\(\int \frac {A+B x^2+C x^4}{x^7 (c+d x^2) \sqrt {a+c x^4}} \, dx\) [27]

Optimal result

Integrand size = 36, antiderivative size = 253 \[ \int \frac {A+B x^2+C x^4}{x^7 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=-\frac {A \sqrt {a+c x^4}}{6 a c x^6}-\frac {(B c-A d) \sqrt {a+c x^4}}{4 a c^2 x^4}-\frac {\left (3 a c (c C-B d)-A \left (2 c^3-3 a d^2\right )\right ) \sqrt {a+c x^4}}{6 a^2 c^3 x^2}-\frac {d^2 \left (c^2 C-B c d+A d^2\right ) \text {arctanh}\left (\frac {a d-c^2 x^2}{\sqrt {c^3+a d^2} \sqrt {a+c x^4}}\right )}{2 c^4 \sqrt {c^3+a d^2}}+\frac {\left (B c^4-A c^3 d+2 a c^2 C d-2 a B c d^2+2 a A d^3\right ) \text {arctanh}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )}{4 a^{3/2} c^4} \] Output:

-1/6*A*(c*x^4+a)^(1/2)/a/c/x^6-1/4*(-A*d+B*c)*(c*x^4+a)^(1/2)/a/c^2/x^4-1/ 
6*(3*a*c*(-B*d+C*c)-A*(-3*a*d^2+2*c^3))*(c*x^4+a)^(1/2)/a^2/c^3/x^2-1/2*d^ 
2*(A*d^2-B*c*d+C*c^2)*arctanh((-c^2*x^2+a*d)/(a*d^2+c^3)^(1/2)/(c*x^4+a)^( 
1/2))/c^4/(a*d^2+c^3)^(1/2)+1/4*(2*A*a*d^3-A*c^3*d-2*B*a*c*d^2+B*c^4+2*C*a 
*c^2*d)*arctanh((c*x^4+a)^(1/2)/a^(1/2))/a^(3/2)/c^4
 

Mathematica [A] (verified)

Time = 1.71 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x^2+C x^4}{x^7 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\frac {\frac {c \sqrt {a+c x^4} \left (4 A c^3 x^4-a \left (3 c x^2 \left (B c+2 c C x^2-2 B d x^2\right )+A \left (2 c^2-3 c d x^2+6 d^2 x^4\right )\right )\right )}{a^2 x^6}-\frac {12 d^2 \left (c^2 C-B c d+A d^2\right ) \arctan \left (\frac {c^{3/2}+\sqrt {c} d x^2-d \sqrt {a+c x^4}}{\sqrt {-c^3-a d^2}}\right )}{\sqrt {-c^3-a d^2}}+\frac {6 \left (-B c^4+A c^3 d-2 a c^2 C d+2 a B c d^2-2 a A d^3\right ) \text {arctanh}\left (\frac {\sqrt {c} x^2-\sqrt {a+c x^4}}{\sqrt {a}}\right )}{a^{3/2}}}{12 c^4} \] Input:

Integrate[(A + B*x^2 + C*x^4)/(x^7*(c + d*x^2)*Sqrt[a + c*x^4]),x]
 

Output:

((c*Sqrt[a + c*x^4]*(4*A*c^3*x^4 - a*(3*c*x^2*(B*c + 2*c*C*x^2 - 2*B*d*x^2 
) + A*(2*c^2 - 3*c*d*x^2 + 6*d^2*x^4))))/(a^2*x^6) - (12*d^2*(c^2*C - B*c* 
d + A*d^2)*ArcTan[(c^(3/2) + Sqrt[c]*d*x^2 - d*Sqrt[a + c*x^4])/Sqrt[-c^3 
- a*d^2]])/Sqrt[-c^3 - a*d^2] + (6*(-(B*c^4) + A*c^3*d - 2*a*c^2*C*d + 2*a 
*B*c*d^2 - 2*a*A*d^3)*ArcTanh[(Sqrt[c]*x^2 - Sqrt[a + c*x^4])/Sqrt[a]])/a^ 
(3/2))/(12*c^4)
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2249, 2009}

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.

