\(\int \frac {1}{x^7 \sqrt {-a+b x^2+c x^4}} \, dx\) [1002]

Optimal result

Integrand size = 22, antiderivative size = 154 \[ \int \frac {1}{x^7 \sqrt {-a+b x^2+c x^4}} \, dx=\frac {\sqrt {-a+b x^2+c x^4}}{6 a x^6}+\frac {5 b \sqrt {-a+b x^2+c x^4}}{24 a^2 x^4}+\frac {\left (15 b^2+16 a c\right ) \sqrt {-a+b x^2+c x^4}}{48 a^3 x^2}-\frac {b \left (5 b^2+12 a c\right ) \arctan \left (\frac {2 a-b x^2}{2 \sqrt {a} \sqrt {-a+b x^2+c x^4}}\right )}{32 a^{7/2}} \] Output:

1/6*(c*x^4+b*x^2-a)^(1/2)/a/x^6+5/24*b*(c*x^4+b*x^2-a)^(1/2)/a^2/x^4+1/48* 
(16*a*c+15*b^2)*(c*x^4+b*x^2-a)^(1/2)/a^3/x^2-1/32*b*(12*a*c+5*b^2)*arctan 
(1/2*(-b*x^2+2*a)/a^(1/2)/(c*x^4+b*x^2-a)^(1/2))/a^(7/2)
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^7 \sqrt {-a+b x^2+c x^4}} \, dx=\frac {\sqrt {-a+b x^2+c x^4} \left (8 a^2+10 a b x^2+15 b^2 x^4+16 a c x^4\right )}{48 a^3 x^6}+\frac {\left (-5 b^3-12 a b c\right ) \arctan \left (\frac {\sqrt {c} x^2-\sqrt {-a+b x^2+c x^4}}{\sqrt {a}}\right )}{16 a^{7/2}} \] Input:

Integrate[1/(x^7*Sqrt[-a + b*x^2 + c*x^4]),x]
 

Output:

(Sqrt[-a + b*x^2 + c*x^4]*(8*a^2 + 10*a*b*x^2 + 15*b^2*x^4 + 16*a*c*x^4))/ 
(48*a^3*x^6) + ((-5*b^3 - 12*a*b*c)*ArcTan[(Sqrt[c]*x^2 - Sqrt[-a + b*x^2 
+ c*x^4])/Sqrt[a]])/(16*a^(7/2))
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1434, 1167, 27, 1237, 27, 1228, 1154, 217}

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.

\(\displaystyle \int \frac {1}{x^7 \sqrt {-a+b x^2+c x^4}} \, dx\)

\(\Big \downarrow \) 1434

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

\(\Big \downarrow \) 1167

\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {4 c x^2+5 b}{2 x^6 \sqrt {c x^4+b x^2-a}}dx^2}{3 a}+\frac {\sqrt {-a+b x^2+c x^4}}{3 a x^6}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {4 c x^2+5 b}{x^6 \sqrt {c x^4+b x^2-a}}dx^2}{6 a}+\frac {\sqrt {-a+b x^2+c x^4}}{3 a x^6}\right )\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {1}{2} \left (\frac {\frac {\int \frac {15 b^2+10 c x^2 b+16 a c}{2 x^4 \sqrt {c x^4+b x^2-a}}dx^2}{2 a}+\frac {5 b \sqrt {-a+b x^2+c x^4}}{2 a x^4}}{6 a}+\frac {\sqrt {-a+b x^2+c x^4}}{3 a x^6}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {\frac {\int \frac {15 b^2+10 c x^2 b+16 a c}{x^4 \sqrt {c x^4+b x^2-a}}dx^2}{4 a}+\frac {5 b \sqrt {-a+b x^2+c x^4}}{2 a x^4}}{6 a}+\frac {\sqrt {-a+b x^2+c x^4}}{3 a x^6}\right )\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {1}{2} \left (\frac {\frac {\frac {3 b \left (12 a c+5 b^2\right ) \int \frac {1}{x^2 \sqrt {c x^4+b x^2-a}}dx^2}{2 a}+\frac {\left (16 a c+15 b^2\right ) \sqrt {-a+b x^2+c x^4}}{a x^2}}{4 a}+\frac {5 b \sqrt {-a+b x^2+c x^4}}{2 a x^4}}{6 a}+\frac {\sqrt {-a+b x^2+c x^4}}{3 a x^6}\right )\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {1}{2} \left (\frac {\frac {\frac {\left (16 a c+15 b^2\right ) \sqrt {-a+b x^2+c x^4}}{a x^2}-\frac {3 b \left (12 a c+5 b^2\right ) \int \frac {1}{-x^4-4 a}d\left (-\frac {2 a-b x^2}{\sqrt {c x^4+b x^2-a}}\right )}{a}}{4 a}+\frac {5 b \sqrt {-a+b x^2+c x^4}}{2 a x^4}}{6 a}+\frac {\sqrt {-a+b x^2+c x^4}}{3 a x^6}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{2} \left (\frac {\frac {\frac {\left (16 a c+15 b^2\right ) \sqrt {-a+b x^2+c x^4}}{a x^2}-\frac {3 b \left (12 a c+5 b^2\right ) \arctan \left (\frac {2 a-b x^2}{2 \sqrt {a} \sqrt {-a+b x^2+c x^4}}\right )}{2 a^{3/2}}}{4 a}+\frac {5 b \sqrt {-a+b x^2+c x^4}}{2 a x^4}}{6 a}+\frac {\sqrt {-a+b x^2+c x^4}}{3 a x^6}\right )\)

