\(\int \frac {4+3 x}{(5-2 x)^{3/2} \sqrt {2+3 x+x^2}} \, dx\) [588]

Optimal result

Integrand size = 27, antiderivative size = 108 \[ \int \frac {4+3 x}{(5-2 x)^{3/2} \sqrt {2+3 x+x^2}} \, dx=\frac {46 \sqrt {2+3 x+x^2}}{63 \sqrt {5-2 x}}+\frac {23 \sqrt {-2-3 x-x^2} E\left (\arcsin \left (\sqrt {2+x}\right )|\frac {2}{9}\right )}{21 \sqrt {2+3 x+x^2}}-\frac {\sqrt {-2-3 x-x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2+x}\right ),\frac {2}{9}\right )}{\sqrt {2+3 x+x^2}} \] Output:

46/63*(x^2+3*x+2)^(1/2)/(5-2*x)^(1/2)+23/21*(-x^2-3*x-2)^(1/2)*EllipticE(( 
2+x)^(1/2),1/3*2^(1/2))/(x^2+3*x+2)^(1/2)-(-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 = 31.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.80 \[ \int \frac {4+3 x}{(5-2 x)^{3/2} \sqrt {2+3 x+x^2}} \, dx=\frac {(1+x) \sqrt {\frac {2+x}{-5+2 x}} \left (23 E\left (\arcsin \left (\frac {3}{\sqrt {5-2 x}}\right )|\frac {7}{9}\right )-2 \operatorname {EllipticF}\left (\arcsin \left (\frac {3}{\sqrt {5-2 x}}\right ),\frac {7}{9}\right )\right )}{21 \sqrt {\frac {1+x}{-5+2 x}} \sqrt {2+3 x+x^2}} \] Input:

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

Output:

((1 + x)*Sqrt[(2 + x)/(-5 + 2*x)]*(23*EllipticE[ArcSin[3/Sqrt[5 - 2*x]], 7 
/9] - 2*EllipticF[ArcSin[3/Sqrt[5 - 2*x]], 7/9]))/(21*Sqrt[(1 + x)/(-5 + 2 
*x)]*Sqrt[2 + 3*x + x^2])
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1237, 27, 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.

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

\(\Big \downarrow \) 1237

\(\displaystyle \frac {46 \sqrt {x^2+3 x+2}}{63 \sqrt {5-2 x}}-\frac {2}{63} \int \frac {23 x+37}{2 \sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {46 \sqrt {x^2+3 x+2}}{63 \sqrt {5-2 x}}-\frac {1}{63} \int \frac {23 x+37}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{63} \left (\frac {23}{2} \int \frac {\sqrt {5-2 x}}{\sqrt {x^2+3 x+2}}dx-\frac {189}{2} \int \frac {1}{\sqrt {5-2 x} \sqrt {x^2+3 x+2}}dx\right )+\frac {46 \sqrt {x^2+3 x+2}}{63 \sqrt {5-2 x}}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {1}{63} \left (\frac {69 \sqrt {-x^2-3 x-2} \int \frac {\sqrt {9-2 (x+2)}}{3 \sqrt {-x-1}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}-\frac {63 \sqrt {-x^2-3 x-2} \int \frac {3}{\sqrt {-x-1} \sqrt {9-2 (x+2)}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}\right )+\frac {46 \sqrt {x^2+3 x+2}}{63 \sqrt {5-2 x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{63} \left (\frac {23 \sqrt {-x^2-3 x-2} \int \frac {\sqrt {9-2 (x+2)}}{\sqrt {-x-1}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}-\frac {189 \sqrt {-x^2-3 x-2} \int \frac {1}{\sqrt {-x-1} \sqrt {9-2 (x+2)}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}\right )+\frac {46 \sqrt {x^2+3 x+2}}{63 \sqrt {5-2 x}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{63} \left (\frac {23 \sqrt {-x^2-3 x-2} \int \frac {\sqrt {9-2 (x+2)}}{\sqrt {-x-1}}d\sqrt {x+2}}{\sqrt {x^2+3 x+2}}-\frac {63 \sqrt {-x^2-3 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x+2}\right ),\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}\right )+\frac {46 \sqrt {x^2+3 x+2}}{63 \sqrt {5-2 x}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{63} \left (\frac {69 \sqrt {-x^2-3 x-2} E\left (\arcsin \left (\sqrt {x+2}\right )|\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}-\frac {63 \sqrt {-x^2-3 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x+2}\right ),\frac {2}{9}\right )}{\sqrt {x^2+3 x+2}}\right )+\frac {46 \sqrt {x^2+3 x+2}}{63 \sqrt {5-2 x}}\)

