\(\int \frac {1}{(5-2 x)^2 (4+3 x)^{3/2} (2+3 x+x^2)} \, dx\) [337]

Optimal result

Integrand size = 27, antiderivative size = 102 \[ \int \frac {1}{(5-2 x)^2 (4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=\frac {563}{11109 \sqrt {4+3 x}}+\frac {4}{1449 (5-2 x) \sqrt {4+3 x}}+\frac {\arctan \left (\frac {\sqrt {4+3 x}}{\sqrt {2}}\right )}{81 \sqrt {2}}-\frac {2}{49} \text {arctanh}\left (\sqrt {4+3 x}\right )+\frac {5212 \sqrt {\frac {2}{23}} \text {arctanh}\left (\sqrt {\frac {2}{23}} \sqrt {4+3 x}\right )}{2099601} \] Output:

563/11109/(4+3*x)^(1/2)+4/1449/(5-2*x)/(4+3*x)^(1/2)+1/162*2^(1/2)*arctan( 
1/2*2^(1/2)*(4+3*x)^(1/2))-2/49*arctanh((4+3*x)^(1/2))+5212/48290823*46^(1 
/2)*arctanh(1/23*46^(1/2)*(4+3*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(5-2 x)^2 (4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=\frac {-8537+3378 x}{33327 (-5+2 x) \sqrt {4+3 x}}+\frac {\arctan \left (\sqrt {2+\frac {3 x}{2}}\right )}{81 \sqrt {2}}-\frac {2}{49} \text {arctanh}\left (\sqrt {4+3 x}\right )+\frac {5212 \sqrt {\frac {2}{23}} \text {arctanh}\left (\sqrt {\frac {2}{23}} \sqrt {4+3 x}\right )}{2099601} \] Input:

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

Output:

(-8537 + 3378*x)/(33327*(-5 + 2*x)*Sqrt[4 + 3*x]) + ArcTan[Sqrt[2 + (3*x)/ 
2]]/(81*Sqrt[2]) - (2*ArcTanh[Sqrt[4 + 3*x]])/49 + (5212*Sqrt[2/23]*ArcTan 
h[Sqrt[2/23]*Sqrt[4 + 3*x]])/2099601
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1199, 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.

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

\(\Big \downarrow \) 1199

\(\displaystyle \frac {2}{3} \int \left (-\frac {81}{1058 (3 x+4)}+\frac {1}{54 (3 x+6)}+\frac {4456}{699867 (23-2 (3 x+4))}+\frac {8}{161 (23-2 (3 x+4))^2}-\frac {3}{49 (-3 x-3)}\right )d\sqrt {3 x+4}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2}{3} \left (\frac {\arctan \left (\frac {\sqrt {3 x+4}}{\sqrt {2}}\right )}{54 \sqrt {2}}-\frac {3}{49} \text {arctanh}\left (\sqrt {3 x+4}\right )+\frac {2606 \sqrt {\frac {2}{23}} \text {arctanh}\left (\sqrt {\frac {2}{23}} \sqrt {3 x+4}\right )}{699867}+\frac {81}{1058 \sqrt {3 x+4}}+\frac {4 \sqrt {3 x+4}}{3703 (23-2 (3 x+4))}\right )\)

Input:

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

Output:

(2*(81/(1058*Sqrt[4 + 3*x]) + (4*Sqrt[4 + 3*x])/(3703*(23 - 2*(4 + 3*x))) 
+ ArcTan[Sqrt[4 + 3*x]/Sqrt[2]]/(54*Sqrt[2]) - (3*ArcTanh[Sqrt[4 + 3*x]])/ 
49 + (2606*Sqrt[2/23]*ArcTanh[Sqrt[2/23]*Sqrt[4 + 3*x]])/699867))/3
 

Defintions of rubi rules used

rule 1199

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Denominator[m]}, Simp[q/e   Subs 
t[Int[ExpandIntegrand[x^(q*(m + 1) - 1)*(((e*f - d*g)/e + g*(x^q/e))^n/((c* 
d^2 - b*d*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))), x], 
 x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Integer 
Q[n] && FractionQ[m]
 

rule 2009

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

Maple [A] (verified)

