\(\int \frac {(5-2 x)^{5/2}}{\sqrt {4+3 x} (2+3 x+x^2)} \, dx\) [360]

Optimal result

Integrand size = 29, antiderivative size = 117 \[ \int \frac {(5-2 x)^{5/2}}{\sqrt {4+3 x} \left (2+3 x+x^2\right )} \, dx=\frac {4}{3} \sqrt {5-2 x} \sqrt {4+3 x}+\frac {238}{3} \sqrt {\frac {2}{3}} \arcsin \left (\sqrt {\frac {2}{23}} \sqrt {4+3 x}\right )-243 \sqrt {2} \arctan \left (\frac {3 \sqrt {4+3 x}}{\sqrt {2} \sqrt {5-2 x}}\right )-98 \sqrt {7} \text {arctanh}\left (\frac {\sqrt {7} \sqrt {4+3 x}}{\sqrt {5-2 x}}\right ) \] Output:

4/3*(5-2*x)^(1/2)*(4+3*x)^(1/2)+238/9*arcsin(1/23*46^(1/2)*(4+3*x)^(1/2))* 
6^(1/2)-243*2^(1/2)*arctan(3/2*(4+3*x)^(1/2)*2^(1/2)/(5-2*x)^(1/2))-98*7^( 
1/2)*arctanh(7^(1/2)*(4+3*x)^(1/2)/(5-2*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.01 \[ \int \frac {(5-2 x)^{5/2}}{\sqrt {4+3 x} \left (2+3 x+x^2\right )} \, dx=\frac {4}{3} \sqrt {5-2 x} \sqrt {4+3 x}+243 \sqrt {2} \arctan \left (\frac {\sqrt {10-4 x}}{3 \sqrt {4+3 x}}\right )-\frac {238}{3} \sqrt {\frac {2}{3}} \arctan \left (\frac {\sqrt {\frac {15}{2}-3 x}}{\sqrt {4+3 x}}\right )-98 \sqrt {7} \text {arctanh}\left (\frac {\sqrt {5-2 x}}{\sqrt {7} \sqrt {4+3 x}}\right ) \] Input:

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

Output:

(4*Sqrt[5 - 2*x]*Sqrt[4 + 3*x])/3 + 243*Sqrt[2]*ArcTan[Sqrt[10 - 4*x]/(3*S 
qrt[4 + 3*x])] - (238*Sqrt[2/3]*ArcTan[Sqrt[15/2 - 3*x]/Sqrt[4 + 3*x]])/3 
- 98*Sqrt[7]*ArcTanh[Sqrt[5 - 2*x]/(Sqrt[7]*Sqrt[4 + 3*x])]
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.21, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1204, 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.

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

\(\Big \downarrow \) 1204

\(\displaystyle \int \left (-\frac {386 x+43}{\sqrt {5-2 x} \sqrt {3 x+4} \left (x^2+3 x+2\right )}-\frac {8 x}{\sqrt {5-2 x} \sqrt {3 x+4}}+\frac {84}{\sqrt {5-2 x} \sqrt {3 x+4}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 28 \sqrt {6} \arcsin \left (\sqrt {\frac {2}{23}} \sqrt {3 x+4}\right )-\frac {14}{3} \sqrt {\frac {2}{3}} \arcsin \left (\sqrt {\frac {2}{23}} \sqrt {3 x+4}\right )-243 \sqrt {2} \arctan \left (\frac {3 \sqrt {3 x+4}}{\sqrt {2} \sqrt {5-2 x}}\right )-98 \sqrt {7} \text {arctanh}\left (\frac {\sqrt {7} \sqrt {3 x+4}}{\sqrt {5-2 x}}\right )+\frac {4}{3} \sqrt {5-2 x} \sqrt {3 x+4}\)

Input:

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

Output:

(4*Sqrt[5 - 2*x]*Sqrt[4 + 3*x])/3 - (14*Sqrt[2/3]*ArcSin[Sqrt[2/23]*Sqrt[4 
 + 3*x]])/3 + 28*Sqrt[6]*ArcSin[Sqrt[2/23]*Sqrt[4 + 3*x]] - 243*Sqrt[2]*Ar 
cTan[(3*Sqrt[4 + 3*x])/(Sqrt[2]*Sqrt[5 - 2*x])] - 98*Sqrt[7]*ArcTanh[(Sqrt 
[7]*Sqrt[4 + 3*x])/Sqrt[5 - 2*x]]
 

