\(\int \frac {(1-2 x)^{3/2} (2+3 x)^4}{3+5 x} \, dx\) [591]

Optimal result

Integrand size = 24, antiderivative size = 108 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{3+5 x} \, dx=\frac {22 \sqrt {1-2 x}}{15625}+\frac {2 (1-2 x)^{3/2}}{9375}-\frac {136419 (1-2 x)^{5/2}}{25000}+\frac {34371 (1-2 x)^{7/2}}{7000}-\frac {321}{200} (1-2 x)^{9/2}+\frac {81}{440} (1-2 x)^{11/2}-\frac {22 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625} \] Output:

22/15625*(1-2*x)^(1/2)+2/9375*(1-2*x)^(3/2)-136419/25000*(1-2*x)^(5/2)+343 
71/7000*(1-2*x)^(7/2)-321/200*(1-2*x)^(9/2)+81/440*(1-2*x)^(11/2)-22/78125 
*55^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{3+5 x} \, dx=\frac {-5 \sqrt {1-2 x} \left (7095688-12144995 x-21433590 x^2+9559125 x^3+39532500 x^4+21262500 x^5\right )-5082 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{18046875} \] Input:

Integrate[((1 - 2*x)^(3/2)*(2 + 3*x)^4)/(3 + 5*x),x]
 

Output:

(-5*Sqrt[1 - 2*x]*(7095688 - 12144995*x - 21433590*x^2 + 9559125*x^3 + 395 
32500*x^4 + 21262500*x^5) - 5082*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x] 
])/18046875
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {99, 2009}

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.

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

\(\Big \downarrow \) 99

\(\displaystyle \int \left (-\frac {81}{40} (1-2 x)^{9/2}+\frac {2889}{200} (1-2 x)^{7/2}-\frac {34371 (1-2 x)^{5/2}}{1000}+\frac {(1-2 x)^{3/2}}{625 (5 x+3)}+\frac {136419 (1-2 x)^{3/2}}{5000}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {22 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625}+\frac {81}{440} (1-2 x)^{11/2}-\frac {321}{200} (1-2 x)^{9/2}+\frac {34371 (1-2 x)^{7/2}}{7000}-\frac {136419 (1-2 x)^{5/2}}{25000}+\frac {2 (1-2 x)^{3/2}}{9375}+\frac {22 \sqrt {1-2 x}}{15625}\)

Input:

Int[((1 - 2*x)^(3/2)*(2 + 3*x)^4)/(3 + 5*x),x]
 

Output:

(22*Sqrt[1 - 2*x])/15625 + (2*(1 - 2*x)^(3/2))/9375 - (136419*(1 - 2*x)^(5 
/2))/25000 + (34371*(1 - 2*x)^(7/2))/7000 - (321*(1 - 2*x)^(9/2))/200 + (8 
1*(1 - 2*x)^(11/2))/440 - (22*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]] 
)/15625
 

Defintions of rubi rules used

rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | 
| (GtQ[m, 0] && GeQ[n, -1]))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.50

method	result	size
pseudoelliptic	\(-\frac {22 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{78125}-\frac {\sqrt {1-2 x}\, \left (21262500 x^{5}[phone] x^{4}+9559125 x^{3}-21433590 x^{2}-12144995 x +7095688\right )}{3609375}\)	\(54\)
risch	\(\frac {\left (21262500 x^{5}[phone] x^{4}+9559125 x^{3}-21433590 x^{2}-12144995 x +7095688\right ) \left (-1+2 x \right )}{3609375 \sqrt {1-2 x}}-\frac {22 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{78125}\)	\(59\)
derivativedivides	\(\frac {22 \sqrt {1-2 x}}{15625}+\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{9375}-\frac {136419 \left (1-2 x \right )^{\frac {5}{2}}}{25000}+\frac {34371 \left (1-2 x \right )^{\frac {7}{2}}}{7000}-\frac {321 \left (1-2 x \right )^{\frac {9}{2}}}{200}+\frac {81 \left (1-2 x \right )^{\frac {11}{2}}}{440}-\frac {22 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{78125}\)	\(74\)
default	\(\frac {22 \sqrt {1-2 x}}{15625}+\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{9375}-\frac {136419 \left (1-2 x \right )^{\frac {5}{2}}}{25000}+\frac {34371 \left (1-2 x \right )^{\frac {7}{2}}}{7000}-\frac {321 \left (1-2 x \right )^{\frac {9}{2}}}{200}+\frac {81 \left (1-2 x \right )^{\frac {11}{2}}}{440}-\frac {22 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{78125}\)	\(74\)
trager	\(\left (-\frac {324}{55} x^{5}-\frac {3012}{275} x^{4}-\frac {25491}{9625} x^{3}+\frac {1428906}{240625} x^{2}+\frac {2428999}{721875} x -\frac {7095688}{3609375}\right ) \sqrt {1-2 x}+\frac {11 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{78125}\)	\(79\)

Input:

int((1-2*x)^(3/2)*(2+3*x)^4/(3+5*x),x,method=_RETURNVERBOSE)
 

Output:

-22/78125*55^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))-1/3609375*(1-2*x)^ 
(1/2)*(21262500*x^5+39532500*x^4+9559125*x^3-21433590*x^2-12144995*x+70956 
88)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{3+5 x} \, dx=-\frac {1}{3609375} \, {\left (21262500 \, x^{5} + 39532500 \, x^{4} + 9559125 \, x^{3} - 21433590 \, x^{2} - 12144995 \, x + 7095688\right )} \sqrt {-2 \, x + 1} + \frac {11}{15625} \, \sqrt {\frac {11}{5}} \log \left (\frac {5 \, x + 5 \, \sqrt {\frac {11}{5}} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) \] Input:

integrate((1-2*x)^(3/2)*(2+3*x)^4/(3+5*x),x, algorithm="fricas")
 

