\(\int \frac {x^2}{(a x^2+b x^3+c x^4)^{3/2}} \, dx\) [77]

Optimal result

Integrand size = 24, antiderivative size = 94 \[ \int \frac {x^2}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\frac {2 x \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {\text {arctanh}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{a^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.15 \[ \int \frac {x^2}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=-\frac {2 x \left (\sqrt {a} \left (b^2-2 a c+b c x\right )+\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)} \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )\right )}{a^{3/2} \left (-b^2+4 a c\right ) \sqrt {x^2 (a+x (b+c x))}} \] Input:

Integrate[x^2/(a*x^2 + b*x^3 + c*x^4)^(3/2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1969, 1951, 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.

\(\displaystyle \int \frac {x^2}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1969

\(\displaystyle \frac {\int \frac {1}{\sqrt {c x^4+b x^3+a x^2}}dx}{a}+\frac {2 x \left (-2 a c+b^2+b c x\right )}{a \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}\)

\(\Big \downarrow \) 1951

\(\displaystyle \frac {2 x \left (-2 a c+b^2+b c x\right )}{a \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 \int \frac {1}{4 a-\frac {x^2 (2 a+b x)^2}{c x^4+b x^3+a x^2}}d\frac {x (2 a+b x)}{\sqrt {c x^4+b x^3+a x^2}}}{a}\)

\(\Big \downarrow \) 219

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

Input:

Int[x^2/(a*x^2 + b*x^3 + c*x^4)^(3/2),x]
 

Output:

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

Defintions of rubi rules used

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 1951

Int[1/Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(r_.)], x_Symbol] : 
> Simp[-2/(n - 2)   Subst[Int[1/(4*a - x^2), x], x, x*((2*a + b*x^(n - 2))/ 
Sqrt[a*x^2 + b*x^n + c*x^r])], x] /; FreeQ[{a, b, c, n, r}, x] && EqQ[r, 2* 
n - 2] && PosQ[n - 2] && NeQ[b^2 - 4*a*c, 0]
 

rule 1969

Int[(x_)^(m_.)*((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(p_ 
), x_Symbol] :> Simp[(-x^(m - q + 1))*(b^2 - 2*a*c + b*c*x^(n - q))*((a*x^q 
 + b*x^n + c*x^(2*n - q))^(p + 1)/(a*(n - q)*(p + 1)*(b^2 - 4*a*c))), x] + 
Simp[(2*a*c - b^2*(p + 2))/(a*(p + 1)*(b^2 - 4*a*c))   Int[x^(m - q)*(a*x^q 
 + b*x^n + c*x^(2*n - q))^(p + 1), x], x] /; FreeQ[{a, b, c}, x] && EqQ[r, 
2*n - q] && PosQ[n - q] &&  !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 
0] && LtQ[p, -1] && RationalQ[m, p, q] && EqQ[m + p*q + 1, (-(n - q))*(2*p 
+ 3)]
 

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.14

method	result	size
pseudoelliptic	\(-\frac {4 \left (-a^{\frac {3}{2}} c +\frac {b \sqrt {a}\, \left (c x +b \right )}{2}+\sqrt {c \,x^{2}+b x +a}\, \left (-\ln \left (2\right )+\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x \sqrt {a}}\right )\right ) \left (a c -\frac {b^{2}}{4}\right )\right )}{a^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}\, \left (4 a c -b^{2}\right )}\)	\(107\)
default	\(\frac {x^{3} \left (c \,x^{2}+b x +a \right ) \left (-2 a^{\frac {3}{2}} b c x +4 a^{\frac {5}{2}} c -2 a^{\frac {3}{2}} b^{2}-4 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{2} c +\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {c \,x^{2}+b x +a}\, a \,b^{2}\right )}{\left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} a^{\frac {5}{2}} \left (4 a c -b^{2}\right )}\)	\(164\)

Input:

int(x^2/(c*x^4+b*x^3+a*x^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (84) = 168\).

Time = 0.12 (sec) , antiderivative size = 411, normalized size of antiderivative = 4.37 \[ \int \frac {x^2}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\left [\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{3} + {\left (b^{3} - 4 \, a b c\right )} x^{2} + {\left (a b^{2} - 4 \, a^{2} c\right )} x\right )} \sqrt {a} \log \left (-\frac {8 \, a b x^{2} + {\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {a}}{x^{3}}\right ) + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (a b c x + a b^{2} - 2 \, a^{2} c\right )}}{2 \, {\left ({\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{3} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2} + {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x\right )}}, \frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{3} + {\left (b^{3} - 4 \, a b c\right )} x^{2} + {\left (a b^{2} - 4 \, a^{2} c\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) + 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (a b c x + a b^{2} - 2 \, a^{2} c\right )}}{{\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{3} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2} + {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x}\right ] \] Input:

integrate(x^2/(c*x^4+b*x^3+a*x^2)^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {x^2}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\int \frac {x^{2}}{\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(x**2/(c*x**4+b*x**3+a*x**2)**(3/2),x)
 

Output:

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

Maxima [F]

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

integrate(x^2/(c*x^4+b*x^3+a*x^2)^(3/2),x, algorithm="maxima")
 

Output:

integrate(x^2/(c*x^4 + b*x^3 + a*x^2)^(3/2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (84) = 168\).