Defintions of rubi rules used

rule 1204

Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)), x_Symbol] :> Int[ExpandIntegrand[1/(Sqrt[d + e*x]*Sqrt[f 
 + g*x]), (d + e*x)^(m + 1/2)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, 
d, e, f, g}, x] && IGtQ[m + 1/2, 0]
 

rule 2009

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

Maple [A] (verified)

Time = 2.08 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.97

method	result	size
default	\(\frac {\sqrt {5-2 x}\, \sqrt {3 x +4}\, \left (238 \sqrt {3}\, \sqrt {2}\, \arcsin \left (\frac {12 x}{23}-\frac {7}{23}\right )-2187 \sqrt {2}\, \arctan \left (\frac {\left (26+31 x \right ) \sqrt {2}}{12 \sqrt {-6 x^{2}+7 x +20}}\right )-882 \sqrt {7}\, \operatorname {arctanh}\left (\frac {\left (33+19 x \right ) \sqrt {7}}{14 \sqrt {-6 x^{2}+7 x +20}}\right )+24 \sqrt {-6 x^{2}+7 x +20}\right )}{18 \sqrt {-6 x^{2}+7 x +20}}\)	\(114\)
risch	\(-\frac {4 \sqrt {3 x +4}\, \left (-5+2 x \right ) \sqrt {\left (5-2 x \right ) \left (3 x +4\right )}}{3 \sqrt {-\left (3 x +4\right ) \left (-5+2 x \right )}\, \sqrt {5-2 x}}-\frac {\left (-\frac {119 \sqrt {6}\, \arcsin \left (\frac {12 x}{23}-\frac {7}{23}\right )}{9}+49 \sqrt {7}\, \operatorname {arctanh}\left (\frac {\left (33+19 x \right ) \sqrt {7}}{14 \sqrt {-6 \left (x +1\right )^{2}+19 x +26}}\right )+\frac {243 \sqrt {2}\, \arctan \left (\frac {\left (26+31 x \right ) \sqrt {2}}{12 \sqrt {-6 \left (2+x \right )^{2}+44+31 x}}\right )}{2}\right ) \sqrt {\left (5-2 x \right ) \left (3 x +4\right )}}{\sqrt {5-2 x}\, \sqrt {3 x +4}}\)	\(151\)

Input:

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

Output:

1/18*(5-2*x)^(1/2)*(3*x+4)^(1/2)*(238*3^(1/2)*2^(1/2)*arcsin(12/23*x-7/23) 
-2187*2^(1/2)*arctan(1/12*(26+31*x)*2^(1/2)/(-6*x^2+7*x+20)^(1/2))-882*7^( 
1/2)*arctanh(1/14*(33+19*x)*7^(1/2)/(-6*x^2+7*x+20)^(1/2))+24*(-6*x^2+7*x+ 
20)^(1/2))/(-6*x^2+7*x+20)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.31 \[ \int \frac {(5-2 x)^{5/2}}{\sqrt {4+3 x} \left (2+3 x+x^2\right )} \, dx=\frac {243}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (31 \, x + 26\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5}}{12 \, {\left (6 \, x^{2} - 7 \, x - 20\right )}}\right ) - \frac {119}{3} \, \sqrt {\frac {2}{3}} \arctan \left (\frac {\sqrt {\frac {2}{3}} {\left (12 \, x - 7\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5}}{4 \, {\left (6 \, x^{2} - 7 \, x - 20\right )}}\right ) + \frac {49}{2} \, \sqrt {7} \log \left (-\frac {4 \, \sqrt {7} {\left (19 \, x + 33\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5} - 193 \, x^{2} - 1450 \, x - 1649}{x^{2} + 2 \, x + 1}\right ) + \frac {4}{3} \, \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5} \] Input:

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

Output:

243/2*sqrt(2)*arctan(1/12*sqrt(2)*(31*x + 26)*sqrt(3*x + 4)*sqrt(-2*x + 5) 
/(6*x^2 - 7*x - 20)) - 119/3*sqrt(2/3)*arctan(1/4*sqrt(2/3)*(12*x - 7)*sqr 
t(3*x + 4)*sqrt(-2*x + 5)/(6*x^2 - 7*x - 20)) + 49/2*sqrt(7)*log(-(4*sqrt( 
7)*(19*x + 33)*sqrt(3*x + 4)*sqrt(-2*x + 5) - 193*x^2 - 1450*x - 1649)/(x^ 
2 + 2*x + 1)) + 4/3*sqrt(3*x + 4)*sqrt(-2*x + 5)
 

