\(\int \frac {(a x^2+b x^3+c x^4)^{3/2}}{x^8} \, dx\) [47]

Optimal result

Integrand size = 24, antiderivative size = 197 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^8} \, dx=-\frac {\left (b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{32 a x^3}+\frac {b \left (3 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{64 a^2 x^2}-\frac {(b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{8 x^4}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}-\frac {3 \left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{128 a^{5/2}} \]

[Out]

-1/4*(c*x^4+b*x^3+a*x^2)^(3/2)/x^7-3/128*(-4*a*c+b^2)^2*arctanh(1/2*x*(b*x+2*a)/a^(1/2)/(c*x^4+b*x^3+a*x^2)^(1
/2))/a^(5/2)-1/32*(-12*a*c+b^2)*(c*x^4+b*x^3+a*x^2)^(1/2)/a/x^3+1/64*b*(-20*a*c+3*b^2)*(c*x^4+b*x^3+a*x^2)^(1/
2)/a^2/x^2-1/8*(6*c*x+b)*(c*x^4+b*x^3+a*x^2)^(1/2)/x^4

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1934, 1955, 1965, 12, 1918, 212} \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^8} \, dx=-\frac {3 \left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{128 a^{5/2}}+\frac {b \left (3 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{64 a^2 x^2}-\frac {\left (b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{32 a x^3}-\frac {(b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{8 x^4}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7} \]

[In]

Int[(a*x^2 + b*x^3 + c*x^4)^(3/2)/x^8,x]

[Out]

-1/32*((b^2 - 12*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(a*x^3) + (b*(3*b^2 - 20*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/
(64*a^2*x^2) - ((b + 6*c*x)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(8*x^4) - (a*x^2 + b*x^3 + c*x^4)^(3/2)/(4*x^7) - (3*
(b^2 - 4*a*c)^2*ArcTanh[(x*(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x^2 + b*x^3 + c*x^4])])/(128*a^(5/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 1918

Int[1/Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(r_.)], x_Symbol] :> Dist[-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 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 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]

Rule 1965

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}+\frac {3}{8} \int \frac {(b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{x^5} \, dx \\ & = -\frac {(b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{8 x^4}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}+\frac {1}{16} \int \frac {b^2-12 a c-4 b c x}{x^2 \sqrt {a x^2+b x^3+c x^4}} \, dx \\ & = -\frac {\left (b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{32 a x^3}-\frac {(b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{8 x^4}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}-\frac {\int \frac {\frac {1}{2} b \left (3 b^2-20 a c\right )+c \left (b^2-12 a c\right ) x}{x \sqrt {a x^2+b x^3+c x^4}} \, dx}{32 a} \\ & = -\frac {\left (b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{32 a x^3}+\frac {b \left (3 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{64 a^2 x^2}-\frac {(b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{8 x^4}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}+\frac {\int \frac {3 \left (b^2-4 a c\right )^2}{4 \sqrt {a x^2+b x^3+c x^4}} \, dx}{32 a^2} \\ & = -\frac {\left (b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{32 a x^3}+\frac {b \left (3 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{64 a^2 x^2}-\frac {(b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{8 x^4}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}+\frac {\left (3 \left (b^2-4 a c\right )^2\right ) \int \frac {1}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{128 a^2} \\ & = -\frac {\left (b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{32 a x^3}+\frac {b \left (3 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{64 a^2 x^2}-\frac {(b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{8 x^4}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}-\frac {\left (3 \left (b^2-4 a c\right )^2\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {x (2 a+b x)}{\sqrt {a x^2+b x^3+c x^4}}\right )}{64 a^2} \\ & = -\frac {\left (b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{32 a x^3}+\frac {b \left (3 b^2-20 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{64 a^2 x^2}-\frac {(b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{8 x^4}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{4 x^7}-\frac {3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{128 a^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.72 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^8} \, dx=\frac {\sqrt {x^2 (a+x (b+c x))} \left (-\sqrt {a} (2 a+b x) \sqrt {a+x (b+c x)} \left (8 a^2-3 b^2 x^2+4 a x (2 b+5 c x)\right )+3 \left (b^2-4 a c\right )^2 x^4 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )\right )}{64 a^{5/2} x^5 \sqrt {a+x (b+c x)}} \]

[In]

Integrate[(a*x^2 + b*x^3 + c*x^4)^(3/2)/x^8,x]

[Out]

(Sqrt[x^2*(a + x*(b + c*x))]*(-(Sqrt[a]*(2*a + b*x)*Sqrt[a + x*(b + c*x)]*(8*a^2 - 3*b^2*x^2 + 4*a*x*(2*b + 5*
c*x))) + 3*(b^2 - 4*a*c)^2*x^4*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x)])/Sqrt[a]]))/(64*a^(5/2)*x^5*Sqrt[a +
x*(b + c*x)])

