\(\int \frac {(a x^2+b x^3+c x^4)^{3/2}}{x^6} \, dx\) [45]

Optimal result

Integrand size = 24, antiderivative size = 219 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^6} \, dx=-\frac {3 (b-2 c x) \sqrt {a x^2+b x^3+c x^4}}{4 x^2}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{2 x^5}-\frac {3 \left (b^2+4 a c\right ) x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}+\frac {3 b \sqrt {c} x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {a x^2+b x^3+c x^4}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1934, 1955, 1947, 857, 635, 212, 738} \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^6} \, dx=-\frac {3 x \left (4 a c+b^2\right ) \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}+\frac {3 b \sqrt {c} x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {a x^2+b x^3+c x^4}}-\frac {3 (b-2 c x) \sqrt {a x^2+b x^3+c x^4}}{4 x^2}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{2 x^5} \]

[In]

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

[Out]

(-3*(b - 2*c*x)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(4*x^2) - (a*x^2 + b*x^3 + c*x^4)^(3/2)/(2*x^5) - (3*(b^2 + 4*a*c
)*x*Sqrt[a + b*x + c*x^2]*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[a]*Sqrt[a*x^2 + b*x^
3 + c*x^4]) + (3*b*Sqrt[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*
Sqrt[a*x^2 + b*x^3 + c*x^4])

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

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

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-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] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 1934

Int[(x_)^(m_.)*((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a*
x^q + b*x^n + c*x^(2*n - q))^p/(m + p*q + 1)), x] - Dist[(n - q)*(p/(m + p*q + 1)), Int[x^(m + n)*(b + 2*c*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] && GtQ[p, 0] && RationalQ[m, q] && LeQ[m + p*q + 1, -
(n - q) + 1] && NeQ[m + p*q + 1, 0]

Rule 1947

Int[((A_) + (B_.)*(x_)^(j_.))/Sqrt[(b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.)], x_Symbol] :> Dist[
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[(A + B*x^(n - q))/(
x^(q/2)*Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))]), x], x] /; FreeQ[{a, b, c, A, B, n, q}, x] && EqQ[j, n - q] &
& EqQ[r, 2*n - q] && PosQ[n - q] && EqQ[n, 3] && EqQ[q, 2]

Rule 1955

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{2 x^5}+\frac {3}{4} \int \frac {(b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{x^3} \, dx \\ & = -\frac {3 (b-2 c x) \sqrt {a x^2+b x^3+c x^4}}{4 x^2}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{2 x^5}-\frac {3}{8} \int \frac {-b^2-4 a c-4 b c x}{\sqrt {a x^2+b x^3+c x^4}} \, dx \\ & = -\frac {3 (b-2 c x) \sqrt {a x^2+b x^3+c x^4}}{4 x^2}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{2 x^5}-\frac {\left (3 x \sqrt {a+b x+c x^2}\right ) \int \frac {-b^2-4 a c-4 b c x}{x \sqrt {a+b x+c x^2}} \, dx}{8 \sqrt {a x^2+b x^3+c x^4}} \\ & = -\frac {3 (b-2 c x) \sqrt {a x^2+b x^3+c x^4}}{4 x^2}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{2 x^5}+\frac {\left (3 b c x \sqrt {a+b x+c x^2}\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2 \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (3 \left (-b^2-4 a c\right ) x \sqrt {a+b x+c x^2}\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{8 \sqrt {a x^2+b x^3+c x^4}} \\ & = -\frac {3 (b-2 c x) \sqrt {a x^2+b x^3+c x^4}}{4 x^2}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{2 x^5}+\frac {\left (3 b c x \sqrt {a+b x+c x^2}\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{\sqrt {a x^2+b x^3+c x^4}}+\frac {\left (3 \left (-b^2-4 a c\right ) x \sqrt {a+b x+c x^2}\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )}{4 \sqrt {a x^2+b x^3+c x^4}} \\ & = -\frac {3 (b-2 c x) \sqrt {a x^2+b x^3+c x^4}}{4 x^2}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{2 x^5}-\frac {3 \left (b^2+4 a c\right ) x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}+\frac {3 b \sqrt {c} x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {a x^2+b x^3+c x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^6} \, dx=\frac {\sqrt {x^2 (a+x (b+c x))} \left (3 \left (b^2+4 a c\right ) x^2 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )-\sqrt {a} \left ((2 a+x (5 b-4 c x)) \sqrt {a+x (b+c x)}+6 b \sqrt {c} x^2 \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )\right )}{4 \sqrt {a} x^3 \sqrt {a+x (b+c x)}} \]

