-2*x)**(5/2)*(2+3*x)**2/(3+5*x)**2,x)
 
output
-9*(1 - 2*x)**(7/2)/175 + 12*(1 - 2*x)**(5/2)/625 + 128*(1 - 2*x)**(3/2)/1 
875 + 1364*sqrt(1 - 2*x)/3125 + 132*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55) 
/5) - log(sqrt(1 - 2*x) + sqrt(55)/5))/3125 - 5324*Piecewise((sqrt(55)*(-l 
og(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)) 
)/3125
 
3.20.82.7 Maxima [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^2} \, dx=-\frac {9}{175} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {12}{625} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {128}{1875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {671}{15625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1364}{3125} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{3125 \, {\left (5 \, x + 3\right )}} \]

input
integrate((1-2*x)^(5/2)*(2+3*x)^2/(3+5*x)^2,x, algorithm="maxima")
 
output
-9/175*(-2*x + 1)^(7/2) + 12/625*(-2*x + 1)^(5/2) + 128/1875*(-2*x + 1)^(3 
/2) + 671/15625*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5* 
sqrt(-2*x + 1))) + 1364/3125*sqrt(-2*x + 1) - 121/3125*sqrt(-2*x + 1)/(5*x 
 + 3)
 
3.20.82.8 Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.04 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^2} \, dx=\frac {9}{175} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {12}{625} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {128}{1875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {671}{15625} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {1364}{3125} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{3125 \, {\left (5 \, x + 3\right )}} \]

input
integrate((1-2*x)^(5/2)*(2+3*x)^2/(3+5*x)^2,x, algorithm="giac")
 
output
9/175*(2*x - 1)^3*sqrt(-2*x + 1) + 12/625*(2*x - 1)^2*sqrt(-2*x + 1) + 128 
/1875*(-2*x + 1)^(3/2) + 671/15625*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*s 
qrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 1364/3125*sqrt(-2*x + 1) - 
 121/3125*sqrt(-2*x + 1)/(5*x + 3)
 
3.20.82.9 Mupad [B] (verification not implemented)

Time = 1.35 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^2} \, dx=\frac {1364\,\sqrt {1-2\,x}}{3125}-\frac {242\,\sqrt {1-2\,x}}{15625\,\left (2\,x+\frac {6}{5}\right )}+\frac {128\,{\left (1-2\,x\right )}^{3/2}}{1875}+\frac {12\,{\left (1-2\,x\right )}^{5/2}}{625}-\frac {9\,{\left (1-2\,x\right )}^{7/2}}{175}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,1342{}\mathrm {i}}{15625} \]

input
int(((1 - 2*x)^(5/2)*(3*x + 2)^2)/(5*x + 3)^2,x)
 
output
(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*1342i)/15625 - (242*(1 - 
2*x)^(1/2))/(15625*(2*x + 6/5)) + (1364*(1 - 2*x)^(1/2))/3125 + (128*(1 - 
2*x)^(3/2))/1875 + (12*(1 - 2*x)^(5/2))/625 - (9*(1 - 2*x)^(7/2))/175