\(\displaystyle \int \frac {A+B x^2+C x^4}{x^7 \sqrt {a+c x^4} \left (c+d x^2\right )} \, dx\)

\(\Big \downarrow \) 2249

\(\displaystyle \int \left (\frac {B c-A d}{c^2 x^5 \sqrt {a+c x^4}}-\frac {d \left (A d^2-B c d+c^2 C\right )}{c^4 x \sqrt {a+c x^4}}+\frac {d^2 x \left (A d^2-B c d+c^2 C\right )}{c^4 \sqrt {a+c x^4} \left (c+d x^2\right )}+\frac {A d^2-B c d+c^2 C}{c^3 x^3 \sqrt {a+c x^4}}+\frac {A}{c x^7 \sqrt {a+c x^4}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\text {arctanh}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right ) (B c-A d)}{4 a^{3/2} c}+\frac {A \sqrt {a+c x^4}}{3 a^2 x^2}+\frac {d \text {arctanh}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right ) \left (A d^2-B c d+c^2 C\right )}{2 \sqrt {a} c^4}-\frac {d^2 \text {arctanh}\left (\frac {a d-c^2 x^2}{\sqrt {a d^2+c^3} \sqrt {a+c x^4}}\right ) \left (A d^2-B c d+c^2 C\right )}{2 c^4 \sqrt {a d^2+c^3}}-\frac {\sqrt {a+c x^4} (B c-A d)}{4 a c^2 x^4}-\frac {\sqrt {a+c x^4} \left (A d^2-B c d+c^2 C\right )}{2 a c^3 x^2}-\frac {A \sqrt {a+c x^4}}{6 a c x^6}\)

Input:

Int[(A + B*x^2 + C*x^4)/(x^7*(c + d*x^2)*Sqrt[a + c*x^4]),x]
 

Output:

-1/6*(A*Sqrt[a + c*x^4])/(a*c*x^6) - ((B*c - A*d)*Sqrt[a + c*x^4])/(4*a*c^ 
2*x^4) + (A*Sqrt[a + c*x^4])/(3*a^2*x^2) - ((c^2*C - B*c*d + A*d^2)*Sqrt[a 
 + c*x^4])/(2*a*c^3*x^2) - (d^2*(c^2*C - B*c*d + A*d^2)*ArcTanh[(a*d - c^2 
*x^2)/(Sqrt[c^3 + a*d^2]*Sqrt[a + c*x^4])])/(2*c^4*Sqrt[c^3 + a*d^2]) + (( 
B*c - A*d)*ArcTanh[Sqrt[a + c*x^4]/Sqrt[a]])/(4*a^(3/2)*c) + (d*(c^2*C - B 
*c*d + A*d^2)*ArcTanh[Sqrt[a + c*x^4]/Sqrt[a]])/(2*Sqrt[a]*c^4)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2249

Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_) 
^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + c*x^4], Px*(f*x)^m*(d 
 + e*x^2)^q*(a + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, c, d, e, f, m}, x] & 
& PolyQ[Px, x] && IntegerQ[p + 1/2] && IntegerQ[q]
 

