\(\int \frac {x^2}{(d+e x) (a+b x+c x^2)^{3/2}} \, dx\) [40]

Optimal result

Integrand size = 25, antiderivative size = 156 \[ \int \frac {x^2}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \left (a (b d-2 a e)+\left (b^2 d-2 a c d-a b e\right ) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}+\frac {d^2 \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^{3/2}} \] Output:

(-2*a*(-2*a*e+b*d)-2*(-a*b*e-2*a*c*d+b^2*d)*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c 
*d^2)/(c*x^2+b*x+a)^(1/2)+d^2*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^ 
2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a*e^2-b*d*e+c*d^2)^(3/2)
 

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.22 \[ \int \frac {x^2}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (c d^2+e (-b d+a e)\right ) \left (2 a^2 e-b^2 d x+a (-b d+2 c d x+b e x)\right )-2 \left (-b^2+4 a c\right ) d^2 \sqrt {-c d^2+b d e-a e^2} \sqrt {a+x (b+c x)} \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )}{\left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )^2 \sqrt {a+x (b+c x)}} \] Input:

Integrate[x^2/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x]
 

Output:

(2*(c*d^2 + e*(-(b*d) + a*e))*(2*a^2*e - b^2*d*x + a*(-(b*d) + 2*c*d*x + b 
*e*x)) - 2*(-b^2 + 4*a*c)*d^2*Sqrt[-(c*d^2) + b*d*e - a*e^2]*Sqrt[a + x*(b 
 + c*x)]*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + x*(b + c*x)])/Sqrt[-(c*d^2 
) + e*(b*d - a*e)]])/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))^2*Sqrt[a + 
x*(b + c*x)])
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1264, 27, 1154, 219}

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.

\(\displaystyle \int \frac {x^2}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1264

\(\displaystyle -\frac {2 \int -\frac {\left (b^2-4 a c\right ) d^2}{2 \left (c d^2-b e d+a e^2\right ) (d+e x) \sqrt {c x^2+b x+a}}dx}{b^2-4 a c}-\frac {2 \left (x \left (-a b e-2 a c d+b^2 d\right )+a (b d-2 a e)\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{a e^2-b d e+c d^2}-\frac {2 \left (x \left (-a b e-2 a c d+b^2 d\right )+a (b d-2 a e)\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {2 d^2 \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{a e^2-b d e+c d^2}-\frac {2 \left (x \left (-a b e-2 a c d+b^2 d\right )+a (b d-2 a e)\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {d^2 \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{3/2}}-\frac {2 \left (x \left (-a b e-2 a c d+b^2 d\right )+a (b d-2 a e)\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\)

Input:

Int[x^2/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x]
 

Output:

(-2*(a*(b*d - 2*a*e) + (b^2*d - 2*a*c*d - a*b*e)*x))/((b^2 - 4*a*c)*(c*d^2 
 - b*d*e + a*e^2)*Sqrt[a + b*x + c*x^2]) + (d^2*ArcTanh[(b*d - 2*a*e + (2* 
c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(c*d 
^2 - b*d*e + a*e^2)^(3/2)
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1154

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

rule 1264

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d + e*x) 
^m*(f + g*x)^n, a + b*x + c*x^2, x], R = Coeff[PolynomialRemainder[(d + e*x 
)^m*(f + g*x)^n, a + b*x + c*x^2, x], x, 0], S = Coeff[PolynomialRemainder[ 
(d + e*x)^m*(f + g*x)^n, a + b*x + c*x^2, x], x, 1]}, Simp[(b*R - 2*a*S + ( 
2*c*R - b*S)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + S 
imp[1/((p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*E 
xpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*R - b*S) 
)/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[n, 1] 
 && LtQ[p, -1] && ILtQ[m, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(497\) vs. \(2(147)=294\).

Time = 1.41 (sec) , antiderivative size = 498, normalized size of antiderivative = 3.19

method	result	size
default	\(\frac {-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}}{e}+\frac {d^{2} \left (\frac {e^{2}}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}-\frac {\left (b e -2 c d \right ) e \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (\frac {4 c \left (a \,e^{2}-b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{3}}-\frac {2 d \left (2 c x +b \right )}{e^{2} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\)	\(498\)

Input:

int(x^2/(e*x+d)/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/e*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2 
))+d^2/e^3*(1/(a*e^2-b*d*e+c*d^2)*e^2/(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+( 
a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(2*c*(x+d/ 
e)+(b*e-2*c*d)/e)/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/(c*(x+d/ 
e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-1/(a*e^2-b*d*e+c 
*d^2)*e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b 
*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+(b*e-2* 
c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e)))-2*d/e^2*(2*c*x+b) 
/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 654 vs. \(2 (146) = 292\).

