\(\int \frac {(5-2 x)^{3/2}}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx\) [565]

Optimal result

Integrand size = 29, antiderivative size = 162 \[ \int \frac {(5-2 x)^{3/2}}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=\frac {23 \sqrt {5-2 x} \sqrt {2+3 x+x^2}}{2 (4+3 x)}-\frac {23 \sqrt {-2-3 x-x^2} E\left (\arcsin \left (\sqrt {2+x}\right )|\frac {2}{9}\right )}{2 \sqrt {2+3 x+x^2}}+\frac {545 \sqrt {-2-3 x-x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2+x}\right ),\frac {2}{9}\right )}{54 \sqrt {2+3 x+x^2}}-\frac {253 \sqrt {-2-3 x-x^2} \operatorname {EllipticPi}\left (\frac {3}{2},\arcsin \left (\sqrt {2+x}\right ),\frac {2}{9}\right )}{108 \sqrt {2+3 x+x^2}} \] Output:

23*(5-2*x)^(1/2)*(x^2+3*x+2)^(1/2)/(8+6*x)-23/2*(-x^2-3*x-2)^(1/2)*Ellipti 
cE((2+x)^(1/2),1/3*2^(1/2))/(x^2+3*x+2)^(1/2)+545/54*(-x^2-3*x-2)^(1/2)*El 
lipticF((2+x)^(1/2),1/3*2^(1/2))/(x^2+3*x+2)^(1/2)-253/108*(-x^2-3*x-2)^(1 
/2)*EllipticPi((2+x)^(1/2),3/2,1/3*2^(1/2))/(x^2+3*x+2)^(1/2)
 

Mathematica [A] (verified)

Time = 20.99 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.95 \[ \int \frac {(5-2 x)^{3/2}}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=\frac {\frac {621 \sqrt {5-2 x} \left (2+3 x+x^2\right )}{4+3 x}-621 \sqrt {1+x} \sqrt {2+x} E\left (\arcsin \left (\frac {\sqrt {5-2 x}}{\sqrt {7}}\right )|\frac {7}{9}\right )+76 \sqrt {1+x} \sqrt {2+x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {5-2 x}}{\sqrt {7}}\right ),\frac {7}{9}\right )-22 \sqrt {1+x} \sqrt {2+x} \operatorname {EllipticPi}\left (\frac {21}{23},\arcsin \left (\frac {\sqrt {5-2 x}}{\sqrt {7}}\right ),\frac {7}{9}\right )}{54 \sqrt {2+3 x+x^2}} \] Input:

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

Output:

((621*Sqrt[5 - 2*x]*(2 + 3*x + x^2))/(4 + 3*x) - 621*Sqrt[1 + x]*Sqrt[2 + 
x]*EllipticE[ArcSin[Sqrt[5 - 2*x]/Sqrt[7]], 7/9] + 76*Sqrt[1 + x]*Sqrt[2 + 
 x]*EllipticF[ArcSin[Sqrt[5 - 2*x]/Sqrt[7]], 7/9] - 22*Sqrt[1 + x]*Sqrt[2 
+ x]*EllipticPi[21/23, ArcSin[Sqrt[5 - 2*x]/Sqrt[7]], 7/9])/(54*Sqrt[2 + 3 
*x + x^2])
 

Rubi [F]

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.

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

\(\Big \downarrow \) 1292

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 1292

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Unintegrable[(d + e*x)^m*(f + g*x)^n* 
(a + b*x + c*x^2)^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x]
 

Maple [A] (verified)

