\(\int \frac {x^6}{(a x^2+b x^3+c x^4)^{3/2}} \, dx\) [73]

Optimal result

Integrand size = 24, antiderivative size = 173 \[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\frac {2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 b \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{c^2 \left (b^2-4 a c\right ) x}-\frac {3 b \text {arctanh}\left (\frac {x (b+2 c x)}{2 \sqrt {c} \sqrt {a x^2+b x^3+c x^4}}\right )}{2 c^{5/2}} \] Output:

2*x^3*(b*x+2*a)/(-4*a*c+b^2)/(c*x^4+b*x^3+a*x^2)^(1/2)-2*b*(c*x^4+b*x^3+a* 
x^2)^(1/2)/c/(-4*a*c+b^2)+(-8*a*c+3*b^2)*(c*x^4+b*x^3+a*x^2)^(1/2)/c^2/(-4 
*a*c+b^2)/x-3/2*b*arctanh(1/2*x*(2*c*x+b)/c^(1/2)/(c*x^4+b*x^3+a*x^2)^(1/2 
))/c^(5/2)
 

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.83 \[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\frac {x \left (2 \sqrt {c} \left (8 a^2 c-b^2 x (3 b+c x)+a \left (-3 b^2+10 b c x+4 c^2 x^2\right )\right )-3 b \left (b^2-4 a c\right ) \sqrt {a+x (b+c x)} \log \left (c^2 \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )\right )}{2 c^{5/2} \left (-b^2+4 a c\right ) \sqrt {x^2 (a+x (b+c x))}} \] Input:

Integrate[x^6/(a*x^2 + b*x^3 + c*x^4)^(3/2),x]
 

Output:

(x*(2*Sqrt[c]*(8*a^2*c - b^2*x*(3*b + c*x) + a*(-3*b^2 + 10*b*c*x + 4*c^2* 
x^2)) - 3*b*(b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)]*Log[c^2*(b + 2*c*x - 2*Sqr 
t[c]*Sqrt[a + x*(b + c*x)])]))/(2*c^(5/2)*(-b^2 + 4*a*c)*Sqrt[x^2*(a + x*( 
b + c*x))])
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.23, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1970, 27, 1996, 27, 1996, 27, 1961, 1092, 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.

\(\displaystyle \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1970

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1996

\(\displaystyle \frac {2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {4 \left (\frac {b \sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {\int \frac {x \left (2 a b+\left (3 b^2-8 a c\right ) x\right )}{2 \sqrt {c x^4+b x^3+a x^2}}dx}{2 c}\right )}{b^2-4 a c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {4 \left (\frac {b \sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {\int \frac {x \left (2 a b+\left (3 b^2-8 a c\right ) x\right )}{\sqrt {c x^4+b x^3+a x^2}}dx}{4 c}\right )}{b^2-4 a c}\)

\(\Big \downarrow \) 1996

\(\displaystyle \frac {2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {4 \left (\frac {b \sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {\frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{c x}-\frac {\int \frac {3 b \left (b^2-4 a c\right ) x}{2 \sqrt {c x^4+b x^3+a x^2}}dx}{c}}{4 c}\right )}{b^2-4 a c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {4 \left (\frac {b \sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {\frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{c x}-\frac {3 b \left (b^2-4 a c\right ) \int \frac {x}{\sqrt {c x^4+b x^3+a x^2}}dx}{2 c}}{4 c}\right )}{b^2-4 a c}\)

\(\Big \downarrow \) 1961

\(\displaystyle \frac {2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {4 \left (\frac {b \sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {\frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{c x}-\frac {3 b x \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{2 c \sqrt {a x^2+b x^3+c x^4}}}{4 c}\right )}{b^2-4 a c}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {4 \left (\frac {b \sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {\frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{c x}-\frac {3 b x \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{c \sqrt {a x^2+b x^3+c x^4}}}{4 c}\right )}{b^2-4 a c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2 x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {4 \left (\frac {b \sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {\frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{c x}-\frac {3 b x \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{3/2} \sqrt {a x^2+b x^3+c x^4}}}{4 c}\right )}{b^2-4 a c}\)

Input:

Int[x^6/(a*x^2 + b*x^3 + c*x^4)^(3/2),x]
 

Output:

(2*x^3*(2*a + b*x))/((b^2 - 4*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4]) - (4*((b*S 
qrt[a*x^2 + b*x^3 + c*x^4])/(2*c) - (((3*b^2 - 8*a*c)*Sqrt[a*x^2 + b*x^3 + 
 c*x^4])/(c*x) - (3*b*(b^2 - 4*a*c)*x*Sqrt[a + b*x + c*x^2]*ArcTanh[(b + 2 
*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*c^(3/2)*Sqrt[a*x^2 + b*x^3 + 
c*x^4]))/(4*c)))/(b^2 - 4*a*c)
 

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 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1961

Int[(x_)^(m_.)/Sqrt[(b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.)] 
, x_Symbol] :> Simp[x^(q/2)*(Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))]/Sqrt[a 
*x^q + b*x^n + c*x^(2*n - q)])   Int[x^(m - q/2)/Sqrt[a + b*x^(n - q) + c*x 
^(2*(n - q))], x], x] /; FreeQ[{a, b, c, m, n, q}, x] && EqQ[r, 2*n - q] && 
 PosQ[n - q] && ((EqQ[m, 1] && EqQ[n, 3] && EqQ[q, 2]) || ((EqQ[m + 1/2] || 
 EqQ[m, 3/2] || EqQ[m, 1/2] || EqQ[m, 5/2]) && EqQ[n, 3] && EqQ[q, 1]))
 

rule 1970

Int[(x_)^(m_.)*((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(p_ 
), x_Symbol] :> Simp[(-x^(m - 2*n + q + 1))*(2*a + b*x^(n - q))*((a*x^q + b 
*x^n + c*x^(2*n - q))^(p + 1)/((n - q)*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1 
/((n - q)*(p + 1)*(b^2 - 4*a*c))   Int[x^(m - 2*n + q)*(2*a*(m + p*q - 2*(n 
 - q) + 1) + b*(m + p*q + (n - q)*(2*p + 1) + 1)*x^(n - 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, q] && GtQ[m + p*q + 1, 2*(n - q)]
 

rule 1996

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

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.71

method	result	size
pseudoelliptic	\(\frac {-\frac {3 \sqrt {c \,x^{2}+b x +a}\, b c \left (a c -\frac {b^{2}}{4}\right ) \ln \left (2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b \right )}{2}+c^{\frac {3}{2}} \left (a \,c^{2} x^{2}+\left (-\frac {1}{4} b^{2} x^{2}+\frac {5}{2} a b x +2 a^{2}\right ) c -\frac {3 b^{2} \left (b x +a \right )}{4}\right )}{\sqrt {c \,x^{2}+b x +a}\, c^{\frac {7}{2}} \left (a c -\frac {b^{2}}{4}\right )}\)	\(122\)
default	\(\frac {x^{3} \left (c \,x^{2}+b x +a \right ) \left (8 c^{\frac {7}{2}} a \,x^{2}-2 c^{\frac {5}{2}} b^{2} x^{2}+20 c^{\frac {5}{2}} a b x -6 c^{\frac {3}{2}} b^{3} x +16 c^{\frac {5}{2}} a^{2}-6 c^{\frac {3}{2}} a \,b^{2}-12 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) \sqrt {c \,x^{2}+b x +a}\, a b \,c^{2}+3 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) \sqrt {c \,x^{2}+b x +a}\, b^{3} c \right )}{2 c^{\frac {7}{2}} \left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (4 a c -b^{2}\right )}\)	\(199\)
risch	\(\frac {\left (c \,x^{2}+b x +a \right ) x}{c^{2} \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}+\frac {\left (\frac {a}{c^{2} \sqrt {c \,x^{2}+b x +a}}-\frac {b^{2}}{4 c^{3} \sqrt {c \,x^{2}+b x +a}}-\frac {b^{3} x}{2 c^{2} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {b^{4}}{4 c^{3} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {3 b x}{2 c^{2} \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {5}{2}}}\right ) x \sqrt {c \,x^{2}+b x +a}}{\sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}\)	\(216\)