Input:

Int[1/(x^7*Sqrt[-a + b*x^2 + c*x^4]),x]
 

Output:

(Sqrt[-a + b*x^2 + c*x^4]/(3*a*x^6) + ((5*b*Sqrt[-a + b*x^2 + c*x^4])/(2*a 
*x^4) + (((15*b^2 + 16*a*c)*Sqrt[-a + b*x^2 + c*x^4])/(a*x^2) - (3*b*(5*b^ 
2 + 12*a*c)*ArcTan[(2*a - b*x^2)/(2*Sqrt[a]*Sqrt[-a + b*x^2 + c*x^4])])/(2 
*a^(3/2)))/(4*a))/(6*a))/2
 

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 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 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 1167

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[1/((m + 1)*(c*d^2 - b*d*e + a*e^2))   Int[ 
(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, 
 x]*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[m 
, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimp 
lerQ[m, 1] && IntegerQ[p]) || ILtQ[Simplify[m + 2*p + 3], 0])
 

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(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, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 1237

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(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[1/((m + 1) 
*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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 = 0.21 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.77

method	result	size
pseudoelliptic	\(\frac {\frac {3 b \,x^{6} \left (a c +\frac {5 b^{2}}{12}\right ) \ln \left (\frac {-2 a +b \,x^{2}+2 \sqrt {-a}\, \sqrt {c \,x^{4}+b \,x^{2}-a}}{x^{2}}\right )}{8}-\frac {5 \left (-\frac {2 \left (\frac {8 c \,x^{2}}{5}+b \right ) x^{2} \left (-a \right )^{\frac {3}{2}}}{3}+b^{2} x^{4} \sqrt {-a}+\frac {8 \left (-a \right )^{\frac {5}{2}}}{15}\right ) \sqrt {c \,x^{4}+b \,x^{2}-a}}{16}}{x^{6} \left (-a \right )^{\frac {7}{2}}}\)	\(119\)
risch	\(-\frac {\left (-c \,x^{4}-b \,x^{2}+a \right ) \left (16 a c \,x^{4}+15 b^{2} x^{4}+10 a b \,x^{2}+8 a^{2}\right )}{48 a^{3} x^{6} \sqrt {c \,x^{4}+b \,x^{2}-a}}-\frac {b \left (12 a c +5 b^{2}\right ) \ln \left (\frac {-2 a +b \,x^{2}+2 \sqrt {-a}\, \sqrt {c \,x^{4}+b \,x^{2}-a}}{x^{2}}\right )}{32 a^{3} \sqrt {-a}}\)	\(126\)
default	\(\frac {\sqrt {c \,x^{4}+b \,x^{2}-a}}{6 a \,x^{6}}+\frac {5 b \sqrt {c \,x^{4}+b \,x^{2}-a}}{24 a^{2} x^{4}}+\frac {5 b^{2} \sqrt {c \,x^{4}+b \,x^{2}-a}}{16 a^{3} x^{2}}-\frac {5 b^{3} \ln \left (\frac {-2 a +b \,x^{2}+2 \sqrt {-a}\, \sqrt {c \,x^{4}+b \,x^{2}-a}}{x^{2}}\right )}{32 a^{3} \sqrt {-a}}-\frac {3 b c \ln \left (\frac {-2 a +b \,x^{2}+2 \sqrt {-a}\, \sqrt {c \,x^{4}+b \,x^{2}-a}}{x^{2}}\right )}{8 a^{2} \sqrt {-a}}+\frac {c \sqrt {c \,x^{4}+b \,x^{2}-a}}{3 a^{2} x^{2}}\)	\(202\)
elliptic	\(\frac {\sqrt {c \,x^{4}+b \,x^{2}-a}}{6 a \,x^{6}}+\frac {5 b \sqrt {c \,x^{4}+b \,x^{2}-a}}{24 a^{2} x^{4}}+\frac {5 b^{2} \sqrt {c \,x^{4}+b \,x^{2}-a}}{16 a^{3} x^{2}}-\frac {5 b^{3} \ln \left (\frac {-2 a +b \,x^{2}+2 \sqrt {-a}\, \sqrt {c \,x^{4}+b \,x^{2}-a}}{x^{2}}\right )}{32 a^{3} \sqrt {-a}}-\frac {3 b c \ln \left (\frac {-2 a +b \,x^{2}+2 \sqrt {-a}\, \sqrt {c \,x^{4}+b \,x^{2}-a}}{x^{2}}\right )}{8 a^{2} \sqrt {-a}}+\frac {c \sqrt {c \,x^{4}+b \,x^{2}-a}}{3 a^{2} x^{2}}\)	\(202\)