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.22

method	result	size
risch	\(-\frac {\sqrt {c \,x^{4}+a}\, \left (6 A a \,d^{2} x^{4}-4 A \,c^{3} x^{4}-6 B a c d \,x^{4}+6 C a \,c^{2} x^{4}-3 A a c d \,x^{2}+3 B a \,c^{2} x^{2}+2 A a \,c^{2}\right )}{12 c^{3} a^{2} x^{6}}+\frac {\frac {\left (2 A a \,d^{3}-A \,c^{3} d -2 B a c \,d^{2}+B \,c^{4}+2 C a \,c^{2} d \right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{4}+a}}{x^{2}}\right )}{2 c \sqrt {a}}-\frac {d a \left (A \,d^{2}-B c d +C \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 c^{3}}{d^{2}}-\frac {2 c^{2} \left (x^{2}+\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+c^{3}}{d^{2}}}\, \sqrt {\left (x^{2}+\frac {c}{d}\right )^{2} c -\frac {2 c^{2} \left (x^{2}+\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+c^{3}}{d^{2}}}}{x^{2}+\frac {c}{d}}\right )}{c \sqrt {\frac {a \,d^{2}+c^{3}}{d^{2}}}}}{2 c^{3} a}\)	\(309\)
default	\(-\frac {A \sqrt {c \,x^{4}+a}\, \left (-2 c \,x^{4}+a \right )}{6 c \,x^{6} a^{2}}-\frac {\left (A d -B c \right ) \left (-\frac {\sqrt {c \,x^{4}+a}}{4 a \,x^{4}}+\frac {c \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{4}+a}}{x^{2}}\right )}{4 a^{\frac {3}{2}}}\right )}{c^{2}}-\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) \sqrt {c \,x^{4}+a}}{2 c^{3} x^{2} a}-\frac {d \left (A \,d^{2}-B c d +C \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 c^{3}}{d^{2}}-\frac {2 c^{2} \left (x^{2}+\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+c^{3}}{d^{2}}}\, \sqrt {\left (x^{2}+\frac {c}{d}\right )^{2} c -\frac {2 c^{2} \left (x^{2}+\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+c^{3}}{d^{2}}}}{x^{2}+\frac {c}{d}}\right )}{2 c^{4} \sqrt {\frac {a \,d^{2}+c^{3}}{d^{2}}}}+\frac {d \left (A \,d^{2}-B c d +C \,c^{2}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{4}+a}}{x^{2}}\right )}{2 c^{4} \sqrt {a}}\)	\(320\)
elliptic	\(\frac {A \left (-\frac {\sqrt {c \,x^{4}+a}}{3 a \,x^{6}}+\frac {2 c \sqrt {c \,x^{4}+a}}{3 a^{2} x^{2}}\right )}{2 c}-\frac {\left (A d -B c \right ) \left (-\frac {\sqrt {c \,x^{4}+a}}{2 a \,x^{4}}+\frac {c \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{4}+a}}{x^{2}}\right )}{2 a^{\frac {3}{2}}}\right )}{2 c^{2}}-\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) \sqrt {c \,x^{4}+a}}{2 c^{3} x^{2} a}-\frac {d \left (A \,d^{2}-B c d +C \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 c^{3}}{d^{2}}-\frac {2 c^{2} \left (x^{2}+\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+c^{3}}{d^{2}}}\, \sqrt {\left (x^{2}+\frac {c}{d}\right )^{2} c -\frac {2 c^{2} \left (x^{2}+\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+c^{3}}{d^{2}}}}{x^{2}+\frac {c}{d}}\right )}{2 c^{4} \sqrt {\frac {a \,d^{2}+c^{3}}{d^{2}}}}+\frac {d \left (A \,d^{2}-B c d +C \,c^{2}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{4}+a}}{x^{2}}\right )}{2 c^{4} \sqrt {a}}\)	\(333\)

Input:

int((C*x^4+B*x^2+A)/x^7/(d*x^2+c)/(c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/12*(c*x^4+a)^(1/2)*(6*A*a*d^2*x^4-4*A*c^3*x^4-6*B*a*c*d*x^4+6*C*a*c^2*x 
^4-3*A*a*c*d*x^2+3*B*a*c^2*x^2+2*A*a*c^2)/c^3/a^2/x^6+1/2/c^3/a*(1/2*(2*A* 
a*d^3-A*c^3*d-2*B*a*c*d^2+B*c^4+2*C*a*c^2*d)/c/a^(1/2)*ln((2*a+2*a^(1/2)*( 
c*x^4+a)^(1/2))/x^2)-d*a*(A*d^2-B*c*d+C*c^2)/c/((a*d^2+c^3)/d^2)^(1/2)*ln( 
(2*(a*d^2+c^3)/d^2-2*c^2/d*(x^2+c/d)+2*((a*d^2+c^3)/d^2)^(1/2)*((x^2+c/d)^ 
2*c-2*c^2/d*(x^2+c/d)+(a*d^2+c^3)/d^2)^(1/2))/(x^2+c/d)))
 

Fricas [A] (verification not implemented)

Time = 2.33 (sec) , antiderivative size = 1609, normalized size of antiderivative = 6.36 \[ \int \frac {A+B x^2+C x^4}{x^7 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\text {Too large to display} \] Input:

integrate((C*x^4+B*x^2+A)/x^7/(d*x^2+c)/(c*x^4+a)^(1/2),x, algorithm="fric 
as")
 

Output:

[1/24*(6*(C*a^2*c^2*d^2 - B*a^2*c*d^3 + A*a^2*d^4)*sqrt(c^3 + a*d^2)*x^6*l 
og((2*a*c^2*d*x^2 - (2*c^4 + a*c*d^2)*x^4 - a*c^3 - 2*a^2*d^2 - 2*sqrt(c*x 
^4 + a)*(c^2*x^2 - a*d)*sqrt(c^3 + a*d^2))/(d^2*x^4 + 2*c*d*x^2 + c^2)) + 
3*(B*c^7 - B*a*c^4*d^2 - 2*B*a^2*c*d^4 + 2*A*a^2*d^5 + (2*C*a^2*c^2 + A*a* 
c^3)*d^3 + (2*C*a*c^5 - A*c^6)*d)*sqrt(a)*x^6*log(-(c*x^4 + 2*sqrt(c*x^4 + 
 a)*sqrt(a) + 2*a)/x^4) - 2*(2*A*a*c^6 + 2*A*a^2*c^3*d^2 + 2*(3*C*a*c^6 - 
2*A*c^7 - 3*B*a*c^5*d - 3*B*a^2*c^2*d^3 + 3*A*a^2*c*d^4 + (3*C*a^2*c^3 + A 
*a*c^4)*d^2)*x^4 + 3*(B*a*c^6 - A*a*c^5*d + B*a^2*c^3*d^2 - A*a^2*c^2*d^3) 
*x^2)*sqrt(c*x^4 + a))/((a^2*c^7 + a^3*c^4*d^2)*x^6), -1/24*(12*(C*a^2*c^2 
*d^2 - B*a^2*c*d^3 + A*a^2*d^4)*sqrt(-c^3 - a*d^2)*x^6*arctan(sqrt(c*x^4 + 
 a)*(c^2*x^2 - a*d)*sqrt(-c^3 - a*d^2)/((c^4 + a*c*d^2)*x^4 + a*c^3 + a^2* 
d^2)) - 3*(B*c^7 - B*a*c^4*d^2 - 2*B*a^2*c*d^4 + 2*A*a^2*d^5 + (2*C*a^2*c^ 
2 + A*a*c^3)*d^3 + (2*C*a*c^5 - A*c^6)*d)*sqrt(a)*x^6*log(-(c*x^4 + 2*sqrt 
(c*x^4 + a)*sqrt(a) + 2*a)/x^4) + 2*(2*A*a*c^6 + 2*A*a^2*c^3*d^2 + 2*(3*C* 
a*c^6 - 2*A*c^7 - 3*B*a*c^5*d - 3*B*a^2*c^2*d^3 + 3*A*a^2*c*d^4 + (3*C*a^2 
*c^3 + A*a*c^4)*d^2)*x^4 + 3*(B*a*c^6 - A*a*c^5*d + B*a^2*c^3*d^2 - A*a^2* 
c^2*d^3)*x^2)*sqrt(c*x^4 + a))/((a^2*c^7 + a^3*c^4*d^2)*x^6), -1/12*(3*(B* 
c^7 - B*a*c^4*d^2 - 2*B*a^2*c*d^4 + 2*A*a^2*d^5 + (2*C*a^2*c^2 + A*a*c^3)* 
d^3 + (2*C*a*c^5 - A*c^6)*d)*sqrt(-a)*x^6*arctan(sqrt(c*x^4 + a)*sqrt(-a)/ 
a) - 3*(C*a^2*c^2*d^2 - B*a^2*c*d^3 + A*a^2*d^4)*sqrt(c^3 + a*d^2)*x^6*...
 

Sympy [F]

\[ \int \frac {A+B x^2+C x^4}{x^7 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\int \frac {A + B x^{2} + C x^{4}}{x^{7} \sqrt {a + c x^{4}} \left (c + d x^{2}\right )}\, dx \] Input:

integrate((C*x**4+B*x**2+A)/x**7/(d*x**2+c)/(c*x**4+a)**(1/2),x)
 

Output:

Integral((A + B*x**2 + C*x**4)/(x**7*sqrt(a + c*x**4)*(c + d*x**2)), x)
 

Maxima [F]

\[ \int \frac {A+B x^2+C x^4}{x^7 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {c x^{4} + a} {\left (d x^{2} + c\right )} x^{7}} \,d x } \] Input:

integrate((C*x^4+B*x^2+A)/x^7/(d*x^2+c)/(c*x^4+a)^(1/2),x, algorithm="maxi 
ma")
 

Output:

integrate((C*x^4 + B*x^2 + A)/(sqrt(c*x^4 + a)*(d*x^2 + c)*x^7), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 556 vs. \(2 (226) = 452\).