Sympy [F]

\[ \int \frac {(5-2 x)^{5/2}}{\sqrt {4+3 x} \left (2+3 x+x^2\right )} \, dx=\int \frac {\left (5 - 2 x\right )^{\frac {5}{2}}}{\left (x + 1\right ) \left (x + 2\right ) \sqrt {3 x + 4}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {(5-2 x)^{5/2}}{\sqrt {4+3 x} \left (2+3 x+x^2\right )} \, dx=\int { \frac {{\left (-2 \, x + 5\right )}^{\frac {5}{2}}}{{\left (x^{2} + 3 \, x + 2\right )} \sqrt {3 \, x + 4}} \,d x } \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (84) = 168\).

Time = 0.51 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.50 \[ \int \frac {(5-2 x)^{5/2}}{\sqrt {4+3 x} \left (2+3 x+x^2\right )} \, dx=-\frac {81}{2} \, \sqrt {6} \sqrt {3} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {3} \sqrt {3 \, x + 4} {\left (\frac {{\left (\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}\right )}^{2}}{3 \, x + 4} - 4\right )}}{18 \, {\left (\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}\right )}}\right )\right )} + \frac {49}{6} \, \sqrt {42} \sqrt {6} \log \left (\frac {{\left | -2 \, \sqrt {42} + \frac {\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}}{\sqrt {3 \, x + 4}} - \frac {4 \, \sqrt {3 \, x + 4}}{\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}} \right |}}{{\left | 2 \, \sqrt {42} + \frac {\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}}{\sqrt {3 \, x + 4}} - \frac {4 \, \sqrt {3 \, x + 4}}{\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}} \right |}}\right ) + \frac {119}{9} \, \sqrt {6} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {3 \, x + 4} {\left (\frac {{\left (\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}\right )}^{2}}{3 \, x + 4} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}\right )}}\right )\right )} + \frac {4}{9} \, \sqrt {3} \sqrt {3 \, x + 4} \sqrt {-6 \, x + 15} \] Input:

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

Output:

-81/2*sqrt(6)*sqrt(3)*(pi + 2*arctan(-1/18*sqrt(3)*sqrt(3*x + 4)*((sqrt(2) 
*sqrt(-6*x + 15) - sqrt(46))^2/(3*x + 4) - 4)/(sqrt(2)*sqrt(-6*x + 15) - s 
qrt(46)))) + 49/6*sqrt(42)*sqrt(6)*log(abs(-2*sqrt(42) + (sqrt(2)*sqrt(-6* 
x + 15) - sqrt(46))/sqrt(3*x + 4) - 4*sqrt(3*x + 4)/(sqrt(2)*sqrt(-6*x + 1 
5) - sqrt(46)))/abs(2*sqrt(42) + (sqrt(2)*sqrt(-6*x + 15) - sqrt(46))/sqrt 
(3*x + 4) - 4*sqrt(3*x + 4)/(sqrt(2)*sqrt(-6*x + 15) - sqrt(46)))) + 119/9 
*sqrt(6)*(pi + 2*arctan(-1/4*sqrt(3*x + 4)*((sqrt(2)*sqrt(-6*x + 15) - sqr 
t(46))^2/(3*x + 4) - 4)/(sqrt(2)*sqrt(-6*x + 15) - sqrt(46)))) + 4/9*sqrt( 
3)*sqrt(3*x + 4)*sqrt(-6*x + 15)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(5-2 x)^{5/2}}{\sqrt {4+3 x} \left (2+3 x+x^2\right )} \, dx=\int \frac {{\left (5-2\,x\right )}^{5/2}}{\sqrt {3\,x+4}\,\left (x^2+3\,x+2\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {(5-2 x)^{5/2}}{\sqrt {4+3 x} \left (2+3 x+x^2\right )} \, dx=\int \frac {\left (-2 x +5\right )^{\frac {5}{2}}}{\sqrt {3 x +4}\, \left (x^{2}+3 x +2\right )}d x \] Input:

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

Output:

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