\(\int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^6} \, dx\) [37]

Optimal result

Integrand size = 24, antiderivative size = 205 \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^6} \, dx=-\frac {\sqrt {a x^2+b x^3+c x^4}}{4 x^5}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{24 a x^4}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{96 a^2 x^3}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{192 a^3 x^2}+\frac {\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \text {arctanh}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{128 a^{7/2}} \]

[Out]

1/128*(-4*a*c+b^2)*(-4*a*c+5*b^2)*arctanh(1/2*x*(b*x+2*a)/a^(1/2)/(c*x^4+b*x^3+a*x^2)^(1/2))/a^(7/2)-1/4*(c*x^
4+b*x^3+a*x^2)^(1/2)/x^5-1/24*b*(c*x^4+b*x^3+a*x^2)^(1/2)/a/x^4+1/96*(-12*a*c+5*b^2)*(c*x^4+b*x^3+a*x^2)^(1/2)
/a^2/x^3-1/192*b*(-52*a*c+15*b^2)*(c*x^4+b*x^3+a*x^2)^(1/2)/a^3/x^2

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1934, 1965, 12, 1918, 212} \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^6} \, dx=\frac {\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \text {arctanh}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{128 a^{7/2}}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{192 a^3 x^2}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{96 a^2 x^3}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{24 a x^4}-\frac {\sqrt {a x^2+b x^3+c x^4}}{4 x^5} \]

[In]

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

[Out]

-1/4*Sqrt[a*x^2 + b*x^3 + c*x^4]/x^5 - (b*Sqrt[a*x^2 + b*x^3 + c*x^4])/(24*a*x^4) + ((5*b^2 - 12*a*c)*Sqrt[a*x
^2 + b*x^3 + c*x^4])/(96*a^2*x^3) - (b*(15*b^2 - 52*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(192*a^3*x^2) + ((b^2 -
4*a*c)*(5*b^2 - 4*a*c)*ArcTanh[(x*(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x^2 + b*x^3 + c*x^4])])/(128*a^(7/2))

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 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 {\sqrt {a x^2+b x^3+c x^4}}{4 x^5}+\frac {1}{8} \int \frac {b+2 c x}{x^3 \sqrt {a x^2+b x^3+c x^4}} \, dx \\ & = -\frac {\sqrt {a x^2+b x^3+c x^4}}{4 x^5}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{24 a x^4}-\frac {\int \frac {\frac {1}{2} \left (5 b^2-12 a c\right )+2 b c x}{x^2 \sqrt {a x^2+b x^3+c x^4}} \, dx}{24 a} \\ & = -\frac {\sqrt {a x^2+b x^3+c x^4}}{4 x^5}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{24 a x^4}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{96 a^2 x^3}+\frac {\int \frac {\frac {1}{4} b \left (15 b^2-52 a c\right )+\frac {1}{2} c \left (5 b^2-12 a c\right ) x}{x \sqrt {a x^2+b x^3+c x^4}} \, dx}{48 a^2} \\ & = -\frac {\sqrt {a x^2+b x^3+c x^4}}{4 x^5}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{24 a x^4}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{96 a^2 x^3}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{192 a^3 x^2}-\frac {\int \frac {3 \left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )}{8 \sqrt {a x^2+b x^3+c x^4}} \, dx}{48 a^3} \\ & = -\frac {\sqrt {a x^2+b x^3+c x^4}}{4 x^5}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{24 a x^4}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{96 a^2 x^3}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{192 a^3 x^2}-\frac {\left (\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )\right ) \int \frac {1}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{128 a^3} \\ & = -\frac {\sqrt {a x^2+b x^3+c x^4}}{4 x^5}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{24 a x^4}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{96 a^2 x^3}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{192 a^3 x^2}+\frac {\left (\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )\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^3} \\ & = -\frac {\sqrt {a x^2+b x^3+c x^4}}{4 x^5}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{24 a x^4}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{96 a^2 x^3}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{192 a^3 x^2}+\frac {\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \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^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^6} \, dx=-\frac {\sqrt {x^2 (a+x (b+c x))} \left (\sqrt {a} \sqrt {a+x (b+c x)} \left (48 a^3+15 b^3 x^3+8 a^2 x (b+3 c x)-2 a b x^2 (5 b+26 c x)\right )+3 \left (5 b^4-24 a b^2 c+16 a^2 c^2\right ) x^4 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )\right )}{192 a^{7/2} x^5 \sqrt {a+x (b+c x)}} \]

[In]

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

[Out]

-1/192*(Sqrt[x^2*(a + x*(b + c*x))]*(Sqrt[a]*Sqrt[a + x*(b + c*x)]*(48*a^3 + 15*b^3*x^3 + 8*a^2*x*(b + 3*c*x)
- 2*a*b*x^2*(5*b + 26*c*x)) + 3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*x^4*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x
)])/Sqrt[a]]))/(a^(7/2)*x^5*Sqrt[a + x*(b + c*x)])