Time = 1.80 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.54

method	result	size
elliptic	\(\frac {\sqrt {-\left (-5+2 x \right ) \left (x^{2}+3 x +2\right )}\, \left (-\frac {50 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )}{63 \sqrt {-2 x^{3}-x^{2}+11 x +10}}-\frac {23 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \left (\frac {7 \operatorname {EllipticE}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )}{2}-\operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )\right )}{42 \sqrt {-2 x^{3}-x^{2}+11 x +10}}+\frac {23 \sqrt {-2 x^{3}-x^{2}+11 x +10}}{2 \left (3 x +4\right )}-\frac {11 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticPi}\left (\frac {\sqrt {5-2 x}}{3}, \frac {27}{23}, \frac {3 \sqrt {7}}{7}\right )}{126 \sqrt {-2 x^{3}-x^{2}+11 x +10}}\right )}{\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}}\)	\(249\)
risch	\(-\frac {23 \left (-5+2 x \right ) \sqrt {x^{2}+3 x +2}\, \sqrt {\left (5-2 x \right ) \left (x^{2}+3 x +2\right )}}{2 \left (3 x +4\right ) \sqrt {-\left (-5+2 x \right ) \left (x^{2}+3 x +2\right )}\, \sqrt {5-2 x}}+\frac {\left (-\frac {50 \sqrt {35-14 x}\, \sqrt {2 x +4}\, \sqrt {14 x +14}\, \operatorname {EllipticF}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )}{189 \sqrt {-2 x^{3}-x^{2}+11 x +10}}-\frac {23 \sqrt {35-14 x}\, \sqrt {2 x +4}\, \sqrt {14 x +14}\, \left (\frac {9 \operatorname {EllipticE}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )}{2}-2 \operatorname {EllipticF}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )\right )}{126 \sqrt {-2 x^{3}-x^{2}+11 x +10}}-\frac {11 \sqrt {35-14 x}\, \sqrt {2 x +4}\, \sqrt {14 x +14}\, \operatorname {EllipticPi}\left (\frac {\sqrt {35-14 x}}{7}, \frac {21}{23}, \frac {\sqrt {7}}{3}\right )}{378 \sqrt {-2 x^{3}-x^{2}+11 x +10}}\right ) \sqrt {\left (5-2 x \right ) \left (x^{2}+3 x +2\right )}}{\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}}\)	\(287\)
default	\(\frac {\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}\, \left (186 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right ) x +1449 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticE}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right ) x +66 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticPi}\left (\frac {\sqrt {5-2 x}}{3}, \frac {27}{23}, \frac {3 \sqrt {7}}{7}\right ) x +248 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )+1932 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticE}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )+88 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticPi}\left (\frac {\sqrt {5-2 x}}{3}, \frac {27}{23}, \frac {3 \sqrt {7}}{7}\right )+5796 x^{3}+2898 x^{2}-31878 x -28980\right )}{252 \left (2 x^{3}+x^{2}-11 x -10\right ) \left (3 x +4\right )}\)	\(290\)

Input:

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

Output:

(-(-5+2*x)*(x^2+3*x+2))^(1/2)/(5-2*x)^(1/2)/(x^2+3*x+2)^(1/2)*(-50/63*(5-2 
*x)^(1/2)*(14*x+14)^(1/2)*(2*x+4)^(1/2)/(-2*x^3-x^2+11*x+10)^(1/2)*Ellipti 
cF(1/3*(5-2*x)^(1/2),3/7*7^(1/2))-23/42*(5-2*x)^(1/2)*(14*x+14)^(1/2)*(2*x 
+4)^(1/2)/(-2*x^3-x^2+11*x+10)^(1/2)*(7/2*EllipticE(1/3*(5-2*x)^(1/2),3/7* 
7^(1/2))-EllipticF(1/3*(5-2*x)^(1/2),3/7*7^(1/2)))+23/2/(3*x+4)*(-2*x^3-x^ 
2+11*x+10)^(1/2)-11/126*(5-2*x)^(1/2)*(14*x+14)^(1/2)*(2*x+4)^(1/2)/(-2*x^ 
3-x^2+11*x+10)^(1/2)*EllipticPi(1/3*(5-2*x)^(1/2),27/23,3/7*7^(1/2)))
 

Fricas [F]

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

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

Output:

integral(sqrt(x^2 + 3*x + 2)*(-2*x + 5)^(3/2)/(9*x^4 + 51*x^3 + 106*x^2 + 
96*x + 32), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(5-2 x)^{3/2}}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=-2 \left (\int \frac {\sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}\, x}{9 x^{4}+51 x^{3}+106 x^{2}+96 x +32}d x \right )+5 \left (\int \frac {\sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}}{9 x^{4}+51 x^{3}+106 x^{2}+96 x +32}d x \right ) \] Input:

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

Output:

 - 2*int((sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2)*x)/(9*x**4 + 51*x**3 + 106 
*x**2 + 96*x + 32),x) + 5*int((sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2))/(9*x 
**4 + 51*x**3 + 106*x**2 + 96*x + 32),x)