Time = 0.13 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.12 \[ \int \frac {x^2}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=-\frac {2 \, {\left (a b^{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) - 4 \, a^{2} c \arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) + \sqrt {-a} \sqrt {a} b^{2} - 2 \, \sqrt {-a} a^{\frac {3}{2}} c\right )} \mathrm {sgn}\left (x\right )}{\sqrt {-a} a^{2} b^{2} - 4 \, \sqrt {-a} a^{3} c} + \frac {2 \, {\left (\frac {a b c x \mathrm {sgn}\left (x\right )}{a^{2} b^{2} - 4 \, a^{3} c} + \frac {a b^{2} \mathrm {sgn}\left (x\right ) - 2 \, a^{2} c \mathrm {sgn}\left (x\right )}{a^{2} b^{2} - 4 \, a^{3} c}\right )}}{\sqrt {c x^{2} + b x + a}} + \frac {2 \, \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a \mathrm {sgn}\left (x\right )} \] Input:

integrate(x^2/(c*x^4+b*x^3+a*x^2)^(3/2),x, algorithm="giac")
 

Output:

-2*(a*b^2*arctan(sqrt(a)/sqrt(-a)) - 4*a^2*c*arctan(sqrt(a)/sqrt(-a)) + sq 
rt(-a)*sqrt(a)*b^2 - 2*sqrt(-a)*a^(3/2)*c)*sgn(x)/(sqrt(-a)*a^2*b^2 - 4*sq 
rt(-a)*a^3*c) + 2*(a*b*c*x*sgn(x)/(a^2*b^2 - 4*a^3*c) + (a*b^2*sgn(x) - 2* 
a^2*c*sgn(x))/(a^2*b^2 - 4*a^3*c))/sqrt(c*x^2 + b*x + a) + 2*arctan(-(sqrt 
(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a*sgn(x))
 

Mupad [F(-1)]

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

int(x^2/(a*x^2 + b*x^3 + c*x^4)^(3/2),x)
 

Output:

int(x^2/(a*x^2 + b*x^3 + c*x^4)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.89 (sec) , antiderivative size = 2247, normalized size of antiderivative = 23.90 \[ \int \frac {x^2}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int(x^2/(c*x^4+b*x^3+a*x^2)^(3/2),x)
 

Output:

( - 2*sqrt(a)*sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b**2)*atan((2*sqrt(c)*sqr 
t(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b**2)) 
*a*b - 2*sqrt(a)*sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b**2)*atan((2*sqrt(c)* 
sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b** 
2))*b**2*x - 2*sqrt(a)*sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b**2)*atan((2*sq 
rt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c 
 - b**2))*b*c*x**2 - 4*sqrt(c)*sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b**2)*at 
an((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*sqrt(c)*sqrt(a)*b 
 - 4*a*c - b**2))*a**2 - 4*sqrt(c)*sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b**2 
)*atan((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*sqrt(c)*sqrt( 
a)*b - 4*a*c - b**2))*a*b*x - 4*sqrt(c)*sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - 
 b**2)*atan((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*sqrt(c)* 
sqrt(a)*b - 4*a*c - b**2))*a*c*x**2 - sqrt(a)*sqrt(4*sqrt(c)*sqrt(a)*b + 4 
*a*c + b**2)*log( - sqrt(4*sqrt(c)*sqrt(a)*b + 4*a*c + b**2) + 2*sqrt(c)*s 
qrt(a + b*x + c*x**2) + b + 2*c*x)*a*b - sqrt(a)*sqrt(4*sqrt(c)*sqrt(a)*b 
+ 4*a*c + b**2)*log( - sqrt(4*sqrt(c)*sqrt(a)*b + 4*a*c + b**2) + 2*sqrt(c 
)*sqrt(a + b*x + c*x**2) + b + 2*c*x)*b**2*x - sqrt(a)*sqrt(4*sqrt(c)*sqrt 
(a)*b + 4*a*c + b**2)*log( - sqrt(4*sqrt(c)*sqrt(a)*b + 4*a*c + b**2) + 2* 
sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)*b*c*x**2 + sqrt(a)*sqrt(4*sqrt 
(c)*sqrt(a)*b + 4*a*c + b**2)*log(sqrt(4*sqrt(c)*sqrt(a)*b + 4*a*c + b*...