Time = 0.21 (sec) , antiderivative size = 556, normalized size of antiderivative = 2.20 \[ \int \frac {A+B x^2+C x^4}{x^7 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\frac {{\left (C c^{2} d^{2} - B c d^{3} + A d^{4}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )} d + c^{\frac {3}{2}}}{\sqrt {-c^{3} - a d^{2}}}\right )}{\sqrt {-c^{3} - a d^{2}} c^{4}} - \frac {{\left (B c^{4} + 2 \, C a c^{2} d - A c^{3} d - 2 \, B a c d^{2} + 2 \, A a d^{3}\right )} \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a} a c^{4}} + \frac {3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{5} B c^{3} - 3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{5} A c^{2} d + 6 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{4} C a c^{\frac {5}{2}} - 6 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{4} B a c^{\frac {3}{2}} d + 6 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{4} A a \sqrt {c} d^{2} - 12 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} C a^{2} c^{\frac {5}{2}} + 12 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} A a c^{\frac {7}{2}} + 12 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} B a^{2} c^{\frac {3}{2}} d - 12 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} A a^{2} \sqrt {c} d^{2} - 3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )} B a^{2} c^{3} + 3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )} A a^{2} c^{2} d + 6 \, C a^{3} c^{\frac {5}{2}} - 4 \, A a^{2} c^{\frac {7}{2}} - 6 \, B a^{3} c^{\frac {3}{2}} d + 6 \, A a^{3} \sqrt {c} d^{2}}{6 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} - a\right )}^{3} a c^{3}} \] Input:

integrate((C*x^4+B*x^2+A)/x^7/(d*x^2+c)/(c*x^4+a)^(1/2),x, algorithm="giac 
")
 

Output:

(C*c^2*d^2 - B*c*d^3 + A*d^4)*arctan(-((sqrt(c)*x^2 - sqrt(c*x^4 + a))*d + 
 c^(3/2))/sqrt(-c^3 - a*d^2))/(sqrt(-c^3 - a*d^2)*c^4) - 1/2*(B*c^4 + 2*C* 
a*c^2*d - A*c^3*d - 2*B*a*c*d^2 + 2*A*a*d^3)*arctan(-(sqrt(c)*x^2 - sqrt(c 
*x^4 + a))/sqrt(-a))/(sqrt(-a)*a*c^4) + 1/6*(3*(sqrt(c)*x^2 - sqrt(c*x^4 + 
 a))^5*B*c^3 - 3*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^5*A*c^2*d + 6*(sqrt(c)*x^ 
2 - sqrt(c*x^4 + a))^4*C*a*c^(5/2) - 6*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^4*B 
*a*c^(3/2)*d + 6*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^4*A*a*sqrt(c)*d^2 - 12*(s 
qrt(c)*x^2 - sqrt(c*x^4 + a))^2*C*a^2*c^(5/2) + 12*(sqrt(c)*x^2 - sqrt(c*x 
^4 + a))^2*A*a*c^(7/2) + 12*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^2*B*a^2*c^(3/2 
)*d - 12*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^2*A*a^2*sqrt(c)*d^2 - 3*(sqrt(c)* 
x^2 - sqrt(c*x^4 + a))*B*a^2*c^3 + 3*(sqrt(c)*x^2 - sqrt(c*x^4 + a))*A*a^2 
*c^2*d + 6*C*a^3*c^(5/2) - 4*A*a^2*c^(7/2) - 6*B*a^3*c^(3/2)*d + 6*A*a^3*s 
qrt(c)*d^2)/(((sqrt(c)*x^2 - sqrt(c*x^4 + a))^2 - a)^3*a*c^3)
 

Mupad [F(-1)]

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

int((A + B*x^2 + C*x^4)/(x^7*(a + c*x^4)^(1/2)*(c + d*x^2)),x)
 

Output:

int((A + B*x^2 + C*x^4)/(x^7*(a + c*x^4)^(1/2)*(c + d*x^2)), x)
 

Reduce [F]

\[ \int \frac {A+B x^2+C x^4}{x^7 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\left (\int \frac {1}{\sqrt {c \,x^{4}+a}\, c \,x^{7}+\sqrt {c \,x^{4}+a}\, d \,x^{9}}d x \right ) a +\left (\int \frac {1}{\sqrt {c \,x^{4}+a}\, c \,x^{5}+\sqrt {c \,x^{4}+a}\, d \,x^{7}}d x \right ) b +\left (\int \frac {1}{\sqrt {c \,x^{4}+a}\, c \,x^{3}+\sqrt {c \,x^{4}+a}\, d \,x^{5}}d x \right ) c \] Input:

int((C*x^4+B*x^2+A)/x^7/(d*x^2+c)/(c*x^4+a)^(1/2),x)
 

Output:

int(1/(sqrt(a + c*x**4)*c*x**7 + sqrt(a + c*x**4)*d*x**9),x)*a + int(1/(sq 
rt(a + c*x**4)*c*x**5 + sqrt(a + c*x**4)*d*x**7),x)*b + int(1/(sqrt(a + c* 
x**4)*c*x**3 + sqrt(a + c*x**4)*d*x**5),x)*c