\(\int \frac {1}{x^7 (a+b x^4) (c+d x^4)} \, dx\) [205]

Optimal result

Integrand size = 22, antiderivative size = 112 \[ \int \frac {1}{x^7 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=-\frac {1}{6 a c x^6}+\frac {b c+a d}{2 a^2 c^2 x^2}+\frac {b^{5/2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 a^{5/2} (b c-a d)}-\frac {d^{5/2} \arctan \left (\frac {\sqrt {d} x^2}{\sqrt {c}}\right )}{2 c^{5/2} (b c-a d)} \] Output:

-1/6/a/c/x^6+1/2*(a*d+b*c)/a^2/c^2/x^2+1/2*b^(5/2)*arctan(b^(1/2)*x^2/a^(1 
/2))/a^(5/2)/(-a*d+b*c)-1/2*d^(5/2)*arctan(d^(1/2)*x^2/c^(1/2))/c^(5/2)/(- 
a*d+b*c)
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.72 \[ \int \frac {1}{x^7 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {\frac {b}{a}-\frac {d}{c}-\frac {3 b^2 x^4}{a^2}+\frac {3 d^2 x^4}{c^2}+\frac {3 b^{5/2} x^6 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{5/2}}+\frac {3 b^{5/2} x^6 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{5/2}}-\frac {3 d^{5/2} x^6 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{c^{5/2}}-\frac {3 d^{5/2} x^6 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{c^{5/2}}}{6 (-b c+a d) x^6} \] Input:

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

Output:

(b/a - d/c - (3*b^2*x^4)/a^2 + (3*d^2*x^4)/c^2 + (3*b^(5/2)*x^6*ArcTan[1 - 
 (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(5/2) + (3*b^(5/2)*x^6*ArcTan[1 + (Sqrt[2 
]*b^(1/4)*x)/a^(1/4)])/a^(5/2) - (3*d^(5/2)*x^6*ArcTan[1 - (Sqrt[2]*d^(1/4 
)*x)/c^(1/4)])/c^(5/2) - (3*d^(5/2)*x^6*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^( 
1/4)])/c^(5/2))/(6*(-(b*c) + a*d)*x^6)
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.19, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {965, 382, 27, 445, 397, 218}

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.

\(\displaystyle \int \frac {1}{x^7 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx\)

\(\Big \downarrow \) 965

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

\(\Big \downarrow \) 382

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

(-1/3*1/(a*c*x^6) - (-((b*c + a*d)/(a*c*x^2)) - ((b^(5/2)*c^2*ArcTan[(Sqrt 
[b]*x^2)/Sqrt[a]])/(Sqrt[a]*(b*c - a*d)) - (a^2*d^(5/2)*ArcTan[(Sqrt[d]*x^ 
2)/Sqrt[c]])/(Sqrt[c]*(b*c - a*d)))/(a*c))/(a*c))/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 218

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

rule 382

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) 
, x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/ 
(a*c*e*(m + 1))), x] - Simp[1/(a*c*e^2*(m + 1))   Int[(e*x)^(m + 2)*(a + b* 
x^2)^p*(c + d*x^2)^q*Simp[(b*c + a*d)*(m + 3) + 2*(b*c*p + a*d*q) + b*d*(m 
+ 2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[ 
b*c - a*d, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
 

rule 397

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ 
Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^2), x], x] - Simp[ 
(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e 
, f}, x]
 

rule 445

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

rule 965

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), 
 x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k   Subst[Int[x^((m + 1)/k - 
 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /; Free 
Q[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]
 

