\(\int \frac {(5-2 x)^{5/2}}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx\) [571]

Optimal result

Integrand size = 29, antiderivative size = 162 \[ \int \frac {(5-2 x)^{5/2}}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=\frac {529 \sqrt {5-2 x} \sqrt {2+3 x+x^2}}{6 (4+3 x)}-\frac {171 \sqrt {-2-3 x-x^2} E\left (\arcsin \left (\sqrt {2+x}\right )|\frac {2}{9}\right )}{2 \sqrt {2+3 x+x^2}}+\frac {4301 \sqrt {-2-3 x-x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2+x}\right ),\frac {2}{9}\right )}{54 \sqrt {2+3 x+x^2}}-\frac {529 \sqrt {-2-3 x-x^2} \operatorname {EllipticPi}\left (\frac {3}{2},\arcsin \left (\sqrt {2+x}\right ),\frac {2}{9}\right )}{108 \sqrt {2+3 x+x^2}} \] Output:

529*(5-2*x)^(1/2)*(x^2+3*x+2)^(1/2)/(24+18*x)-171/2*(-x^2-3*x-2)^(1/2)*Ell 
ipticE((2+x)^(1/2),1/3*2^(1/2))/(x^2+3*x+2)^(1/2)+4301/54*(-x^2-3*x-2)^(1/ 
2)*EllipticF((2+x)^(1/2),1/3*2^(1/2))/(x^2+3*x+2)^(1/2)-529/108*(-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 = 20.82 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.04 \[ \int \frac {(5-2 x)^{5/2}}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=\frac {4761 \sqrt {5-2 x} \left (2+3 x+x^2\right )-4617 \sqrt {1+x} \sqrt {2+x} (4+3 x) E\left (\arcsin \left (\frac {\sqrt {5-2 x}}{\sqrt {7}}\right )|\frac {7}{9}\right )+316 \sqrt {1+x} \sqrt {2+x} (4+3 x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {5-2 x}}{\sqrt {7}}\right ),\frac {7}{9}\right )-46 \sqrt {1+x} \sqrt {2+x} (4+3 x) \operatorname {EllipticPi}\left (\frac {21}{23},\arcsin \left (\frac {\sqrt {5-2 x}}{\sqrt {7}}\right ),\frac {7}{9}\right )}{54 (4+3 x) \sqrt {2+3 x+x^2}} \] Input:

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

Output:

(4761*Sqrt[5 - 2*x]*(2 + 3*x + x^2) - 4617*Sqrt[1 + x]*Sqrt[2 + x]*(4 + 3* 
x)*EllipticE[ArcSin[Sqrt[5 - 2*x]/Sqrt[7]], 7/9] + 316*Sqrt[1 + x]*Sqrt[2 
+ x]*(4 + 3*x)*EllipticF[ArcSin[Sqrt[5 - 2*x]/Sqrt[7]], 7/9] - 46*Sqrt[1 + 
 x]*Sqrt[2 + x]*(4 + 3*x)*EllipticPi[21/23, ArcSin[Sqrt[5 - 2*x]/Sqrt[7]], 
 7/9])/(54*(4 + 3*x)*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.

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

\(\Big \downarrow \) 1292

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

Input:

Int[(5 - 2*x)^(5/2)/((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 = 1.78 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.54

