\(\int \frac {\sqrt {a+b x^2+c x^4}}{x^7} \, dx\) [955]

Optimal result

Integrand size = 20, antiderivative size = 116 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^7} \, dx=\frac {b \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{16 a^2 x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{6 a x^6}-\frac {b \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{32 a^{5/2}} \] Output:

1/16*b*(b*x^2+2*a)*(c*x^4+b*x^2+a)^(1/2)/a^2/x^4-1/6*(c*x^4+b*x^2+a)^(3/2) 
/a/x^6-1/32*b*(-4*a*c+b^2)*arctanh(1/2*(b*x^2+2*a)/a^(1/2)/(c*x^4+b*x^2+a) 
^(1/2))/a^(5/2)
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^7} \, dx=\frac {\sqrt {a+b x^2+c x^4} \left (-8 a^2-2 a b x^2+3 b^2 x^4-8 a c x^4\right )}{48 a^2 x^6}+\frac {\left (b^3-4 a b c\right ) \text {arctanh}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{16 a^{5/2}} \] Input:

Integrate[Sqrt[a + b*x^2 + c*x^4]/x^7,x]
 

Output:

(Sqrt[a + b*x^2 + c*x^4]*(-8*a^2 - 2*a*b*x^2 + 3*b^2*x^4 - 8*a*c*x^4))/(48 
*a^2*x^6) + ((b^3 - 4*a*b*c)*ArcTanh[(Sqrt[c]*x^2 - Sqrt[a + b*x^2 + c*x^4 
])/Sqrt[a]])/(16*a^(5/2))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1434, 1157, 1152, 1154, 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.

\(\displaystyle \int \frac {\sqrt {a+b x^2+c x^4}}{x^7} \, dx\)

\(\Big \downarrow \) 1434

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

\(\Big \downarrow \) 1157

\(\displaystyle \frac {1}{2} \left (-\frac {b \int \frac {\sqrt {c x^4+b x^2+a}}{x^6}dx^2}{2 a}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 a x^6}\right )\)

\(\Big \downarrow \) 1152

\(\displaystyle \frac {1}{2} \left (-\frac {b \left (-\frac {\left (b^2-4 a c\right ) \int \frac {1}{x^2 \sqrt {c x^4+b x^2+a}}dx^2}{8 a}-\frac {\left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{4 a x^4}\right )}{2 a}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 a x^6}\right )\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 a-x^4}d\frac {b x^2+2 a}{\sqrt {c x^4+b x^2+a}}}{4 a}-\frac {\left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{4 a x^4}\right )}{2 a}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 a x^6}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{8 a^{3/2}}-\frac {\left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{4 a x^4}\right )}{2 a}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 a x^6}\right )\)

Input:

Int[Sqrt[a + b*x^2 + c*x^4]/x^7,x]
 

Output:

(-1/3*(a + b*x^2 + c*x^4)^(3/2)/(a*x^6) - (b*(-1/4*((2*a + b*x^2)*Sqrt[a + 
 b*x^2 + c*x^4])/(a*x^4) + ((b^2 - 4*a*c)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a] 
*Sqrt[a + b*x^2 + c*x^4])])/(8*a^(3/2))))/(2*a))/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 1152

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b 
*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a 
*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)))   Int[(d + e*x)^(m + 2)*(a + b*x + 
 c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] 
 && GtQ[p, 0]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-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]
 

rule 1157

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d 
^2 - b*d*e + a*e^2))), x] + Simp[(2*c*d - b*e)/(2*(c*d^2 - b*d*e + a*e^2)) 
  Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e 
, m, p}, x] && EqQ[m + 2*p + 3, 0]
 

rule 1434

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp 
[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; Free 
Q[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
 

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.87

method	result	size
risch	\(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (8 a c \,x^{4}-3 b^{2} x^{4}+2 a b \,x^{2}+8 a^{2}\right )}{48 x^{6} a^{2}}+\frac {b \left (4 a c -b^{2}\right ) \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {5}{2}}}\)	\(101\)
pseudoelliptic	\(\frac {b \,x^{6} \left (a c -\frac {b^{2}}{4}\right ) \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )+\frac {\left (-\frac {2 x^{2} \left (4 c \,x^{2}+b \right ) a^{\frac {3}{2}}}{3}+\sqrt {a}\, b^{2} x^{4}-\frac {8 a^{\frac {5}{2}}}{3}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{2}}{8 a^{\frac {5}{2}} x^{6}}\)	\(105\)
default	\(-\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{6 a \,x^{6}}+\frac {b \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{8 a^{2} x^{4}}-\frac {b^{2} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{16 a^{3} x^{2}}+\frac {b^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{16 a^{3}}-\frac {b^{3} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {5}{2}}}+\frac {b^{2} c \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{16 a^{3}}-\frac {b c \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 a^{2}}+\frac {b c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{8 a^{\frac {3}{2}}}\)	\(222\)
elliptic	\(-\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{6 a \,x^{6}}+\frac {b \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{8 a^{2} x^{4}}-\frac {b^{2} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{16 a^{3} x^{2}}+\frac {b^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{16 a^{3}}-\frac {b^{3} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {5}{2}}}+\frac {b^{2} c \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{16 a^{3}}-\frac {b c \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 a^{2}}+\frac {b c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{8 a^{\frac {3}{2}}}\)	\(222\)

Input:

int((c*x^4+b*x^2+a)^(1/2)/x^7,x,method=_RETURNVERBOSE)
 

Output:

-1/48*(c*x^4+b*x^2+a)^(1/2)*(8*a*c*x^4-3*b^2*x^4+2*a*b*x^2+8*a^2)/x^6/a^2+ 
1/32*b*(4*a*c-b^2)/a^(5/2)*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2))/ 
x^2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 261, normalized size of antiderivative = 2.25 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^7} \, dx=\left [-\frac {3 \, {\left (b^{3} - 4 \, a b c\right )} \sqrt {a} x^{6} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{4}}\right ) + 4 \, {\left (2 \, a^{2} b x^{2} - {\left (3 \, a b^{2} - 8 \, a^{2} c\right )} x^{4} + 8 \, a^{3}\right )} \sqrt {c x^{4} + b x^{2} + a}}{192 \, a^{3} x^{6}}, \frac {3 \, {\left (b^{3} - 4 \, a b c\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) - 2 \, {\left (2 \, a^{2} b x^{2} - {\left (3 \, a b^{2} - 8 \, a^{2} c\right )} x^{4} + 8 \, a^{3}\right )} \sqrt {c x^{4} + b x^{2} + a}}{96 \, a^{3} x^{6}}\right ] \] Input:

integrate((c*x^4+b*x^2+a)^(1/2)/x^7,x, algorithm="fricas")
 

Output:

[-1/192*(3*(b^3 - 4*a*b*c)*sqrt(a)*x^6*log(-((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 
 + 4*sqrt(c*x^4 + b*x^2 + a)*(b*x^2 + 2*a)*sqrt(a) + 8*a^2)/x^4) + 4*(2*a^ 
2*b*x^2 - (3*a*b^2 - 8*a^2*c)*x^4 + 8*a^3)*sqrt(c*x^4 + b*x^2 + a))/(a^3*x 
^6), 1/96*(3*(b^3 - 4*a*b*c)*sqrt(-a)*x^6*arctan(1/2*sqrt(c*x^4 + b*x^2 + 
a)*(b*x^2 + 2*a)*sqrt(-a)/(a*c*x^4 + a*b*x^2 + a^2)) - 2*(2*a^2*b*x^2 - (3 
*a*b^2 - 8*a^2*c)*x^4 + 8*a^3)*sqrt(c*x^4 + b*x^2 + a))/(a^3*x^6)]
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^7} \, dx=\int \frac {\sqrt {a + b x^{2} + c x^{4}}}{x^{7}}\, dx \] Input:

integrate((c*x**4+b*x**2+a)**(1/2)/x**7,x)
 

Output:

Integral(sqrt(a + b*x**2 + c*x**4)/x**7, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^7} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^4+b*x^2+a)^(1/2)/x^7,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (98) = 196\).

Time = 0.13 (sec) , antiderivative size = 359, normalized size of antiderivative = 3.09 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^7} \, dx=\frac {{\left (b^{3} - 4 \, a b c\right )} \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{16 \, \sqrt {-a} a^{2}} - \frac {3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} b^{3} - 12 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a b c - 48 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{4} a^{2} c^{\frac {3}{2}} - 8 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a b^{3} - 48 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a^{2} b c - 48 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} a^{2} b^{2} \sqrt {c} - 3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{2} b^{3} - 36 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{3} b c - 16 \, a^{4} c^{\frac {3}{2}}}{48 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} - a\right )}^{3} a^{2}} \] Input:

integrate((c*x^4+b*x^2+a)^(1/2)/x^7,x, algorithm="giac")
 

Output:

1/16*(b^3 - 4*a*b*c)*arctan(-(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))/sqrt( 
-a))/(sqrt(-a)*a^2) - 1/48*(3*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^5*b^ 
3 - 12*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^5*a*b*c - 48*(sqrt(c)*x^2 - 
 sqrt(c*x^4 + b*x^2 + a))^4*a^2*c^(3/2) - 8*(sqrt(c)*x^2 - sqrt(c*x^4 + b* 
x^2 + a))^3*a*b^3 - 48*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^3*a^2*b*c - 
 48*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^2*a^2*b^2*sqrt(c) - 3*(sqrt(c) 
*x^2 - sqrt(c*x^4 + b*x^2 + a))*a^2*b^3 - 36*(sqrt(c)*x^2 - sqrt(c*x^4 + b 
*x^2 + a))*a^3*b*c - 16*a^4*c^(3/2))/(((sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + 
 a))^2 - a)^3*a^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^7} \, dx=\int \frac {\sqrt {c\,x^4+b\,x^2+a}}{x^7} \,d x \] Input:

int((a + b*x^2 + c*x^4)^(1/2)/x^7,x)
 

Output:

int((a + b*x^2 + c*x^4)^(1/2)/x^7, x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^7} \, dx=\frac {-8 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{2}-2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a b \,x^{2}-8 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a c \,x^{4}+3 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{2} x^{4}-12 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{5}+b \,x^{3}+a x}d x \right ) a b c \,x^{6}+3 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{5}+b \,x^{3}+a x}d x \right ) b^{3} x^{6}}{48 a^{2} x^{6}} \] Input:

int((c*x^4+b*x^2+a)^(1/2)/x^7,x)
 

Output:

( - 8*sqrt(a + b*x**2 + c*x**4)*a**2 - 2*sqrt(a + b*x**2 + c*x**4)*a*b*x** 
2 - 8*sqrt(a + b*x**2 + c*x**4)*a*c*x**4 + 3*sqrt(a + b*x**2 + c*x**4)*b** 
2*x**4 - 12*int(sqrt(a + b*x**2 + c*x**4)/(a*x + b*x**3 + c*x**5),x)*a*b*c 
*x**6 + 3*int(sqrt(a + b*x**2 + c*x**4)/(a*x + b*x**3 + c*x**5),x)*b**3*x* 
*6)/(48*a**2*x**6)