Input:

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

Output:

3/8/(-a)^(7/2)*(b*x^6*(a*c+5/12*b^2)*ln((-2*a+b*x^2+2*(-a)^(1/2)*(c*x^4+b* 
x^2-a)^(1/2))/x^2)-5/6*(-2/3*(8/5*c*x^2+b)*x^2*(-a)^(3/2)+b^2*x^4*(-a)^(1/ 
2)+8/15*(-a)^(5/2))*(c*x^4+b*x^2-a)^(1/2))/x^6
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.77 \[ \int \frac {1}{x^7 \sqrt {-a+b x^2+c x^4}} \, dx=\left [-\frac {3 \, {\left (5 \, b^{3} + 12 \, a b c\right )} \sqrt {-a} x^{6} \log \left (\frac {{\left (b^{2} - 4 \, a c\right )} x^{4} - 8 \, a b x^{2} - 4 \, \sqrt {c x^{4} + b x^{2} - a} {\left (b x^{2} - 2 \, a\right )} \sqrt {-a} + 8 \, a^{2}}{x^{4}}\right ) - 4 \, {\left (10 \, a^{2} b x^{2} + {\left (15 \, a b^{2} + 16 \, a^{2} c\right )} x^{4} + 8 \, a^{3}\right )} \sqrt {c x^{4} + b x^{2} - a}}{192 \, a^{4} x^{6}}, \frac {3 \, {\left (5 \, b^{3} + 12 \, a b c\right )} \sqrt {a} x^{6} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} - a} {\left (b x^{2} - 2 \, a\right )} \sqrt {a}}{2 \, {\left (a c x^{4} + a b x^{2} - a^{2}\right )}}\right ) + 2 \, {\left (10 \, a^{2} b x^{2} + {\left (15 \, a b^{2} + 16 \, a^{2} c\right )} x^{4} + 8 \, a^{3}\right )} \sqrt {c x^{4} + b x^{2} - a}}{96 \, a^{4} x^{6}}\right ] \] Input:

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

Output:

[-1/192*(3*(5*b^3 + 12*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*( 
10*a^2*b*x^2 + (15*a*b^2 + 16*a^2*c)*x^4 + 8*a^3)*sqrt(c*x^4 + b*x^2 - a)) 
/(a^4*x^6), 1/96*(3*(5*b^3 + 12*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*(10*a^2*b 
*x^2 + (15*a*b^2 + 16*a^2*c)*x^4 + 8*a^3)*sqrt(c*x^4 + b*x^2 - a))/(a^4*x^ 
6)]
 