Time = 2.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.81

method	result	size
risch	\(\frac {-8537+3378 x}{33327 \left (-5+2 x \right ) \sqrt {3 x +4}}+\frac {5212 \sqrt {46}\, \operatorname {arctanh}\left (\frac {\sqrt {46}\, \sqrt {3 x +4}}{23}\right )}{48290823}-\frac {\ln \left (\sqrt {3 x +4}+1\right )}{49}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3 x +4}}{2}\right )}{162}+\frac {\ln \left (\sqrt {3 x +4}-1\right )}{49}\)	\(83\)
derivativedivides	\(\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3 x +4}}{2}\right )}{162}-\frac {4 \sqrt {3 x +4}}{11109 \left (3 x -\frac {15}{2}\right )}+\frac {5212 \sqrt {46}\, \operatorname {arctanh}\left (\frac {\sqrt {46}\, \sqrt {3 x +4}}{23}\right )}{48290823}+\frac {27}{529 \sqrt {3 x +4}}-\frac {\ln \left (\sqrt {3 x +4}+1\right )}{49}+\frac {\ln \left (\sqrt {3 x +4}-1\right )}{49}\)	\(87\)
default	\(\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3 x +4}}{2}\right )}{162}-\frac {4 \sqrt {3 x +4}}{11109 \left (3 x -\frac {15}{2}\right )}+\frac {5212 \sqrt {46}\, \operatorname {arctanh}\left (\frac {\sqrt {46}\, \sqrt {3 x +4}}{23}\right )}{48290823}+\frac {27}{529 \sqrt {3 x +4}}-\frac {\ln \left (\sqrt {3 x +4}+1\right )}{49}+\frac {\ln \left (\sqrt {3 x +4}-1\right )}{49}\)	\(87\)
pseudoelliptic	\(\frac {20848 \sqrt {2}\, \sqrt {23}\, \sqrt {3 x +4}\, \left (x -\frac {5}{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {46}\, \sqrt {3 x +4}}{23}\right )+1192366 \sqrt {3 x +4}\, \sqrt {2}\, \left (x -\frac {5}{2}\right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {3 x +4}}{2}\right )+3942108 \sqrt {3 x +4}\, \left (x -\frac {5}{2}\right ) \ln \left (\sqrt {3 x +4}-1\right )-3942108 \sqrt {3 x +4}\, \left (x -\frac {5}{2}\right ) \ln \left (\sqrt {3 x +4}+1\right )+9789444 x -24740226}{\sqrt {3 x +4}\, \left (-482908230+193163292 x \right )}\)	\(124\)
trager	\(\frac {\left (-8537+3378 x \right ) \sqrt {3 x +4}}{199962 x^{2}-233289 x -666540}-\frac {\ln \left (\frac {2 \sqrt {3 x +4}+5+3 x}{x +1}\right )}{49}+\frac {2606 \operatorname {RootOf}\left (\textit {\_Z}^{2}-46\right ) \ln \left (-\frac {6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-46\right ) x +92 \sqrt {3 x +4}+31 \operatorname {RootOf}\left (\textit {\_Z}^{2}-46\right )}{-5+2 x}\right )}{48290823}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {3 x \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )-4 \sqrt {3 x +4}}{2+x}\right )}{324}\)	\(139\)

Input:

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

Output:

1/33327*(-8537+3378*x)/(-5+2*x)/(3*x+4)^(1/2)+5212/48290823*46^(1/2)*arcta 
nh(1/23*46^(1/2)*(3*x+4)^(1/2))-1/49*ln((3*x+4)^(1/2)+1)+1/162*2^(1/2)*arc 
tan(1/2*2^(1/2)*(3*x+4)^(1/2))+1/49*ln((3*x+4)^(1/2)-1)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.39 \[ \int \frac {1}{(5-2 x)^2 (4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=\frac {25921 \, \sqrt {2} {\left (6 \, x^{2} - 7 \, x - 20\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {3 \, x + 4}\right ) + 5212 \, \sqrt {\frac {2}{23}} {\left (6 \, x^{2} - 7 \, x - 20\right )} \log \left (\frac {46 \, \sqrt {\frac {2}{23}} \sqrt {3 \, x + 4} + 6 \, x + 31}{2 \, x - 5}\right ) - 85698 \, {\left (6 \, x^{2} - 7 \, x - 20\right )} \log \left (\sqrt {3 \, x + 4} + 1\right ) + 85698 \, {\left (6 \, x^{2} - 7 \, x - 20\right )} \log \left (\sqrt {3 \, x + 4} - 1\right ) + 126 \, {\left (3378 \, x - 8537\right )} \sqrt {3 \, x + 4}}{4199202 \, {\left (6 \, x^{2} - 7 \, x - 20\right )}} \] Input:

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

Output:

1/4199202*(25921*sqrt(2)*(6*x^2 - 7*x - 20)*arctan(1/2*sqrt(2)*sqrt(3*x + 
4)) + 5212*sqrt(2/23)*(6*x^2 - 7*x - 20)*log((46*sqrt(2/23)*sqrt(3*x + 4) 
+ 6*x + 31)/(2*x - 5)) - 85698*(6*x^2 - 7*x - 20)*log(sqrt(3*x + 4) + 1) + 
 85698*(6*x^2 - 7*x - 20)*log(sqrt(3*x + 4) - 1) + 126*(3378*x - 8537)*sqr 
t(3*x + 4))/(6*x^2 - 7*x - 20)
 

