\(\int \frac {(4+3 x) \sqrt {2+3 x+x^2}}{\sqrt {5-2 x}} \, dx\) [514]

Optimal result

Integrand size = 27, antiderivative size = 115 \[ \int \frac {(4+3 x) \sqrt {2+3 x+x^2}}{\sqrt {5-2 x}} \, dx=-\frac {1}{15} \sqrt {5-2 x} (59+9 x) \sqrt {2+3 x+x^2}-\frac {719 \sqrt {-2-3 x-x^2} E\left (\arcsin \left (\sqrt {2+x}\right )|\frac {2}{9}\right )}{10 \sqrt {2+3 x+x^2}}+\frac {637 \sqrt {-2-3 x-x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2+x}\right ),\frac {2}{9}\right )}{10 \sqrt {2+3 x+x^2}} \] Output:

-1/15*(5-2*x)^(1/2)*(59+9*x)*(x^2+3*x+2)^(1/2)-719/10*(-x^2-3*x-2)^(1/2)*E 
llipticE((2+x)^(1/2),1/3*2^(1/2))/(x^2+3*x+2)^(1/2)+637/10*(-x^2-3*x-2)^(1 
/2)*EllipticF((2+x)^(1/2),1/3*2^(1/2))/(x^2+3*x+2)^(1/2)
 

Mathematica [A] (verified)

Time = 28.24 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.43 \[ \int \frac {(4+3 x) \sqrt {2+3 x+x^2}}{\sqrt {5-2 x}} \, dx=\frac {-2 \sqrt {5-2 x} \left (848+1418 x+679 x^2+127 x^3+18 x^4\right )-2157 (5-2 x)^2 \sqrt {\frac {1+x}{-5+2 x}} \sqrt {\frac {2+x}{-5+2 x}} E\left (\arcsin \left (\frac {3}{\sqrt {5-2 x}}\right )|\frac {7}{9}\right )+246 (5-2 x)^2 \sqrt {\frac {1+x}{-5+2 x}} \sqrt {\frac {2+x}{-5+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {3}{\sqrt {5-2 x}}\right ),\frac {7}{9}\right )}{30 (-5+2 x) \sqrt {2+3 x+x^2}} \] Input:

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

Output:

(-2*Sqrt[5 - 2*x]*(848 + 1418*x + 679*x^2 + 127*x^3 + 18*x^4) - 2157*(5 - 
2*x)^2*Sqrt[(1 + x)/(-5 + 2*x)]*Sqrt[(2 + x)/(-5 + 2*x)]*EllipticE[ArcSin[ 
3/Sqrt[5 - 2*x]], 7/9] + 246*(5 - 2*x)^2*Sqrt[(1 + x)/(-5 + 2*x)]*Sqrt[(2 
+ x)/(-5 + 2*x)]*EllipticF[ArcSin[3/Sqrt[5 - 2*x]], 7/9])/(30*(-5 + 2*x)*S 
qrt[2 + 3*x + x^2])
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1231, 25, 1269, 1172, 27, 321, 327}

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.

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

\(\Big \downarrow \) 1231

\(\displaystyle -\frac {1}{30} \int -\frac {719 x+1069}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx-\frac {1}{15} \sqrt {5-2 x} \sqrt {x^2+3 x+2} (9 x+59)\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{30} \int \frac {719 x+1069}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx-\frac {1}{15} \sqrt {5-2 x} (9 x+59) \sqrt {x^2+3 x+2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{30} \left (\frac {5733}{2} \int \frac {1}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx-\frac {719}{2} \int \frac {\sqrt {5-2 x}}{\sqrt {x^2+3 x+2}}dx\right )-\frac {1}{15} \sqrt {5-2 x} (9 x+59) \sqrt {x^2+3 x+2}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {1}{30} \left (\frac {1911 \sqrt {-x^2-3 x-2} \int \frac {3}{\sqrt {-x-1} \sqrt {9-2 (x+2)}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}-\frac {2157 \sqrt {-x^2-3 x-2} \int \frac {\sqrt {9-2 (x+2)}}{3 \sqrt {-x-1}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}\right )-\frac {1}{15} \sqrt {5-2 x} (9 x+59) \sqrt {x^2+3 x+2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{30} \left (\frac {5733 \sqrt {-x^2-3 x-2} \int \frac {1}{\sqrt {-x-1} \sqrt {9-2 (x+2)}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}-\frac {719 \sqrt {-x^2-3 x-2} \int \frac {\sqrt {9-2 (x+2)}}{\sqrt {-x-1}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}\right )-\frac {1}{15} \sqrt {5-2 x} (9 x+59) \sqrt {x^2+3 x+2}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{30} \left (\frac {1911 \sqrt {-x^2-3 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x+2}\right ),\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}-\frac {719 \sqrt {-x^2-3 x-2} \int \frac {\sqrt {9-2 (x+2)}}{\sqrt {-x-1}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}\right )-\frac {1}{15} \sqrt {5-2 x} (9 x+59) \sqrt {x^2+3 x+2}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{30} \left (\frac {1911 \sqrt {-x^2-3 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x+2}\right ),\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}-\frac {2157 \sqrt {-x^2-3 x-2} E\left (\arcsin \left (\sqrt {x+2}\right )|\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}\right )-\frac {1}{15} \sqrt {5-2 x} (9 x+59) \sqrt {x^2+3 x+2}\)

Input:

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

Output:

-1/15*(Sqrt[5 - 2*x]*(59 + 9*x)*Sqrt[2 + 3*x + x^2]) + ((-2157*Sqrt[-2 - 3 
*x - x^2]*EllipticE[ArcSin[Sqrt[2 + x]], 2/9])/Sqrt[2 + 3*x + x^2] + (1911 
*Sqrt[-2 - 3*x - x^2]*EllipticF[ArcSin[Sqrt[2 + x]], 2/9])/Sqrt[2 + 3*x + 
x^2])/30
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 1172

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy 
mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 
)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e 
*Rt[b^2 - 4*a*c, 2])))^m))   Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* 
c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 
2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[m^2, 1/4]
 

rule 1231

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) 
 - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ 
(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 
 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* 
a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* 
c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c 
^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  !R 
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Integer 
Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1269

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

Maple [A] (verified)

Time = 1.29 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.14

method	result	size
default	\(\frac {\sqrt {x^{2}+3 x +2}\, \sqrt {5-2 x}\, \left (100 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )+719 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticE}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )-72 x^{4}-508 x^{3}+160 x^{2}+2956 x +2360\right )}{120 x^{3}+60 x^{2}-660 x -600}\)	\(131\)
elliptic	\(\frac {\sqrt {-\left (-5+2 x \right ) \left (x^{2}+3 x +2\right )}\, \left (-\frac {3 x \sqrt {-2 x^{3}-x^{2}+11 x +10}}{5}-\frac {59 \sqrt {-2 x^{3}-x^{2}+11 x +10}}{15}-\frac {1069 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )}{210 \sqrt {-2 x^{3}-x^{2}+11 x +10}}-\frac {719 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \left (\frac {7 \operatorname {EllipticE}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )}{2}-\operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )\right )}{210 \sqrt {-2 x^{3}-x^{2}+11 x +10}}\right )}{\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}}\)	\(206\)
risch	\(\frac {\left (59+9 x \right ) \left (-5+2 x \right ) \sqrt {x^{2}+3 x +2}\, \sqrt {\left (5-2 x \right ) \left (x^{2}+3 x +2\right )}}{15 \sqrt {-\left (-5+2 x \right ) \left (x^{2}+3 x +2\right )}\, \sqrt {5-2 x}}+\frac {\left (-\frac {1069 \sqrt {35-14 x}\, \sqrt {2 x +4}\, \sqrt {14 x +14}\, \operatorname {EllipticF}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )}{630 \sqrt {-2 x^{3}-x^{2}+11 x +10}}-\frac {719 \sqrt {35-14 x}\, \sqrt {2 x +4}\, \sqrt {14 x +14}\, \left (\frac {9 \operatorname {EllipticE}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )}{2}-2 \operatorname {EllipticF}\left (\frac {\sqrt {35-14 x}}{7}, \frac {\sqrt {7}}{3}\right )\right )}{630 \sqrt {-2 x^{3}-x^{2}+11 x +10}}\right ) \sqrt {\left (5-2 x \right ) \left (x^{2}+3 x +2\right )}}{\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}}\)	\(229\)

Input:

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

Output:

1/60*(x^2+3*x+2)^(1/2)*(5-2*x)^(1/2)*(100*(5-2*x)^(1/2)*(14*x+14)^(1/2)*(2 
*x+4)^(1/2)*EllipticF(1/3*(5-2*x)^(1/2),3/7*7^(1/2))+719*(5-2*x)^(1/2)*(14 
*x+14)^(1/2)*(2*x+4)^(1/2)*EllipticE(1/3*(5-2*x)^(1/2),3/7*7^(1/2))-72*x^4 
-508*x^3+160*x^2+2956*x+2360)/(2*x^3+x^2-11*x-10)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.43 \[ \int \frac {(4+3 x) \sqrt {2+3 x+x^2}}{\sqrt {5-2 x}} \, dx=-\frac {1}{15} \, \sqrt {x^{2} + 3 \, x + 2} {\left (9 \, x + 59\right )} \sqrt {-2 \, x + 5} - \frac {1139}{36} \, \sqrt {-2} {\rm weierstrassPInverse}\left (\frac {67}{3}, \frac {440}{27}, x + \frac {1}{6}\right ) + \frac {719}{30} \, \sqrt {-2} {\rm weierstrassZeta}\left (\frac {67}{3}, \frac {440}{27}, {\rm weierstrassPInverse}\left (\frac {67}{3}, \frac {440}{27}, x + \frac {1}{6}\right )\right ) \] Input:

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

Output:

-1/15*sqrt(x^2 + 3*x + 2)*(9*x + 59)*sqrt(-2*x + 5) - 1139/36*sqrt(-2)*wei 
erstrassPInverse(67/3, 440/27, x + 1/6) + 719/30*sqrt(-2)*weierstrassZeta( 
67/3, 440/27, weierstrassPInverse(67/3, 440/27, x + 1/6))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(4+3 x) \sqrt {2+3 x+x^2}}{\sqrt {5-2 x}} \, dx=-\frac {3 \sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}\, x}{5}-\frac {279 \sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}}{10}+\frac {719 \left (\int \frac {\sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}\, x^{2}}{2 x^{3}+x^{2}-11 x -10}d x \right )}{10}-\frac {3349 \left (\int \frac {\sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}}{2 x^{3}+x^{2}-11 x -10}d x \right )}{20} \] Input:

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

Output:

( - 12*sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2)*x - 558*sqrt( - 2*x + 5)*sqrt 
(x**2 + 3*x + 2) + 1438*int((sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2)*x**2)/( 
2*x**3 + x**2 - 11*x - 10),x) - 3349*int((sqrt( - 2*x + 5)*sqrt(x**2 + 3*x 
 + 2))/(2*x**3 + x**2 - 11*x - 10),x))/20