Sympy [F]

\[ \int \frac {1}{x^7 \sqrt {-a+b x^2+c x^4}} \, dx=\int \frac {1}{x^{7} \sqrt {- a + b x^{2} + c x^{4}}}\, dx \] Input:

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

Output:

Integral(1/(x**7*sqrt(-a + b*x**2 + c*x**4)), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x^7 \sqrt {-a+b x^2+c x^4}} \, dx=-\frac {5 \, b^{3} \arcsin \left (-\frac {b}{\sqrt {b^{2} + 4 \, a c}} + \frac {2 \, a}{\sqrt {b^{2} + 4 \, a c} x^{2}}\right )}{32 \, a^{\frac {7}{2}}} - \frac {3 \, b c \arcsin \left (-\frac {b}{\sqrt {b^{2} + 4 \, a c}} + \frac {2 \, a}{\sqrt {b^{2} + 4 \, a c} x^{2}}\right )}{8 \, a^{\frac {5}{2}}} + \frac {5 \, \sqrt {c x^{4} + b x^{2} - a} b^{2}}{16 \, a^{3} x^{2}} + \frac {\sqrt {c x^{4} + b x^{2} - a} c}{3 \, a^{2} x^{2}} + \frac {5 \, \sqrt {c x^{4} + b x^{2} - a} b}{24 \, a^{2} x^{4}} + \frac {\sqrt {c x^{4} + b x^{2} - a}}{6 \, a x^{6}} \] Input:

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

Output:

-5/32*b^3*arcsin(-b/sqrt(b^2 + 4*a*c) + 2*a/(sqrt(b^2 + 4*a*c)*x^2))/a^(7/ 
2) - 3/8*b*c*arcsin(-b/sqrt(b^2 + 4*a*c) + 2*a/(sqrt(b^2 + 4*a*c)*x^2))/a^ 
(5/2) + 5/16*sqrt(c*x^4 + b*x^2 - a)*b^2/(a^3*x^2) + 1/3*sqrt(c*x^4 + b*x^ 
2 - a)*c/(a^2*x^2) + 5/24*sqrt(c*x^4 + b*x^2 - a)*b/(a^2*x^4) + 1/6*sqrt(c 
*x^4 + b*x^2 - a)/(a*x^6)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 344 vs. \(2 (131) = 262\).

Time = 0.14 (sec) , antiderivative size = 344, normalized size of antiderivative = 2.23 \[ \int \frac {1}{x^7 \sqrt {-a+b x^2+c x^4}} \, dx=\frac {{\left (5 \, b^{3} + 12 \, a b c\right )} \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}}{\sqrt {a}}\right )}{16 \, a^{\frac {7}{2}}} - \frac {15 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}\right )}^{5} b^{3} + 36 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}\right )}^{5} a b c + 40 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}\right )}^{3} a b^{3} + 96 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}\right )}^{3} a^{2} b c - 96 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}\right )}^{2} a^{3} c^{\frac {3}{2}} + 33 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}\right )} a^{2} b^{3} - 36 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}\right )} a^{3} b c - 48 \, a^{3} b^{2} \sqrt {c} - 32 \, a^{4} c^{\frac {3}{2}}}{48 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}\right )}^{2} + a\right )}^{3} a^{3}} \] Input:

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

Output:

1/16*(5*b^3 + 12*a*b*c)*arctan(-(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 - a))/sq 
rt(a))/a^(7/2) - 1/48*(15*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 - a))^5*b^3 + 
36*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 - a))^5*a*b*c + 40*(sqrt(c)*x^2 - sqr 
t(c*x^4 + b*x^2 - a))^3*a*b^3 + 96*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 - a)) 
^3*a^2*b*c - 96*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 - a))^2*a^3*c^(3/2) + 33 
*(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 - 48*a^3*b^2*sqrt(c) - 32*a^4*c^(3/2))/(((sqrt 
(c)*x^2 - sqrt(c*x^4 + b*x^2 - a))^2 + a)^3*a^3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^7 \sqrt {-a+b x^2+c x^4}} \, dx=\int \frac {1}{x^7\,\sqrt {c\,x^4+b\,x^2-a}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{x^7 \sqrt {-a+b x^2+c x^4}} \, dx=\int \frac {1}{\sqrt {c \,x^{4}+b \,x^{2}-a}\, x^{7}}d x \] Input:

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

Output:

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