Sympy [A] (verification not implemented)

Time = 15.74 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.20 \[ \int \frac {1}{(5-2 x)^2 (4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=- \frac {2228 \sqrt {46} \left (\log {\left (\sqrt {3 x + 4} - \frac {\sqrt {46}}{2} \right )} - \log {\left (\sqrt {3 x + 4} + \frac {\sqrt {46}}{2} \right )}\right )}{48290823} + \frac {16 \left (\begin {cases} \frac {\sqrt {46} \left (- \frac {\log {\left (\frac {\sqrt {46} \sqrt {3 x + 4}}{23} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {46} \sqrt {3 x + 4}}{23} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {46} \sqrt {3 x + 4}}{23} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {46} \sqrt {3 x + 4}}{23} - 1\right )}\right )}{1058} & \text {for}\: \sqrt {3 x + 4} > - \frac {\sqrt {46}}{2} \wedge \sqrt {3 x + 4} < \frac {\sqrt {46}}{2} \end {cases}\right )}{483} + \frac {\log {\left (\sqrt {3 x + 4} - 1 \right )}}{49} - \frac {\log {\left (\sqrt {3 x + 4} + 1 \right )}}{49} + \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {3 x + 4}}{2} \right )}}{162} + \frac {27}{529 \sqrt {3 x + 4}} \] Input:

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

Output:

-2228*sqrt(46)*(log(sqrt(3*x + 4) - sqrt(46)/2) - log(sqrt(3*x + 4) + sqrt 
(46)/2))/48290823 + 16*Piecewise((sqrt(46)*(-log(sqrt(46)*sqrt(3*x + 4)/23 
 - 1)/4 + log(sqrt(46)*sqrt(3*x + 4)/23 + 1)/4 - 1/(4*(sqrt(46)*sqrt(3*x + 
 4)/23 + 1)) - 1/(4*(sqrt(46)*sqrt(3*x + 4)/23 - 1)))/1058, (sqrt(3*x + 4) 
 > -sqrt(46)/2) & (sqrt(3*x + 4) < sqrt(46)/2)))/483 + log(sqrt(3*x + 4) - 
 1)/49 - log(sqrt(3*x + 4) + 1)/49 + sqrt(2)*atan(sqrt(2)*sqrt(3*x + 4)/2) 
/162 + 27/(529*sqrt(3*x + 4))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(5-2 x)^2 (4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=\frac {1}{162} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {3 \, x + 4}\right ) - \frac {2606}{48290823} \, \sqrt {46} \log \left (-\frac {\sqrt {46} - 2 \, \sqrt {3 \, x + 4}}{\sqrt {46} + 2 \, \sqrt {3 \, x + 4}}\right ) + \frac {3378 \, x - 8537}{11109 \, {\left (2 \, {\left (3 \, x + 4\right )}^{\frac {3}{2}} - 23 \, \sqrt {3 \, x + 4}\right )}} - \frac {1}{49} \, \log \left (\sqrt {3 \, x + 4} + 1\right ) + \frac {1}{49} \, \log \left (\sqrt {3 \, x + 4} - 1\right ) \] Input:

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

Output:

1/162*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(3*x + 4)) - 2606/48290823*sqrt(46)*l 
og(-(sqrt(46) - 2*sqrt(3*x + 4))/(sqrt(46) + 2*sqrt(3*x + 4))) + 1/11109*( 
3378*x - 8537)/(2*(3*x + 4)^(3/2) - 23*sqrt(3*x + 4)) - 1/49*log(sqrt(3*x 
+ 4) + 1) + 1/49*log(sqrt(3*x + 4) - 1)
 

Giac [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(5-2 x)^2 (4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=\frac {1}{162} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {3 \, x + 4}\right ) - \frac {2606}{48290823} \, \sqrt {46} \log \left (\frac {{\left | -2 \, \sqrt {46} + 4 \, \sqrt {3 \, x + 4} \right |}}{2 \, {\left (\sqrt {46} + 2 \, \sqrt {3 \, x + 4}\right )}}\right ) + \frac {3378 \, x - 8537}{11109 \, {\left (2 \, {\left (3 \, x + 4\right )}^{\frac {3}{2}} - 23 \, \sqrt {3 \, x + 4}\right )}} - \frac {1}{49} \, \log \left (\sqrt {3 \, x + 4} + 1\right ) + \frac {1}{49} \, \log \left ({\left | \sqrt {3 \, x + 4} - 1 \right |}\right ) \] Input:

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

Output:

1/162*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(3*x + 4)) - 2606/48290823*sqrt(46)*l 
og(1/2*abs(-2*sqrt(46) + 4*sqrt(3*x + 4))/(sqrt(46) + 2*sqrt(3*x + 4))) + 
1/11109*(3378*x - 8537)/(2*(3*x + 4)^(3/2) - 23*sqrt(3*x + 4)) - 1/49*log( 
sqrt(3*x + 4) + 1) + 1/49*log(abs(sqrt(3*x + 4) - 1))
 

Mupad [B] (verification not implemented)

Time = 12.82 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(5-2 x)^2 (4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=\frac {\mathrm {atan}\left (\sqrt {3\,x+4}\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{49}-\frac {\frac {563\,x}{3703}-\frac {8537}{22218}}{\frac {23\,\sqrt {3\,x+4}}{2}-{\left (3\,x+4\right )}^{3/2}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {3\,x+4}}{2}\right )}{162}-\frac {\sqrt {46}\,\mathrm {atan}\left (\frac {\sqrt {46}\,\sqrt {3\,x+4}\,1{}\mathrm {i}}{23}\right )\,5212{}\mathrm {i}}{48290823} \] Input:

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

Output:

(atan((3*x + 4)^(1/2)*1i)*2i)/49 - ((563*x)/3703 - 8537/22218)/((23*(3*x + 
 4)^(1/2))/2 - (3*x + 4)^(3/2)) + (2^(1/2)*atan((2^(1/2)*(3*x + 4)^(1/2))/ 
2))/162 - (46^(1/2)*atan((46^(1/2)*(3*x + 4)^(1/2)*1i)/23)*5212i)/48290823
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.26 \[ \int \frac {1}{(5-2 x)^2 (4+3 x)^{3/2} \left (2+3 x+x^2\right )} \, dx=\frac {1192366 \sqrt {3 x +4}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {3 x +4}}{\sqrt {2}}\right ) x -2980915 \sqrt {3 x +4}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {3 x +4}}{\sqrt {2}}\right )-10424 \sqrt {3 x +4}\, \sqrt {46}\, \mathrm {log}\left (2 \sqrt {3 x +4}-\sqrt {46}\right ) x +26060 \sqrt {3 x +4}\, \sqrt {46}\, \mathrm {log}\left (2 \sqrt {3 x +4}-\sqrt {46}\right )+10424 \sqrt {3 x +4}\, \sqrt {46}\, \mathrm {log}\left (2 \sqrt {3 x +4}+\sqrt {46}\right ) x -26060 \sqrt {3 x +4}\, \sqrt {46}\, \mathrm {log}\left (2 \sqrt {3 x +4}+\sqrt {46}\right )+3942108 \sqrt {3 x +4}\, \mathrm {log}\left (\sqrt {3 x +4}-1\right ) x -9855270 \sqrt {3 x +4}\, \mathrm {log}\left (\sqrt {3 x +4}-1\right )-3942108 \sqrt {3 x +4}\, \mathrm {log}\left (\sqrt {3 x +4}+1\right ) x +9855270 \sqrt {3 x +4}\, \mathrm {log}\left (\sqrt {3 x +4}+1\right )+9789444 x -24740226}{96581646 \sqrt {3 x +4}\, \left (2 x -5\right )} \] Input:

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

Output:

(1192366*sqrt(3*x + 4)*sqrt(2)*atan(sqrt(3*x + 4)/sqrt(2))*x - 2980915*sqr 
t(3*x + 4)*sqrt(2)*atan(sqrt(3*x + 4)/sqrt(2)) - 10424*sqrt(3*x + 4)*sqrt( 
46)*log(2*sqrt(3*x + 4) - sqrt(46))*x + 26060*sqrt(3*x + 4)*sqrt(46)*log(2 
*sqrt(3*x + 4) - sqrt(46)) + 10424*sqrt(3*x + 4)*sqrt(46)*log(2*sqrt(3*x + 
 4) + sqrt(46))*x - 26060*sqrt(3*x + 4)*sqrt(46)*log(2*sqrt(3*x + 4) + sqr 
t(46)) + 3942108*sqrt(3*x + 4)*log(sqrt(3*x + 4) - 1)*x - 9855270*sqrt(3*x 
 + 4)*log(sqrt(3*x + 4) - 1) - 3942108*sqrt(3*x + 4)*log(sqrt(3*x + 4) + 1 
)*x + 9855270*sqrt(3*x + 4)*log(sqrt(3*x + 4) + 1) + 9789444*x - 24740226) 
/(96581646*sqrt(3*x + 4)*(2*x - 5))