[In]

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

[Out]

(Sqrt[x^2*(a + x*(b + c*x))]*(3*(b^2 + 4*a*c)*x^2*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x)])/Sqrt[a]] - Sqrt[
a]*((2*a + x*(5*b - 4*c*x))*Sqrt[a + x*(b + c*x)] + 6*b*Sqrt[c]*x^2*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b +
c*x)]])))/(4*Sqrt[a]*x^3*Sqrt[a + x*(b + c*x)])

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.64

method	result	size
pseudoelliptic	\(-\frac {3 \left (x^{2} \left (a c +\frac {b^{2}}{4}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x \sqrt {a}}\right )-\ln \left (2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b \right ) b \,x^{2} \sqrt {c}\, \sqrt {a}+\frac {\left (a^{\frac {3}{2}}+\left (-2 c \,x^{2}+\frac {5}{2} b x \right ) \sqrt {a}\right ) \sqrt {c \,x^{2}+b x +a}}{3}-\ln \left (2\right ) \left (a c +\frac {b^{2}}{4}\right ) x^{2}\right )}{2 \sqrt {a}\, x^{2}}\)	\(140\)
risch	\(-\frac {\left (5 b x +2 a \right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{4 x^{3}}+\frac {\left (c \sqrt {c \,x^{2}+b x +a}+\frac {3 b \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2}-\frac {3 \sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) c}{2}-\frac {3 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b^{2}}{8 \sqrt {a}}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{x \sqrt {c \,x^{2}+b x +a}}\)	\(181\)
default	\(-\frac {\left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (12 a^{\frac {5}{2}} \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) c^{\frac {5}{2}} x^{2}-2 c^{\frac {5}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b \,x^{3}+3 a^{\frac {3}{2}} \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) c^{\frac {3}{2}} b^{2} x^{2}-4 c^{\frac {5}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,x^{2}-6 c^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}\, a b \,x^{3}-12 c^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}\, a^{2} x^{2}+2 c^{\frac {3}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b x -2 c^{\frac {3}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{2} x^{2}+4 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a \,c^{\frac {3}{2}}-6 c^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} x^{2}-12 c^{2} \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a^{2} b \,x^{2}\right )}{8 x^{5} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2} c^{\frac {3}{2}}}\)	\(338\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 757, normalized size of antiderivative = 3.46 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^6} \, dx=\left [\frac {12 \, a b \sqrt {c} x^{3} \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 ) + 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {a} x^{3} \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 (4 \, a c x^{2} - 5 \, a b x - 2 \, a^{2}\right )}}{16 \, a x^{3}}, -\frac {24 \, a b \sqrt {-c} x^{3} \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 ) - 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {a} x^{3} \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 (4 \, a c x^{2} - 5 \, a b x - 2 \, a^{2}\right )}}{16 \, a x^{3}}, \frac {6 \, a b \sqrt {c} x^{3} \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 ) + 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {-a} x^{3} \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 (4 \, a c x^{2} - 5 \, a b x - 2 \, a^{2}\right )}}{8 \, a x^{3}}, -\frac {12 \, a b \sqrt {-c} x^{3} \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 ) - 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {-a} x^{3} \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 (4 \, a c x^{2} - 5 \, a b x - 2 \, a^{2}\right )}}{8 \, a x^{3}}\right ] \]

[In]

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

[Out]

[1/16*(12*a*b*sqrt(c)*x^3*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) + 3*(b^2 + 4*a*c)*sqrt(a)*x^3*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)*(4*a*c*x^2 - 5*a*b*x - 2*a^2))/(a*x^
3), -1/16*(24*a*b*sqrt(-c)*x^3*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)) - 3*(b^2 + 4*a*c)*sqrt(a)*x^3*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)*(4*a*c*x^2 - 5*a*b*x - 2*a^2))/(a*x^3), 1/8*
(6*a*b*sqrt(c)*x^3*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) + 3*(b^2 + 4*a*c)*sqrt(-a)*x^3*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)*(4*a*c*x^2 - 5*a*b*x - 2*a^2))/(a*x^3), -1/8*(12*a*b*sqrt
(-c)*x^3*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)) - 3*(b^2 + 4
*a*c)*sqrt(-a)*x^3*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)*(4*a*c*x^2 - 5*a*b*x - 2*a^2))/(a*x^3)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^6} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Not invertible Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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