method	result	size
elliptic	\(\frac {\sqrt {-\left (-5+2 x \right ) \left (x^{2}+3 x +2\right )}\, \left (-\frac {62 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )}{9 \sqrt {-2 x^{3}-x^{2}+11 x +10}}-\frac {57 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \left (\frac {7 \operatorname {EllipticE}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )}{2}-\operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )\right )}{14 \sqrt {-2 x^{3}-x^{2}+11 x +10}}+\frac {529 \sqrt {-2 x^{3}-x^{2}+11 x +10}}{6 \left (3 x +4\right )}-\frac {23 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticPi}\left (\frac {\sqrt {5-2 x}}{3}, \frac {27}{23}, \frac {3 \sqrt {7}}{7}\right )}{126 \sqrt {-2 x^{3}-x^{2}+11 x +10}}\right )}{\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}}\)	\(249\)
risch	\(-\frac {529 \left (-5+2 x \right ) \sqrt {x^{2}+3 x +2}\, \sqrt {\left (5-2 x \right ) \left (x^{2}+3 x +2\right )}}{6 \left (3 x +4\right ) \sqrt {-\left (-5+2 x \right ) \left (x^{2}+3 x +2\right )}\, \sqrt {5-2 x}}-\frac {\left (\frac {62 \sqrt {35-14 x}\, \sqrt {2 x +4}\, \sqrt {14 x +14}\, \operatorname {EllipticF}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )}{27 \sqrt {-2 x^{3}-x^{2}+11 x +10}}+\frac {19 \sqrt {35-14 x}\, \sqrt {2 x +4}\, \sqrt {14 x +14}\, \left (\frac {9 \operatorname {EllipticE}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )}{2}-2 \operatorname {EllipticF}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )\right )}{14 \sqrt {-2 x^{3}-x^{2}+11 x +10}}+\frac {23 \sqrt {35-14 x}\, \sqrt {2 x +4}\, \sqrt {14 x +14}\, \operatorname {EllipticPi}\left (\frac {\sqrt {35-14 x}}{7}, \frac {21}{23}, \frac {\sqrt {7}}{3}\right )}{378 \sqrt {-2 x^{3}-x^{2}+11 x +10}}\right ) \sqrt {\left (5-2 x \right ) \left (x^{2}+3 x +2\right )}}{\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}}\)	\(288\)
default	\(\frac {\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}\, \left (2130 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right ) x +10773 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticE}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right ) x +138 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticPi}\left (\frac {\sqrt {5-2 x}}{3}, \frac {27}{23}, \frac {3 \sqrt {7}}{7}\right ) x +2840 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )+14364 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticE}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )+184 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticPi}\left (\frac {\sqrt {5-2 x}}{3}, \frac {27}{23}, \frac {3 \sqrt {7}}{7}\right )+44436 x^{3}+22218 x^{2}-244398 x -222180\right )}{252 \left (2 x^{3}+x^{2}-11 x -10\right ) \left (3 x +4\right )}\)	\(290\)

Input:

int((5-2*x)^(5/2)/(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)*(-62/9*(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)*Elliptic 
F(1/3*(5-2*x)^(1/2),3/7*7^(1/2))-57/14*(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)))+529/6/(3*x+4)*(-2*x^3-x^ 
2+11*x+10)^(1/2)-23/126*(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]

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

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

Output:

integral((4*x^2 - 20*x + 25)*sqrt(x^2 + 3*x + 2)*sqrt(-2*x + 5)/(9*x^4 + 5 
1*x^3 + 106*x^2 + 96*x + 32), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(5-2 x)^{5/2}}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=\frac {-10 \sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}+138 \left (\int \frac {\sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}\, x^{3}}{18 x^{5}+57 x^{4}-43 x^{3}-338 x^{2}-416 x -160}d x \right ) x +184 \left (\int \frac {\sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}\, x^{3}}{18 x^{5}+57 x^{4}-43 x^{3}-338 x^{2}-416 x -160}d x \right )+1515 \left (\int \frac {\sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}\, x}{18 x^{5}+57 x^{4}-43 x^{3}-338 x^{2}-416 x -160}d x \right ) x +2020 \left (\int \frac {\sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}\, x}{18 x^{5}+57 x^{4}-43 x^{3}-338 x^{2}-416 x -160}d x \right )-510 \left (\int \frac {\sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}}{18 x^{5}+57 x^{4}-43 x^{3}-338 x^{2}-416 x -160}d x \right ) x -680 \left (\int \frac {\sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}}{18 x^{5}+57 x^{4}-43 x^{3}-338 x^{2}-416 x -160}d x \right )}{6 x +8} \] Input:

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

Output:

( - 10*sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2) + 138*int((sqrt( - 2*x + 5)*s 
qrt(x**2 + 3*x + 2)*x**3)/(18*x**5 + 57*x**4 - 43*x**3 - 338*x**2 - 416*x 
- 160),x)*x + 184*int((sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2)*x**3)/(18*x** 
5 + 57*x**4 - 43*x**3 - 338*x**2 - 416*x - 160),x) + 1515*int((sqrt( - 2*x 
 + 5)*sqrt(x**2 + 3*x + 2)*x)/(18*x**5 + 57*x**4 - 43*x**3 - 338*x**2 - 41 
6*x - 160),x)*x + 2020*int((sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2)*x)/(18*x 
**5 + 57*x**4 - 43*x**3 - 338*x**2 - 416*x - 160),x) - 510*int((sqrt( - 2* 
x + 5)*sqrt(x**2 + 3*x + 2))/(18*x**5 + 57*x**4 - 43*x**3 - 338*x**2 - 416 
*x - 160),x)*x - 680*int((sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2))/(18*x**5 
+ 57*x**4 - 43*x**3 - 338*x**2 - 416*x - 160),x))/(2*(3*x + 4))