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.90

method	result	size
default	\(-\frac {b^{3} \arctan \left (\frac {b \,x^{2}}{\sqrt {a b}}\right )}{2 a^{2} \left (a d -c b \right ) \sqrt {a b}}-\frac {1}{6 a c \,x^{6}}-\frac {-a d -c b}{2 a^{2} c^{2} x^{2}}+\frac {d^{3} \arctan \left (\frac {d \,x^{2}}{\sqrt {c d}}\right )}{2 c^{2} \left (a d -c b \right ) \sqrt {c d}}\)	\(101\)
risch	\(\frac {\frac {\left (a d +c b \right ) x^{4}}{2 a^{2} c^{2}}-\frac {1}{6 a c}}{x^{6}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (d^{2} c^{5} a^{2}-2 c^{6} d a b +b^{2} c^{7}\right ) \textit {\_Z}^{2}+d^{5}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (5 c^{5} a^{9} d^{4}-18 c^{6} a^{8} b \,d^{3}+26 c^{7} a^{7} b^{2} d^{2}-18 c^{8} a^{6} b^{3} d +5 c^{9} a^{5} b^{4}\right ) \textit {\_R}^{4}+\left (4 a^{7} d^{7}-8 c \,a^{6} b \,d^{6}+5 c^{2} a^{5} b^{2} d^{5}+5 c^{5} a^{2} b^{5} d^{2}-8 c^{6} a \,b^{6} d +4 c^{7} b^{7}\right ) \textit {\_R}^{2}+4 b^{5} d^{5}\right ) x^{2}+\left (-a^{8} c^{3} d^{5}+a^{7} b \,c^{4} d^{4}+a^{4} b^{4} c^{7} d -a^{3} b^{5} c^{8}\right ) \textit {\_R}^{3}+a^{2} b^{4} c^{2} d^{4} \textit {\_R} \right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (d^{2} a^{7}-2 a^{6} b c d +c^{2} a^{5} b^{2}\right ) \textit {\_Z}^{2}+b^{5}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (5 c^{5} a^{9} d^{4}-18 c^{6} a^{8} b \,d^{3}+26 c^{7} a^{7} b^{2} d^{2}-18 c^{8} a^{6} b^{3} d +5 c^{9} a^{5} b^{4}\right ) \textit {\_R}^{4}+\left (4 a^{7} d^{7}-8 c \,a^{6} b \,d^{6}+5 c^{2} a^{5} b^{2} d^{5}+5 c^{5} a^{2} b^{5} d^{2}-8 c^{6} a \,b^{6} d +4 c^{7} b^{7}\right ) \textit {\_R}^{2}+4 b^{5} d^{5}\right ) x^{2}+\left (-a^{8} c^{3} d^{5}+a^{7} b \,c^{4} d^{4}+a^{4} b^{4} c^{7} d -a^{3} b^{5} c^{8}\right ) \textit {\_R}^{3}+a^{2} b^{4} c^{2} d^{4} \textit {\_R} \right )\right )}{4}\)	\(541\)

Input:

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

Output:

-1/2*b^3/a^2/(a*d-b*c)/(a*b)^(1/2)*arctan(b*x^2/(a*b)^(1/2))-1/6/a/c/x^6-1 
/2/a^2/c^2*(-a*d-b*c)/x^2+1/2*d^3/c^2/(a*d-b*c)/(c*d)^(1/2)*arctan(d*x^2/( 
c*d)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 1.82 (sec) , antiderivative size = 576, normalized size of antiderivative = 5.14 \[ \int \frac {1}{x^7 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\left [-\frac {3 \, b^{2} c^{2} x^{6} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{4} - 2 \, a x^{2} \sqrt {-\frac {b}{a}} - a}{b x^{4} + a}\right ) + 3 \, a^{2} d^{2} x^{6} \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{4} + 2 \, c x^{2} \sqrt {-\frac {d}{c}} - c}{d x^{4} + c}\right ) - 6 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{4} + 2 \, a b c^{2} - 2 \, a^{2} c d}{12 \, {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{6}}, -\frac {6 \, a^{2} d^{2} x^{6} \sqrt {\frac {d}{c}} \arctan \left (x^{2} \sqrt {\frac {d}{c}}\right ) + 3 \, b^{2} c^{2} x^{6} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{4} - 2 \, a x^{2} \sqrt {-\frac {b}{a}} - a}{b x^{4} + a}\right ) - 6 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{4} + 2 \, a b c^{2} - 2 \, a^{2} c d}{12 \, {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{6}}, \frac {6 \, b^{2} c^{2} x^{6} \sqrt {\frac {b}{a}} \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right ) - 3 \, a^{2} d^{2} x^{6} \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{4} + 2 \, c x^{2} \sqrt {-\frac {d}{c}} - c}{d x^{4} + c}\right ) + 6 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{4} - 2 \, a b c^{2} + 2 \, a^{2} c d}{12 \, {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{6}}, \frac {3 \, b^{2} c^{2} x^{6} \sqrt {\frac {b}{a}} \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right ) - 3 \, a^{2} d^{2} x^{6} \sqrt {\frac {d}{c}} \arctan \left (x^{2} \sqrt {\frac {d}{c}}\right ) + 3 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{4} - a b c^{2} + a^{2} c d}{6 \, {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{6}}\right ] \] Input:

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

Output:

[-1/12*(3*b^2*c^2*x^6*sqrt(-b/a)*log((b*x^4 - 2*a*x^2*sqrt(-b/a) - a)/(b*x 
^4 + a)) + 3*a^2*d^2*x^6*sqrt(-d/c)*log((d*x^4 + 2*c*x^2*sqrt(-d/c) - c)/( 
d*x^4 + c)) - 6*(b^2*c^2 - a^2*d^2)*x^4 + 2*a*b*c^2 - 2*a^2*c*d)/((a^2*b*c 
^3 - a^3*c^2*d)*x^6), -1/12*(6*a^2*d^2*x^6*sqrt(d/c)*arctan(x^2*sqrt(d/c)) 
 + 3*b^2*c^2*x^6*sqrt(-b/a)*log((b*x^4 - 2*a*x^2*sqrt(-b/a) - a)/(b*x^4 + 
a)) - 6*(b^2*c^2 - a^2*d^2)*x^4 + 2*a*b*c^2 - 2*a^2*c*d)/((a^2*b*c^3 - a^3 
*c^2*d)*x^6), 1/12*(6*b^2*c^2*x^6*sqrt(b/a)*arctan(x^2*sqrt(b/a)) - 3*a^2* 
d^2*x^6*sqrt(-d/c)*log((d*x^4 + 2*c*x^2*sqrt(-d/c) - c)/(d*x^4 + c)) + 6*( 
b^2*c^2 - a^2*d^2)*x^4 - 2*a*b*c^2 + 2*a^2*c*d)/((a^2*b*c^3 - a^3*c^2*d)*x 
^6), 1/6*(3*b^2*c^2*x^6*sqrt(b/a)*arctan(x^2*sqrt(b/a)) - 3*a^2*d^2*x^6*sq 
rt(d/c)*arctan(x^2*sqrt(d/c)) + 3*(b^2*c^2 - a^2*d^2)*x^4 - a*b*c^2 + a^2* 
c*d)/((a^2*b*c^3 - a^3*c^2*d)*x^6)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^7 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^7 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {b^{3} \arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, {\left (a^{2} b c - a^{3} d\right )} \sqrt {a b}} - \frac {d^{3} \arctan \left (\frac {d x^{2}}{\sqrt {c d}}\right )}{2 \, {\left (b c^{3} - a c^{2} d\right )} \sqrt {c d}} + \frac {3 \, {\left (b c + a d\right )} x^{4} - a c}{6 \, a^{2} c^{2} x^{6}} \] Input:

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

Output:

1/2*b^3*arctan(b*x^2/sqrt(a*b))/((a^2*b*c - a^3*d)*sqrt(a*b)) - 1/2*d^3*ar 
ctan(d*x^2/sqrt(c*d))/((b*c^3 - a*c^2*d)*sqrt(c*d)) + 1/6*(3*(b*c + a*d)*x 
^4 - a*c)/(a^2*c^2*x^6)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^7 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {b^{3} \arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, {\left (a^{2} b c - a^{3} d\right )} \sqrt {a b}} - \frac {d^{3} \arctan \left (\frac {d x^{2}}{\sqrt {c d}}\right )}{2 \, {\left (b c^{3} - a c^{2} d\right )} \sqrt {c d}} + \frac {3 \, b c x^{4} + 3 \, a d x^{4} - a c}{6 \, a^{2} c^{2} x^{6}} \] Input:

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

Output:

1/2*b^3*arctan(b*x^2/sqrt(a*b))/((a^2*b*c - a^3*d)*sqrt(a*b)) - 1/2*d^3*ar 
ctan(d*x^2/sqrt(c*d))/((b*c^3 - a*c^2*d)*sqrt(c*d)) + 1/6*(3*b*c*x^4 + 3*a 
*d*x^4 - a*c)/(a^2*c^2*x^6)
 

Mupad [B] (verification not implemented)

Time = 4.97 (sec) , antiderivative size = 535, normalized size of antiderivative = 4.78 \[ \int \frac {1}{x^7 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {\ln \left (c^{10}\,{\left (-a^5\,b^5\right )}^{5/2}+a^{20}\,d^{10}\,\sqrt {-a^5\,b^5}-a^{12}\,b^{13}\,c^{10}\,x^2-a^{22}\,b^3\,d^{10}\,x^2+2\,a^{10}\,c^5\,d^5\,{\left (-a^5\,b^5\right )}^{3/2}+2\,a^{17}\,b^8\,c^5\,d^5\,x^2\right )\,\sqrt {-a^5\,b^5}}{4\,a^6\,d-4\,a^5\,b\,c}-\frac {\ln \left (c^{10}\,{\left (-a^5\,b^5\right )}^{5/2}+a^{20}\,d^{10}\,\sqrt {-a^5\,b^5}+a^{12}\,b^{13}\,c^{10}\,x^2+a^{22}\,b^3\,d^{10}\,x^2+2\,a^{10}\,c^5\,d^5\,{\left (-a^5\,b^5\right )}^{3/2}-2\,a^{17}\,b^8\,c^5\,d^5\,x^2\right )\,\sqrt {-a^5\,b^5}}{4\,\left (a^6\,d-a^5\,b\,c\right )}-\frac {\frac {1}{6\,a\,c}-\frac {x^4\,\left (a\,d+b\,c\right )}{2\,a^2\,c^2}}{x^6}-\frac {\ln \left (a^{10}\,{\left (-c^5\,d^5\right )}^{5/2}+b^{10}\,c^{20}\,\sqrt {-c^5\,d^5}+a^{10}\,c^{12}\,d^{13}\,x^2+b^{10}\,c^{22}\,d^3\,x^2+2\,a^5\,b^5\,c^{10}\,{\left (-c^5\,d^5\right )}^{3/2}-2\,a^5\,b^5\,c^{17}\,d^8\,x^2\right )\,\sqrt {-c^5\,d^5}}{4\,\left (b\,c^6-a\,c^5\,d\right )}+\frac {\ln \left (a^{10}\,{\left (-c^5\,d^5\right )}^{5/2}+b^{10}\,c^{20}\,\sqrt {-c^5\,d^5}-a^{10}\,c^{12}\,d^{13}\,x^2-b^{10}\,c^{22}\,d^3\,x^2+2\,a^5\,b^5\,c^{10}\,{\left (-c^5\,d^5\right )}^{3/2}+2\,a^5\,b^5\,c^{17}\,d^8\,x^2\right )\,\sqrt {-c^5\,d^5}}{4\,b\,c^6-4\,a\,c^5\,d} \] Input:

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

Output:

(log(c^10*(-a^5*b^5)^(5/2) + a^20*d^10*(-a^5*b^5)^(1/2) - a^12*b^13*c^10*x 
^2 - a^22*b^3*d^10*x^2 + 2*a^10*c^5*d^5*(-a^5*b^5)^(3/2) + 2*a^17*b^8*c^5* 
d^5*x^2)*(-a^5*b^5)^(1/2))/(4*a^6*d - 4*a^5*b*c) - (log(c^10*(-a^5*b^5)^(5 
/2) + a^20*d^10*(-a^5*b^5)^(1/2) + a^12*b^13*c^10*x^2 + a^22*b^3*d^10*x^2 
+ 2*a^10*c^5*d^5*(-a^5*b^5)^(3/2) - 2*a^17*b^8*c^5*d^5*x^2)*(-a^5*b^5)^(1/ 
2))/(4*(a^6*d - a^5*b*c)) - (1/(6*a*c) - (x^4*(a*d + b*c))/(2*a^2*c^2))/x^ 
6 - (log(a^10*(-c^5*d^5)^(5/2) + b^10*c^20*(-c^5*d^5)^(1/2) + a^10*c^12*d^ 
13*x^2 + b^10*c^22*d^3*x^2 + 2*a^5*b^5*c^10*(-c^5*d^5)^(3/2) - 2*a^5*b^5*c 
^17*d^8*x^2)*(-c^5*d^5)^(1/2))/(4*(b*c^6 - a*c^5*d)) + (log(a^10*(-c^5*d^5 
)^(5/2) + b^10*c^20*(-c^5*d^5)^(1/2) - a^10*c^12*d^13*x^2 - b^10*c^22*d^3* 
x^2 + 2*a^5*b^5*c^10*(-c^5*d^5)^(3/2) + 2*a^5*b^5*c^17*d^8*x^2)*(-c^5*d^5) 
^(1/2))/(4*b*c^6 - 4*a*c^5*d)
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.06 \[ \int \frac {1}{x^7 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx=\frac {3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {b}\, x}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) b^{2} c^{3} x^{6}+3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {b}\, x}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) b^{2} c^{3} x^{6}-3 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}-2 \sqrt {d}\, x}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right ) a^{3} d^{2} x^{6}-3 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}+2 \sqrt {d}\, x}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right ) a^{3} d^{2} x^{6}-a^{3} c^{2} d +3 a^{3} c \,d^{2} x^{4}+a^{2} b \,c^{3}-3 a \,b^{2} c^{3} x^{4}}{6 a^{3} c^{3} x^{6} \left (a d -b c \right )} \] Input:

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

Output:

(3*sqrt(b)*sqrt(a)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(b)*x)/(b**(1/4 
)*a**(1/4)*sqrt(2)))*b**2*c**3*x**6 + 3*sqrt(b)*sqrt(a)*atan((b**(1/4)*a** 
(1/4)*sqrt(2) + 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*b**2*c**3*x**6 - 
 3*sqrt(d)*sqrt(c)*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(d)*x)/(d**(1/4 
)*c**(1/4)*sqrt(2)))*a**3*d**2*x**6 - 3*sqrt(d)*sqrt(c)*atan((d**(1/4)*c** 
(1/4)*sqrt(2) + 2*sqrt(d)*x)/(d**(1/4)*c**(1/4)*sqrt(2)))*a**3*d**2*x**6 - 
 a**3*c**2*d + 3*a**3*c*d**2*x**4 + a**2*b*c**3 - 3*a*b**2*c**3*x**4)/(6*a 
**3*c**3*x**6*(a*d - b*c))