Output:

-1/3609375*(21262500*x^5 + 39532500*x^4 + 9559125*x^3 - 21433590*x^2 - 121 
44995*x + 7095688)*sqrt(-2*x + 1) + 11/15625*sqrt(11/5)*log((5*x + 5*sqrt( 
11/5)*sqrt(-2*x + 1) - 8)/(5*x + 3))
 

Sympy [A] (verification not implemented)

Time = 2.35 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.04 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{3+5 x} \, dx=\frac {81 \left (1 - 2 x\right )^{\frac {11}{2}}}{440} - \frac {321 \left (1 - 2 x\right )^{\frac {9}{2}}}{200} + \frac {34371 \left (1 - 2 x\right )^{\frac {7}{2}}}{7000} - \frac {136419 \left (1 - 2 x\right )^{\frac {5}{2}}}{25000} + \frac {2 \left (1 - 2 x\right )^{\frac {3}{2}}}{9375} + \frac {22 \sqrt {1 - 2 x}}{15625} + \frac {11 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{78125} \] Input:

integrate((1-2*x)**(3/2)*(2+3*x)**4/(3+5*x),x)
                                                                                    
                                                                                    
 

Output:

81*(1 - 2*x)**(11/2)/440 - 321*(1 - 2*x)**(9/2)/200 + 34371*(1 - 2*x)**(7/ 
2)/7000 - 136419*(1 - 2*x)**(5/2)/25000 + 2*(1 - 2*x)**(3/2)/9375 + 22*sqr 
t(1 - 2*x)/15625 + 11*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt 
(1 - 2*x) + sqrt(55)/5))/78125
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{3+5 x} \, dx=\frac {81}{440} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {321}{200} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {34371}{7000} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {136419}{25000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2}{9375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{78125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {22}{15625} \, \sqrt {-2 \, x + 1} \] Input:

integrate((1-2*x)^(3/2)*(2+3*x)^4/(3+5*x),x, algorithm="maxima")
 

Output:

81/440*(-2*x + 1)^(11/2) - 321/200*(-2*x + 1)^(9/2) + 34371/7000*(-2*x + 1 
)^(7/2) - 136419/25000*(-2*x + 1)^(5/2) + 2/9375*(-2*x + 1)^(3/2) + 11/781 
25*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1 
))) + 22/15625*sqrt(-2*x + 1)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.13 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{3+5 x} \, dx=-\frac {81}{440} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {321}{200} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {34371}{7000} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {136419}{25000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2}{9375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{78125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {22}{15625} \, \sqrt {-2 \, x + 1} \] Input:

integrate((1-2*x)^(3/2)*(2+3*x)^4/(3+5*x),x, algorithm="giac")
 

Output:

-81/440*(2*x - 1)^5*sqrt(-2*x + 1) - 321/200*(2*x - 1)^4*sqrt(-2*x + 1) - 
34371/7000*(2*x - 1)^3*sqrt(-2*x + 1) - 136419/25000*(2*x - 1)^2*sqrt(-2*x 
 + 1) + 2/9375*(-2*x + 1)^(3/2) + 11/78125*sqrt(55)*log(1/2*abs(-2*sqrt(55 
) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 22/15625*sqrt(-2*x 
 + 1)
 

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{3+5 x} \, dx=\frac {22\,\sqrt {1-2\,x}}{15625}+\frac {2\,{\left (1-2\,x\right )}^{3/2}}{9375}-\frac {136419\,{\left (1-2\,x\right )}^{5/2}}{25000}+\frac {34371\,{\left (1-2\,x\right )}^{7/2}}{7000}-\frac {321\,{\left (1-2\,x\right )}^{9/2}}{200}+\frac {81\,{\left (1-2\,x\right )}^{11/2}}{440}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,22{}\mathrm {i}}{78125} \] Input:

int(((1 - 2*x)^(3/2)*(3*x + 2)^4)/(5*x + 3),x)
 

Output:

(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*22i)/78125 + (22*(1 - 2*x 
)^(1/2))/15625 + (2*(1 - 2*x)^(3/2))/9375 - (136419*(1 - 2*x)^(5/2))/25000 
 + (34371*(1 - 2*x)^(7/2))/7000 - (321*(1 - 2*x)^(9/2))/200 + (81*(1 - 2*x 
)^(11/2))/440
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.89 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{3+5 x} \, dx=-\frac {324 \sqrt {-2 x +1}\, x^{5}}{55}-\frac {3012 \sqrt {-2 x +1}\, x^{4}}{275}-\frac {25491 \sqrt {-2 x +1}\, x^{3}}{9625}+\frac {1428906 \sqrt {-2 x +1}\, x^{2}}{240625}+\frac {2428999 \sqrt {-2 x +1}\, x}{721875}-\frac {7095688 \sqrt {-2 x +1}}{3609375}+\frac {11 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right )}{78125}-\frac {11 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right )}{78125} \] Input:

int((1-2*x)^(3/2)*(2+3*x)^4/(3+5*x),x)
 

Output:

( - 106312500*sqrt( - 2*x + 1)*x**5 - 197662500*sqrt( - 2*x + 1)*x**4 - 47 
795625*sqrt( - 2*x + 1)*x**3 + 107167950*sqrt( - 2*x + 1)*x**2 + 60724975* 
sqrt( - 2*x + 1)*x - 35478440*sqrt( - 2*x + 1) + 2541*sqrt(55)*log(5*sqrt( 
 - 2*x + 1) - sqrt(55)) - 2541*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55)) 
)/18046875