Input:

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

Output:

(46*Sqrt[2 + 3*x + x^2])/(63*Sqrt[5 - 2*x]) + ((69*Sqrt[-2 - 3*x - x^2]*El 
lipticE[ArcSin[Sqrt[2 + x]], 2/9])/Sqrt[2 + 3*x + x^2] - (63*Sqrt[-2 - 3*x 
 - x^2]*EllipticF[ArcSin[Sqrt[2 + x]], 2/9])/Sqrt[2 + 3*x + x^2])/63
 

Defintions of rubi rules used

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 1237

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* 
x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) 
*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 
] && (IntegerQ[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 = 1.25 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.12

method	result	size
default	\(-\frac {\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}\, \left (4 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )+23 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticE}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )+92 x^{2}+276 x +184\right )}{126 \left (2 x^{3}+x^{2}-11 x -10\right )}\)	\(121\)
elliptic	\(\frac {\sqrt {-\left (-5+2 x \right ) \left (x^{2}+3 x +2\right )}\, \left (-\frac {23 \left (-2 x^{2}-6 x -4\right )}{63 \sqrt {\left (x -\frac {5}{2}\right ) \left (-2 x^{2}-6 x -4\right )}}+\frac {37 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )}{441 \sqrt {-2 x^{3}-x^{2}+11 x +10}}+\frac {23 \sqrt {5-2 x}\, \sqrt {14 x +14}\, \sqrt {2 x +4}\, \left (\frac {7 \operatorname {EllipticE}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )}{2}-\operatorname {EllipticF}\left (\frac {\sqrt {5-2 x}}{3}, \frac {3 \sqrt {7}}{7}\right )\right )}{441 \sqrt {-2 x^{3}-x^{2}+11 x +10}}\right )}{\sqrt {5-2 x}\, \sqrt {x^{2}+3 x +2}}\)	\(195\)

Input:

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

Output:

-1/126*(5-2*x)^(1/2)*(x^2+3*x+2)^(1/2)*(4*(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))+23*(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))+92*x^2+ 
276*x+184)/(2*x^3+x^2-11*x-10)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.59 \[ \int \frac {4+3 x}{(5-2 x)^{3/2} \sqrt {2+3 x+x^2}} \, dx=\frac {199 \, \sqrt {-2} {\left (2 \, x - 5\right )} {\rm weierstrassPInverse}\left (\frac {67}{3}, \frac {440}{27}, x + \frac {1}{6}\right ) - 138 \, \sqrt {-2} {\left (2 \, x - 5\right )} {\rm weierstrassZeta}\left (\frac {67}{3}, \frac {440}{27}, {\rm weierstrassPInverse}\left (\frac {67}{3}, \frac {440}{27}, x + \frac {1}{6}\right )\right ) - 276 \, \sqrt {x^{2} + 3 \, x + 2} \sqrt {-2 \, x + 5}}{378 \, {\left (2 \, x - 5\right )}} \] Input:

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

Output:

1/378*(199*sqrt(-2)*(2*x - 5)*weierstrassPInverse(67/3, 440/27, x + 1/6) - 
 138*sqrt(-2)*(2*x - 5)*weierstrassZeta(67/3, 440/27, weierstrassPInverse( 
67/3, 440/27, x + 1/6)) - 276*sqrt(x^2 + 3*x + 2)*sqrt(-2*x + 5))/(2*x - 5 
)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {4+3 x}{(5-2 x)^{3/2} \sqrt {2+3 x+x^2}} \, dx=3 \left (\int \frac {\sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}\, x}{4 x^{4}-8 x^{3}-27 x^{2}+35 x +50}d x \right )+4 \left (\int \frac {\sqrt {-2 x +5}\, \sqrt {x^{2}+3 x +2}}{4 x^{4}-8 x^{3}-27 x^{2}+35 x +50}d x \right ) \] Input:

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

Output:

3*int((sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2)*x)/(4*x**4 - 8*x**3 - 27*x**2 
 + 35*x + 50),x) + 4*int((sqrt( - 2*x + 5)*sqrt(x**2 + 3*x + 2))/(4*x**4 - 
 8*x**3 - 27*x**2 + 35*x + 50),x)