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.73

method	result	size
pseudoelliptic	\(\frac {x^{4} \left (a c -\frac {5 b^{2}}{4}\right ) \left (a c -\frac {b^{2}}{4}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x \sqrt {a}}\right )+\left (\frac {5 \left (\frac {26 c x}{5}+b \right ) x^{2} b \,a^{\frac {3}{2}}}{12}+\left (-c \,x^{2}-\frac {1}{3} b x \right ) a^{\frac {5}{2}}-\frac {5 \sqrt {a}\, b^{3} x^{3}}{8}-2 a^{\frac {7}{2}}\right ) \sqrt {c \,x^{2}+b x +a}-\ln \left (2\right ) x^{4} \left (a c -\frac {5 b^{2}}{4}\right ) \left (a c -\frac {b^{2}}{4}\right )}{8 a^{\frac {7}{2}} x^{4}}\)	\(149\)
risch	\(-\frac {\left (-52 a b c \,x^{3}+15 b^{3} x^{3}+24 a^{2} c \,x^{2}-10 a \,b^{2} x^{2}+8 a^{2} b x +48 a^{3}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{192 x^{5} a^{3}}+\frac {\left (16 a^{2} c^{2}-24 a \,b^{2} c +5 b^{4}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{128 a^{\frac {7}{2}} x \sqrt {c \,x^{2}+b x +a}}\)	\(159\)
default	\(\frac {\sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}\, \left (48 c^{2} a^{\frac {5}{2}} \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) x^{4}+24 c^{2} \sqrt {c \,x^{2}+b x +a}\, a b \,x^{5}-72 c \,a^{\frac {3}{2}} \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b^{2} x^{4}-48 c^{2} \sqrt {c \,x^{2}+b x +a}\, a^{2} x^{4}-30 c \sqrt {c \,x^{2}+b x +a}\, b^{3} x^{5}-24 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a b \,x^{3}+84 c \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} x^{4}+15 \sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b^{4} x^{4}+48 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2} x^{2}+30 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{3} x^{3}-30 \sqrt {c \,x^{2}+b x +a}\, b^{4} x^{4}-60 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,b^{2} x^{2}+80 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2} b x -96 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{3}\right )}{384 x^{5} \sqrt {c \,x^{2}+b x +a}\, a^{4}}\)	\(387\)

[In]

int((c*x^4+b*x^3+a*x^2)^(1/2)/x^6,x,method=_RETURNVERBOSE)

[Out]

1/8/a^(7/2)*(x^4*(a*c-5/4*b^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))+(5/12*(26/5
*c*x+b)*x^2*b*a^(3/2)+(-c*x^2-1/3*b*x)*a^(5/2)-5/8*a^(1/2)*b^3*x^3-2*a^(7/2))*(c*x^2+b*x+a)^(1/2)-ln(2)*x^4*(a
*c-5/4*b^2)*(a*c-1/4*b^2))/x^4

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.64 \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^6} \, dx=\left [\frac {3 \, {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {a} x^{5} \log \left (-\frac {8 \, a b x^{2} + {\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {a}}{x^{3}}\right ) - 4 \, {\left (8 \, a^{3} b x + 48 \, a^{4} + {\left (15 \, a b^{3} - 52 \, a^{2} b c\right )} x^{3} - 2 \, {\left (5 \, a^{2} b^{2} - 12 \, a^{3} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{768 \, a^{4} x^{5}}, -\frac {3 \, {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-a} x^{5} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) + 2 \, {\left (8 \, a^{3} b x + 48 \, a^{4} + {\left (15 \, a b^{3} - 52 \, a^{2} b c\right )} x^{3} - 2 \, {\left (5 \, a^{2} b^{2} - 12 \, a^{3} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{384 \, a^{4} x^{5}}\right ] \]

[In]

integrate((c*x^4+b*x^3+a*x^2)^(1/2)/x^6,x, algorithm="fricas")

[Out]

[1/768*(3*(5*b^4 - 24*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*(8*a^3*b*x + 48*a^4 + (15*a*b^3 - 52*a^2*b*c)*x^3 - 2*(5
*a^2*b^2 - 12*a^3*c)*x^2)*sqrt(c*x^4 + b*x^3 + a*x^2))/(a^4*x^5), -1/384*(3*(5*b^4 - 24*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*(8*a
^3*b*x + 48*a^4 + (15*a*b^3 - 52*a^2*b*c)*x^3 - 2*(5*a^2*b^2 - 12*a^3*c)*x^2)*sqrt(c*x^4 + b*x^3 + a*x^2))/(a^
4*x^5)]

Sympy [F]

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

[In]

integrate((c*x**4+b*x**3+a*x**2)**(1/2)/x**6,x)

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^6} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{3} + a x^{2}}}{x^{6}} \,d x } \]

[In]

integrate((c*x^4+b*x^3+a*x^2)^(1/2)/x^6,x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

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

[In]

integrate((c*x^4+b*x^3+a*x^2)^(1/2)/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. \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^6} \, dx=\int \frac {\sqrt {c\,x^4+b\,x^3+a\,x^2}}{x^6} \,d x \]

[In]

int((a*x^2 + b*x^3 + c*x^4)^(1/2)/x^6,x)

[Out]

int((a*x^2 + b*x^3 + c*x^4)^(1/2)/x^6, x)