Time = 0.34 (sec) , antiderivative size = 1351, normalized size of antiderivative = 8.66 \[ \int \frac {x^2}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(x^2/(e*x+d)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
 

Output:

[1/2*(((b^2*c - 4*a*c^2)*d^2*x^2 + (b^3 - 4*a*b*c)*d^2*x + (a*b^2 - 4*a^2* 
c)*d^2)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4* 
a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 4*sqrt(c*d^2 
- b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2 
*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2 
)) - 4*(a*b*c*d^3 + 3*a^2*b*d*e^2 - 2*a^3*e^3 - (a*b^2 + 2*a^2*c)*d^2*e - 
(a^2*b*e^3 - (b^2*c - 2*a*c^2)*d^3 + (b^3 - a*b*c)*d^2*e - 2*(a*b^2 - a^2* 
c)*d*e^2)*x)*sqrt(c*x^2 + b*x + a))/((a*b^2*c^2 - 4*a^2*c^3)*d^4 - 2*(a*b^ 
3*c - 4*a^2*b*c^2)*d^3*e + (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2*e^2 - 2*( 
a^2*b^3 - 4*a^3*b*c)*d*e^3 + (a^3*b^2 - 4*a^4*c)*e^4 + ((b^2*c^3 - 4*a*c^4 
)*d^4 - 2*(b^3*c^2 - 4*a*b*c^3)*d^3*e + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)* 
d^2*e^2 - 2*(a*b^3*c - 4*a^2*b*c^2)*d*e^3 + (a^2*b^2*c - 4*a^3*c^2)*e^4)*x 
^2 + ((b^3*c^2 - 4*a*b*c^3)*d^4 - 2*(b^4*c - 4*a*b^2*c^2)*d^3*e + (b^5 - 2 
*a*b^3*c - 8*a^2*b*c^2)*d^2*e^2 - 2*(a*b^4 - 4*a^2*b^2*c)*d*e^3 + (a^2*b^3 
 - 4*a^3*b*c)*e^4)*x), (((b^2*c - 4*a*c^2)*d^2*x^2 + (b^3 - 4*a*b*c)*d^2*x 
 + (a*b^2 - 4*a^2*c)*d^2)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(-c 
*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x 
)/(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c* 
d^2 - b^2*d*e + a*b*e^2)*x)) - 2*(a*b*c*d^3 + 3*a^2*b*d*e^2 - 2*a^3*e^3 - 
(a*b^2 + 2*a^2*c)*d^2*e - (a^2*b*e^3 - (b^2*c - 2*a*c^2)*d^3 + (b^3 - a...
 

Sympy [F]

\[ \int \frac {x^2}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {x^{2}}{\left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(x**2/(e*x+d)/(c*x**2+b*x+a)**(3/2),x)
 

Output:

Integral(x**2/((d + e*x)*(a + b*x + c*x**2)**(3/2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^2/(e*x+d)/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume((b/e-(2*c*d)/e^2)^2>0)', see `as 
sume?` for
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (146) = 292\).

Time = 0.26 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.96 \[ \int \frac {x^2}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \, d^{2} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e - a e^{2}}}\right )}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {-c d^{2} + b d e - a e^{2}}} - \frac {2 \, {\left (\frac {{\left (b^{2} c d^{3} - 2 \, a c^{2} d^{3} - b^{3} d^{2} e + a b c d^{2} e + 2 \, a b^{2} d e^{2} - 2 \, a^{2} c d e^{2} - a^{2} b e^{3}\right )} x}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}} + \frac {a b c d^{3} - a b^{2} d^{2} e - 2 \, a^{2} c d^{2} e + 3 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3}}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}}\right )}}{\sqrt {c x^{2} + b x + a}} \] Input:

integrate(x^2/(e*x+d)/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
 

Output:

2*d^2*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c* 
d^2 + b*d*e - a*e^2))/((c*d^2 - b*d*e + a*e^2)*sqrt(-c*d^2 + b*d*e - a*e^2 
)) - 2*((b^2*c*d^3 - 2*a*c^2*d^3 - b^3*d^2*e + a*b*c*d^2*e + 2*a*b^2*d*e^2 
 - 2*a^2*c*d*e^2 - a^2*b*e^3)*x/(b^2*c^2*d^4 - 4*a*c^3*d^4 - 2*b^3*c*d^3*e 
 + 8*a*b*c^2*d^3*e + b^4*d^2*e^2 - 2*a*b^2*c*d^2*e^2 - 8*a^2*c^2*d^2*e^2 - 
 2*a*b^3*d*e^3 + 8*a^2*b*c*d*e^3 + a^2*b^2*e^4 - 4*a^3*c*e^4) + (a*b*c*d^3 
 - a*b^2*d^2*e - 2*a^2*c*d^2*e + 3*a^2*b*d*e^2 - 2*a^3*e^3)/(b^2*c^2*d^4 - 
 4*a*c^3*d^4 - 2*b^3*c*d^3*e + 8*a*b*c^2*d^3*e + b^4*d^2*e^2 - 2*a*b^2*c*d 
^2*e^2 - 8*a^2*c^2*d^2*e^2 - 2*a*b^3*d*e^3 + 8*a^2*b*c*d*e^3 + a^2*b^2*e^4 
 - 4*a^3*c*e^4))/sqrt(c*x^2 + b*x + a)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {x^2}{\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \] Input:

int(x^2/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x)
 

Output:

int(x^2/((d + e*x)*(a + b*x + c*x^2)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {x^2}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {x^{2}}{\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}d x \] Input:

int(x^2/(e*x+d)/(c*x^2+b*x+a)^(3/2),x)
 

Output:

int(x^2/(e*x+d)/(c*x^2+b*x+a)^(3/2),x)