\(\int \frac {(a+b x^2+c x^4)^{3/2}}{x^7} \, dx\) [971]

Optimal result

Integrand size = 20, antiderivative size = 163 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^7} \, dx=-\frac {\left (2 a b+\left (b^2+8 a c\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{16 a x^4}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{6 x^6}+\frac {b \left (b^2-12 a c\right ) \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{32 a^{3/2}}+\frac {1}{2} c^{3/2} \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right ) \] Output:

-1/16*(2*a*b+(8*a*c+b^2)*x^2)*(c*x^4+b*x^2+a)^(1/2)/a/x^4-1/6*(c*x^4+b*x^2 
+a)^(3/2)/x^6+1/32*b*(-12*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^(3/2)+1/2*c^(3/2)*arctanh(1/2*(2*c*x^2+b)/c^(1/2)/(c*x^4 
+b*x^2+a)^(1/2))
 

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^7} \, dx=\frac {\sqrt {a+b x^2+c x^4} \left (-8 a^2-14 a b x^2-3 b^2 x^4-32 a c x^4\right )}{48 a x^6}+\frac {\left (b^3-12 a b c\right ) \text {arctanh}\left (\frac {-\sqrt {c} x^2+\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{16 a^{3/2}}-\frac {1}{2} c^{3/2} \log \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right ) \] Input:

Integrate[(a + b*x^2 + c*x^4)^(3/2)/x^7,x]
 

Output:

(Sqrt[a + b*x^2 + c*x^4]*(-8*a^2 - 14*a*b*x^2 - 3*b^2*x^4 - 32*a*c*x^4))/( 
48*a*x^6) + ((b^3 - 12*a*b*c)*ArcTanh[(-(Sqrt[c]*x^2) + Sqrt[a + b*x^2 + c 
*x^4])/Sqrt[a]])/(16*a^(3/2)) - (c^(3/2)*Log[b + 2*c*x^2 - 2*Sqrt[c]*Sqrt[ 
a + b*x^2 + c*x^4]])/2
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {1434, 1161, 1229, 27, 1269, 1092, 219, 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.

\(\displaystyle \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^7} \, dx\)

\(\Big \downarrow \) 1434

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

\(\Big \downarrow \) 1161

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

\(\Big \downarrow \) 1229

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (-\frac {\int \frac {b \left (b^2-12 a c\right )-16 a c^2 x^2}{2 x^2 \sqrt {c x^4+b x^2+a}}dx^2}{4 a}-\frac {\sqrt {a+b x^2+c x^4} \left (x^2 \left (8 a c+b^2\right )+2 a b\right )}{4 a x^4}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 x^6}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (-\frac {\int \frac {b \left (b^2-12 a c\right )-16 a c^2 x^2}{x^2 \sqrt {c x^4+b x^2+a}}dx^2}{8 a}-\frac {\sqrt {a+b x^2+c x^4} \left (x^2 \left (8 a c+b^2\right )+2 a b\right )}{4 a x^4}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 x^6}\right )\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (-\frac {b \left (b^2-12 a c\right ) \int \frac {1}{x^2 \sqrt {c x^4+b x^2+a}}dx^2-16 a c^2 \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx^2}{8 a}-\frac {\sqrt {a+b x^2+c x^4} \left (x^2 \left (8 a c+b^2\right )+2 a b\right )}{4 a x^4}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 x^6}\right )\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (-\frac {b \left (b^2-12 a c\right ) \int \frac {1}{x^2 \sqrt {c x^4+b x^2+a}}dx^2-32 a c^2 \int \frac {1}{4 c-x^4}d\frac {2 c x^2+b}{\sqrt {c x^4+b x^2+a}}}{8 a}-\frac {\sqrt {a+b x^2+c x^4} \left (x^2 \left (8 a c+b^2\right )+2 a b\right )}{4 a x^4}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 x^6}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (-\frac {b \left (b^2-12 a c\right ) \int \frac {1}{x^2 \sqrt {c x^4+b x^2+a}}dx^2-16 a c^{3/2} \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 a}-\frac {\sqrt {a+b x^2+c x^4} \left (x^2 \left (8 a c+b^2\right )+2 a b\right )}{4 a x^4}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 x^6}\right )\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (-\frac {-2 b \left (b^2-12 a c\right ) \int \frac {1}{4 a-x^4}d\frac {b x^2+2 a}{\sqrt {c x^4+b x^2+a}}-16 a c^{3/2} \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 a}-\frac {\sqrt {a+b x^2+c x^4} \left (x^2 \left (8 a c+b^2\right )+2 a b\right )}{4 a x^4}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 x^6}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (-\frac {-\frac {b \left (b^2-12 a c\right ) \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{\sqrt {a}}-16 a c^{3/2} \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 a}-\frac {\sqrt {a+b x^2+c x^4} \left (x^2 \left (8 a c+b^2\right )+2 a b\right )}{4 a x^4}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 x^6}\right )\)