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.66

method	result	size
pseudoelliptic	\(-\frac {3 \left (x^{4} \left (a c -\frac {b^{2}}{4}\right )^{2} \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x \sqrt {a}}\right )+\left (\frac {b \,x^{2} \left (10 c x +b \right ) a^{\frac {3}{2}}}{12}+x \left (\frac {5 c x}{3}+b \right ) a^{\frac {5}{2}}-\frac {\sqrt {a}\, b^{3} x^{3}}{8}+\frac {2 a^{\frac {7}{2}}}{3}\right ) \sqrt {c \,x^{2}+b x +a}-\ln \left (2\right ) x^{4} \left (a c -\frac {b^{2}}{4}\right )^{2}\right )}{8 a^{\frac {5}{2}} x^{4}}\)	\(131\)
risch	\(-\frac {\left (20 a b c \,x^{3}-3 b^{3} x^{3}+40 a^{2} c \,x^{2}+2 a \,b^{2} x^{2}+24 a^{2} b x +16 a^{3}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{64 x^{5} a^{2}}-\frac {3 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{128 a^{\frac {5}{2}} x \sqrt {c \,x^{2}+b x +a}}\)	\(157\)
default	\(-\frac {\left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (48 c^{2} a^{\frac {7}{2}} \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) x^{4}+24 c^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a b \,x^{5}-24 c \,a^{\frac {5}{2}} \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b^{2} x^{4}-16 c^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2} x^{4}+24 c^{2} \sqrt {c \,x^{2}+b x +a}\, a^{2} b \,x^{5}-2 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{3} x^{5}-48 c^{2} \sqrt {c \,x^{2}+b x +a}\, a^{3} x^{4}-24 c \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a b \,x^{3}+20 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,b^{2} x^{4}-6 c \sqrt {c \,x^{2}+b x +a}\, a \,b^{3} x^{5}+3 a^{\frac {3}{2}} \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b^{4} x^{4}+16 c \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a^{2} x^{2}+36 c \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2} x^{4}+2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b^{3} x^{3}-2 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{4} x^{4}+4 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a \,b^{2} x^{2}-6 \sqrt {c \,x^{2}+b x +a}\, a \,b^{4} x^{4}-16 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a^{2} b x +32 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a^{3}\right )}{128 x^{7} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{4}}\)	\(501\)

[In]

int((c*x^4+b*x^3+a*x^2)^(3/2)/x^8,x,method=_RETURNVERBOSE)

[Out]

-3/8*(x^4*(a*c-1/4*b^2)^2*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x/a^(1/2))+(1/12*b*x^2*(10*c*x+b)*a^(3/2)
+x*(5/3*c*x+b)*a^(5/2)-1/8*a^(1/2)*b^3*x^3+2/3*a^(7/2))*(c*x^2+b*x+a)^(1/2)-ln(2)*x^4*(a*c-1/4*b^2)^2)/a^(5/2)
/x^4

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.69 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^8} \, dx=\left [\frac {3 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {a} x^{5} \log \left (-\frac {8 \, a b x^{2} + {\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {a}}{x^{3}}\right ) - 4 \, {\left (24 \, a^{3} b x + 16 \, a^{4} - {\left (3 \, a b^{3} - 20 \, a^{2} b c\right )} x^{3} + 2 \, {\left (a^{2} b^{2} + 20 \, a^{3} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{256 \, a^{3} x^{5}}, \frac {3 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-a} x^{5} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) - 2 \, {\left (24 \, a^{3} b x + 16 \, a^{4} - {\left (3 \, a b^{3} - 20 \, a^{2} b c\right )} x^{3} + 2 \, {\left (a^{2} b^{2} + 20 \, a^{3} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{128 \, a^{3} x^{5}}\right ] \]

[In]

integrate((c*x^4+b*x^3+a*x^2)^(3/2)/x^8,x, algorithm="fricas")

[Out]

[1/256*(3*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(a)*x^5*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*(24*a^3*b*x + 16*a^4 - (3*a*b^3 - 20*a^2*b*c)*x^3 + 2*(a^2*
b^2 + 20*a^3*c)*x^2)*sqrt(c*x^4 + b*x^3 + a*x^2))/(a^3*x^5), 1/128*(3*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(-a)*
x^5*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*(24*a^3*b*x +
16*a^4 - (3*a*b^3 - 20*a^2*b*c)*x^3 + 2*(a^2*b^2 + 20*a^3*c)*x^2)*sqrt(c*x^4 + b*x^3 + a*x^2))/(a^3*x^5)]

Sympy [F]

\[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^8} \, dx=\int \frac {\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}}{x^{8}}\, dx \]

[In]

integrate((c*x**4+b*x**3+a*x**2)**(3/2)/x**8,x)

[Out]

Integral((x**2*(a + b*x + c*x**2))**(3/2)/x**8, x)

Maxima [F]

\[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^8} \, dx=\int { \frac {{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}{x^{8}} \,d x } \]

[In]

integrate((c*x^4+b*x^3+a*x^2)^(3/2)/x^8,x, algorithm="maxima")

[Out]

integrate((c*x^4 + b*x^3 + a*x^2)^(3/2)/x^8, x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^8} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((c*x^4+b*x^3+a*x^2)^(3/2)/x^8,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. \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^8} \, dx=\int \frac {{\left (c\,x^4+b\,x^3+a\,x^2\right )}^{3/2}}{x^8} \,d x \]

[In]

int((a*x^2 + b*x^3 + c*x^4)^(3/2)/x^8,x)

[Out]

int((a*x^2 + b*x^3 + c*x^4)^(3/2)/x^8, x)