Input:

int(x^6/(c*x^4+b*x^3+a*x^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/(c*x^2+b*x+a)^(1/2)*(-3/2*(c*x^2+b*x+a)^(1/2)*b*c*(a*c-1/4*b^2)*ln(2*(c* 
x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)+c^(3/2)*(a*c^2*x^2+(-1/4*b^2*x^2+5/2*a*b 
*x+2*a^2)*c-3/4*b^2*(b*x+a)))/c^(7/2)/(a*c-1/4*b^2)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 486, normalized size of antiderivative = 2.81 \[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\left [\frac {3 \, {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3} + {\left (b^{4} - 4 \, a b^{2} c\right )} x^{2} + {\left (a b^{3} - 4 \, a^{2} b c\right )} x\right )} \sqrt {c} \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (3 \, a b^{2} c - 8 \, a^{2} c^{2} + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + {\left (3 \, b^{3} c - 10 \, a b c^{2}\right )} x\right )}}{4 \, {\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{3} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{2} + {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4}\right )} x\right )}}, \frac {3 \, {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3} + {\left (b^{4} - 4 \, a b^{2} c\right )} x^{2} + {\left (a b^{3} - 4 \, a^{2} b c\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) + 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (3 \, a b^{2} c - 8 \, a^{2} c^{2} + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + {\left (3 \, b^{3} c - 10 \, a b c^{2}\right )} x\right )}}{2 \, {\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{3} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{2} + {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4}\right )} x\right )}}\right ] \] Input:

integrate(x^6/(c*x^4+b*x^3+a*x^2)^(3/2),x, algorithm="fricas")
 

Output:

[1/4*(3*((b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 4*a*b^2*c)*x^2 + (a*b^3 - 4*a^2* 
b*c)*x)*sqrt(c)*log(-(8*c^2*x^3 + 8*b*c*x^2 - 4*sqrt(c*x^4 + b*x^3 + a*x^2 
)*(2*c*x + b)*sqrt(c) + (b^2 + 4*a*c)*x)/x) + 4*sqrt(c*x^4 + b*x^3 + a*x^2 
)*(3*a*b^2*c - 8*a^2*c^2 + (b^2*c^2 - 4*a*c^3)*x^2 + (3*b^3*c - 10*a*b*c^2 
)*x))/((b^2*c^4 - 4*a*c^5)*x^3 + (b^3*c^3 - 4*a*b*c^4)*x^2 + (a*b^2*c^3 - 
4*a^2*c^4)*x), 1/2*(3*((b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 4*a*b^2*c)*x^2 + ( 
a*b^3 - 4*a^2*b*c)*x)*sqrt(-c)*arctan(1/2*sqrt(c*x^4 + b*x^3 + a*x^2)*(2*c 
*x + b)*sqrt(-c)/(c^2*x^3 + b*c*x^2 + a*c*x)) + 2*sqrt(c*x^4 + b*x^3 + a*x 
^2)*(3*a*b^2*c - 8*a^2*c^2 + (b^2*c^2 - 4*a*c^3)*x^2 + (3*b^3*c - 10*a*b*c 
^2)*x))/((b^2*c^4 - 4*a*c^5)*x^3 + (b^3*c^3 - 4*a*b*c^4)*x^2 + (a*b^2*c^3 
- 4*a^2*c^4)*x)]
 

Sympy [F]

\[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\int \frac {x^{6}}{\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(x**6/(c*x**4+b*x**3+a*x**2)**(3/2),x)
 

Output:

Integral(x**6/(x**2*(a + b*x + c*x**2))**(3/2), x)
 

Maxima [F]

\[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\int { \frac {x^{6}}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^6/(c*x^4+b*x^3+a*x^2)^(3/2),x, algorithm="maxima")
 

Output:

integrate(x^6/(c*x^4 + b*x^3 + a*x^2)^(3/2), x)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.34 \[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=-\frac {{\left (3 \, b^{3} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 12 \, a b c \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 6 \, \sqrt {a} b^{2} \sqrt {c} - 16 \, a^{\frac {3}{2}} c^{\frac {3}{2}}\right )} \mathrm {sgn}\left (x\right )}{2 \, {\left (b^{2} c^{\frac {5}{2}} - 4 \, a c^{\frac {7}{2}}\right )}} + \frac {{\left (\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} x}{b^{2} c^{2} \mathrm {sgn}\left (x\right ) - 4 \, a c^{3} \mathrm {sgn}\left (x\right )} + \frac {3 \, b^{3} - 10 \, a b c}{b^{2} c^{2} \mathrm {sgn}\left (x\right ) - 4 \, a c^{3} \mathrm {sgn}\left (x\right )}\right )} x + \frac {3 \, a b^{2} - 8 \, a^{2} c}{b^{2} c^{2} \mathrm {sgn}\left (x\right ) - 4 \, a c^{3} \mathrm {sgn}\left (x\right )}}{\sqrt {c x^{2} + b x + a}} + \frac {3 \, b \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{2 \, c^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} \] Input:

integrate(x^6/(c*x^4+b*x^3+a*x^2)^(3/2),x, algorithm="giac")
 