Input:

Int[(a + b*x^2 + c*x^4)^(3/2)/x^7,x]
 

Output:

(-1/3*(a + b*x^2 + c*x^4)^(3/2)/x^6 + (-1/4*((2*a*b + (b^2 + 8*a*c)*x^2)*S 
qrt[a + b*x^2 + c*x^4])/(a*x^4) - (-((b*(b^2 - 12*a*c)*ArcTanh[(2*a + b*x^ 
2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/Sqrt[a]) - 16*a*c^(3/2)*ArcTanh[( 
b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(8*a))/2)/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 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 1092

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

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 1161

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si 
mp[p/(e*(m + 1))   Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 
 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || 
 LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, 
 c, d, e, m, p, x]
 

rule 1229

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

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(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] &&  !IGtQ[m, 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 = 0.24 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.87

method	result	size
risch	\(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (32 a c \,x^{4}+3 b^{2} x^{4}+14 a b \,x^{2}+8 a^{2}\right )}{48 x^{6} a}+\frac {-\frac {b \left (12 a c -b^{2}\right ) \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{2 \sqrt {a}}+8 c^{\frac {3}{2}} a \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{16 a}\)	\(142\)
pseudoelliptic	\(-\frac {3 \left (b \,x^{6} \left (a c -\frac {b^{2}}{12}\right ) \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )-\frac {4 c^{\frac {3}{2}} a^{\frac {3}{2}} x^{6} \ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{\sqrt {c}}\right )}{3}+\left (\frac {7 \left (\frac {16 c \,x^{2}}{7}+b \right ) x^{2} a^{\frac {3}{2}}}{9}+\frac {\sqrt {a}\, b^{2} x^{4}}{6}+\frac {4 a^{\frac {5}{2}}}{9}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}+\frac {4 c^{\frac {3}{2}} a^{\frac {3}{2}} x^{6} \ln \left (2\right )}{3}\right )}{8 a^{\frac {3}{2}} x^{6}}\)	\(161\)
default	\(\frac {c^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2}-\frac {a \sqrt {c \,x^{4}+b \,x^{2}+a}}{6 x^{6}}-\frac {7 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{24 x^{4}}-\frac {b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{16 a \,x^{2}}+\frac {b^{3} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {3}{2}}}-\frac {3 b c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{8 \sqrt {a}}-\frac {2 c \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 x^{2}}\)	\(202\)
elliptic	\(\frac {c^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2}-\frac {a \sqrt {c \,x^{4}+b \,x^{2}+a}}{6 x^{6}}-\frac {7 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{24 x^{4}}-\frac {b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{16 a \,x^{2}}+\frac {b^{3} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{32 a^{\frac {3}{2}}}-\frac {3 b c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{8 \sqrt {a}}-\frac {2 c \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 x^{2}}\)	\(202\)

Input:

int((c*x^4+b*x^2+a)^(3/2)/x^7,x,method=_RETURNVERBOSE)
 

Output:

-1/48*(c*x^4+b*x^2+a)^(1/2)*(32*a*c*x^4+3*b^2*x^4+14*a*b*x^2+8*a^2)/x^6/a+ 
1/16/a*(-1/2*b*(12*a*c-b^2)/a^(1/2)*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a 
)^(1/2))/x^2)+8*c^(3/2)*a*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 771, normalized size of antiderivative = 4.73 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^7} \, dx =\text {Too large to display} \] Input:

integrate((c*x^4+b*x^2+a)^(3/2)/x^7,x, algorithm="fricas")
 

Output:

[1/192*(48*a^2*c^(3/2)*x^6*log(-8*c^2*x^4 - 8*b*c*x^2 - b^2 - 4*sqrt(c*x^4 
 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(c) - 4*a*c) - 3*(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*(14*a^2*b*x^2 + (3*a*b^2 + 32*a^2*c)*x^4 
 + 8*a^3)*sqrt(c*x^4 + b*x^2 + a))/(a^2*x^6), -1/192*(96*a^2*sqrt(-c)*c*x^ 
6*arctan(1/2*sqrt(c*x^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(-c)/(c^2*x^4 + b*c 
*x^2 + a*c)) + 3*(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*(14*a^2*b*x^2 + (3*a*b^2 + 32*a^2*c)*x^4 + 8*a^3)*sqrt(c*x^4 + b*x^2 + a 
))/(a^2*x^6), 1/96*(24*a^2*c^(3/2)*x^6*log(-8*c^2*x^4 - 8*b*c*x^2 - b^2 - 
4*sqrt(c*x^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(c) - 4*a*c) - 3*(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*(14*a^2*b*x^2 + (3*a*b^2 + 32*a^2*c)*x^4 + 
 8*a^3)*sqrt(c*x^4 + b*x^2 + a))/(a^2*x^6), -1/96*(48*a^2*sqrt(-c)*c*x^6*a 
rctan(1/2*sqrt(c*x^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(-c)/(c^2*x^4 + b*c*x^ 
2 + a*c)) + 3*(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*(14*a^2*b*x^2 + 
 (3*a*b^2 + 32*a^2*c)*x^4 + 8*a^3)*sqrt(c*x^4 + b*x^2 + a))/(a^2*x^6)]
 

Sympy [F]

\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^7} \, dx=\int \frac {\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}{x^{7}}\, dx \] Input:

integrate((c*x**4+b*x**2+a)**(3/2)/x**7,x)
 

Output:

Integral((a + b*x**2 + c*x**4)**(3/2)/x**7, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^7} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^4+b*x^2+a)^(3/2)/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. 398 vs. \(2 (135) = 270\).

Time = 0.20 (sec) , antiderivative size = 398, normalized size of antiderivative = 2.44 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^7} \, dx=-\frac {1}{2} \, c^{\frac {3}{2}} \log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} - b \right |}\right ) - \frac {{\left (b^{3} - 12 \, a b c\right )} \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{16 \, \sqrt {-a} a} + \frac {3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} b^{3} + 60 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a b c + 48 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{4} a b^{2} \sqrt {c} + 96 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{4} a^{2} c^{\frac {3}{2}} + 8 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a b^{3} - 96 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} a^{3} c^{\frac {3}{2}} - 3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{2} b^{3} + 36 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{3} b c + 64 \, a^{4} c^{\frac {3}{2}}}{48 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} - a\right )}^{3} a} \] Input:

integrate((c*x^4+b*x^2+a)^(3/2)/x^7,x, algorithm="giac")
 

Output:

-1/2*c^(3/2)*log(abs(-2*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*sqrt(c) - 
b)) - 1/16*(b^3 - 12*a*b*c)*arctan(-(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a) 
)/sqrt(-a))/(sqrt(-a)*a) + 1/48*(3*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a)) 
^5*b^3 + 60*(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*b^2*sqrt(c) + 96*(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 - 96*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^2*a^3*c^(3/2) - 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 + 64*a^4*c^(3/2))/(((sqrt(c)*x^2 - sqrt(c*x^4 + 
b*x^2 + a))^2 - a)^3*a)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^7} \, dx=\int \frac {{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{x^7} \,d x \] Input:

int((a + b*x^2 + c*x^4)^(3/2)/x^7,x)
 

Output:

int((a + b*x^2 + c*x^4)^(3/2)/x^7, x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^7} \, dx=\frac {-8 \sqrt {c}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{2}-14 \sqrt {c}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, a b \,x^{2}-32 \sqrt {c}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, a c \,x^{4}-3 \sqrt {c}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{2} x^{4}+36 \sqrt {c}\, \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{5}+b \,x^{3}+a x}d x \right ) a b c \,x^{6}-3 \sqrt {c}\, \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{5}+b \,x^{3}+a x}d x \right ) b^{3} x^{6}+24 \,\mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{4}+b \,x^{2}+a}+b +2 c \,x^{2}}{\sqrt {4 a c -b^{2}}}\right ) a \,c^{2} x^{6}}{48 \sqrt {c}\, a \,x^{6}} \] Input:

int((c*x^4+b*x^2+a)^(3/2)/x^7,x)
 

Output:

( - 8*sqrt(c)*sqrt(a + b*x**2 + c*x**4)*a**2 - 14*sqrt(c)*sqrt(a + b*x**2 
+ c*x**4)*a*b*x**2 - 32*sqrt(c)*sqrt(a + b*x**2 + c*x**4)*a*c*x**4 - 3*sqr 
t(c)*sqrt(a + b*x**2 + c*x**4)*b**2*x**4 + 36*sqrt(c)*int(sqrt(a + b*x**2 
+ c*x**4)/(a*x + b*x**3 + c*x**5),x)*a*b*c*x**6 - 3*sqrt(c)*int(sqrt(a + b 
*x**2 + c*x**4)/(a*x + b*x**3 + c*x**5),x)*b**3*x**6 + 24*log((2*sqrt(c)*s 
qrt(a + b*x**2 + c*x**4) + b + 2*c*x**2)/sqrt(4*a*c - b**2))*a*c**2*x**6)/ 
(48*sqrt(c)*a*x**6)