Output:

-1/2*(3*b^3*log(abs(b - 2*sqrt(a)*sqrt(c))) - 12*a*b*c*log(abs(b - 2*sqrt( 
a)*sqrt(c))) + 6*sqrt(a)*b^2*sqrt(c) - 16*a^(3/2)*c^(3/2))*sgn(x)/(b^2*c^( 
5/2) - 4*a*c^(7/2)) + (((b^2*c - 4*a*c^2)*x/(b^2*c^2*sgn(x) - 4*a*c^3*sgn( 
x)) + (3*b^3 - 10*a*b*c)/(b^2*c^2*sgn(x) - 4*a*c^3*sgn(x)))*x + (3*a*b^2 - 
 8*a^2*c)/(b^2*c^2*sgn(x) - 4*a*c^3*sgn(x)))/sqrt(c*x^2 + b*x + a) + 3/2*b 
*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/(c^(5/2)*sgn( 
x))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\int \frac {x^6}{{\left (c\,x^4+b\,x^3+a\,x^2\right )}^{3/2}} \,d x \] Input:

int(x^6/(a*x^2 + b*x^3 + c*x^4)^(3/2),x)
 

Output:

int(x^6/(a*x^2 + b*x^3 + c*x^4)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 499, normalized size of antiderivative = 2.88 \[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\frac {16 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{2}-6 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c +20 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{2} x +8 \sqrt {c \,x^{2}+b x +a}\, a \,c^{3} x^{2}-6 \sqrt {c \,x^{2}+b x +a}\, b^{3} c x -2 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{2} x^{2}-12 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b c +3 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{3}-12 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{2} c x -12 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a b \,c^{2} x^{2}+3 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{4} x +3 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{3} c \,x^{2}+12 \sqrt {c}\, a^{2} b c -4 \sqrt {c}\, a \,b^{3}+12 \sqrt {c}\, a \,b^{2} c x +12 \sqrt {c}\, a b \,c^{2} x^{2}-4 \sqrt {c}\, b^{4} x -4 \sqrt {c}\, b^{3} c \,x^{2}}{2 c^{3} \left (4 a \,c^{2} x^{2}-b^{2} c \,x^{2}+4 a b c x -b^{3} x +4 a^{2} c -a \,b^{2}\right )} \] Input:

int(x^6/(c*x^4+b*x^3+a*x^2)^(3/2),x)
 

Output:

(16*sqrt(a + b*x + c*x**2)*a**2*c**2 - 6*sqrt(a + b*x + c*x**2)*a*b**2*c + 
 20*sqrt(a + b*x + c*x**2)*a*b*c**2*x + 8*sqrt(a + b*x + c*x**2)*a*c**3*x* 
*2 - 6*sqrt(a + b*x + c*x**2)*b**3*c*x - 2*sqrt(a + b*x + c*x**2)*b**2*c** 
2*x**2 - 12*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqr 
t(4*a*c - b**2))*a**2*b*c + 3*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2 
) + b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**3 - 12*sqrt(c)*log((2*sqrt(c)*sqrt 
(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**2*c*x - 12*sqrt(c 
)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a 
*b*c**2*x**2 + 3*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x 
)/sqrt(4*a*c - b**2))*b**4*x + 3*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x 
**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**3*c*x**2 + 12*sqrt(c)*a**2*b*c - 
4*sqrt(c)*a*b**3 + 12*sqrt(c)*a*b**2*c*x + 12*sqrt(c)*a*b*c**2*x**2 - 4*sq 
rt(c)*b**4*x - 4*sqrt(c)*b**3*c*x**2)/(2*c**3*(4*a**2*c - a*b**2 + 4*a*b*c 
*x + 4*a*c**2*x**2 - b**3*x - b**2*c*x**2))