\(\int \frac {1}{(a+b x^4)^{9/4} (c+d x^4)^2} \, dx\) [120]

Optimal result

Integrand size = 21, antiderivative size = 266 \[ \int \frac {1}{\left (a+b x^4\right )^{9/4} \left (c+d x^4\right )^2} \, dx=\frac {b (4 b c+5 a d) x}{20 a c (b c-a d)^2 \left (a+b x^4\right )^{5/4}}+\frac {b \left (16 b^2 c^2-56 a b c d-5 a^2 d^2\right ) x}{20 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^4}}-\frac {d x}{4 c (b c-a d) \left (a+b x^4\right )^{5/4} \left (c+d x^4\right )}+\frac {3 d^2 (4 b c-a d) \arctan \left (\frac {\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{13/4}}+\frac {3 d^2 (4 b c-a d) \text {arctanh}\left (\frac {\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{13/4}} \] Output:

1/20*b*(5*a*d+4*b*c)*x/a/c/(-a*d+b*c)^2/(b*x^4+a)^(5/4)+1/20*b*(-5*a^2*d^2 
-56*a*b*c*d+16*b^2*c^2)*x/a^2/c/(-a*d+b*c)^3/(b*x^4+a)^(1/4)-1/4*d*x/c/(-a 
*d+b*c)/(b*x^4+a)^(5/4)/(d*x^4+c)+3/8*d^2*(-a*d+4*b*c)*arctan((-a*d+b*c)^( 
1/4)*x/c^(1/4)/(b*x^4+a)^(1/4))/c^(7/4)/(-a*d+b*c)^(13/4)+3/8*d^2*(-a*d+4* 
b*c)*arctanh((-a*d+b*c)^(1/4)*x/c^(1/4)/(b*x^4+a)^(1/4))/c^(7/4)/(-a*d+b*c 
)^(13/4)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 14.53 (sec) , antiderivative size = 1216, normalized size of antiderivative = 4.57 \[ \int \frac {1}{\left (a+b x^4\right )^{9/4} \left (c+d x^4\right )^2} \, dx =\text {Too large to display} \] Input:

Integrate[1/((a + b*x^4)^(9/4)*(c + d*x^4)^2),x]
 

Output:

-1/198900*(285532*c^5*(b*c - a*d)^2*x^8*(a + b*x^4)^2 + 933504*c^4*d*(b*c 
- a*d)^2*x^12*(a + b*x^4)^2 + 891072*c^3*d^2*(b*c - a*d)^2*x^16*(a + b*x^4 
)^2 + 282880*c^2*d^3*(b*c - a*d)^2*x^20*(a + b*x^4)^2 + 9793836*c^6*(b*c - 
 a*d)*x^4*(a + b*x^4)^3 + 27973296*c^5*d*(b*c - a*d)*x^8*(a + b*x^4)^3 + 2 
5968384*c^4*d^2*(b*c - a*d)*x^12*(a + b*x^4)^3 + 8146944*c^3*d^3*(b*c - a* 
d)*x^16*(a + b*x^4)^3 - 23529870*c^7*(a + b*x^4)^4 - 65547495*c^6*d*x^4*(a 
 + b*x^4)^4 - 60505380*c^5*d^2*x^8*(a + b*x^4)^4 - 18935280*c^4*d^3*x^12*( 
a + b*x^4)^4 - 14499810*c^6*(b*c - a*d)*x^4*(a + b*x^4)^3*Hypergeometric2F 
1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] - 41082795*c^5*d*(b*c - 
a*d)*x^8*(a + b*x^4)^3*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c 
*(a + b*x^4))] - 38069460*c^4*d^2*(b*c - a*d)*x^12*(a + b*x^4)^3*Hypergeom 
etric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] - 11934000*c^3*d^ 
3*(b*c - a*d)*x^16*(a + b*x^4)^3*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a* 
d)*x^4)/(c*(a + b*x^4))] + 23529870*c^7*(a + b*x^4)^4*Hypergeometric2F1[1/ 
4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 65547495*c^6*d*x^4*(a + b* 
x^4)^4*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 
 60505380*c^5*d^2*x^8*(a + b*x^4)^4*Hypergeometric2F1[1/4, 1, 5/4, ((b*c - 
 a*d)*x^4)/(c*(a + b*x^4))] + 18935280*c^4*d^3*x^12*(a + b*x^4)^4*Hypergeo 
metric2F1[1/4, 1, 5/4, ((b*c - a*d)*x^4)/(c*(a + b*x^4))] + 77760*c^3*(b*c 
 - a*d)^4*x^16*HypergeometricPFQ[{2, 2, 13/4}, {1, 21/4}, ((b*c - a*d)*...
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {931, 1024, 25, 1024, 27, 902, 756, 218, 221}

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.

\(\displaystyle \int \frac {1}{\left (a+b x^4\right )^{9/4} \left (c+d x^4\right )^2} \, dx\)

\(\Big \downarrow \) 931

\(\displaystyle \frac {\int \frac {-8 b d x^4+4 b c-3 a d}{\left (b x^4+a\right )^{9/4} \left (d x^4+c\right )}dx}{4 c (b c-a d)}-\frac {d x}{4 c \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) (b c-a d)}\)

\(\Big \downarrow \) 1024

\(\displaystyle \frac {\frac {b x (5 a d+4 b c)}{5 a \left (a+b x^4\right )^{5/4} (b c-a d)}-\frac {\int -\frac {4 b d (4 b c+5 a d) x^4+16 b^2 c^2+15 a^2 d^2-40 a b c d}{\left (b x^4+a\right )^{5/4} \left (d x^4+c\right )}dx}{5 a (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{4 c \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) (b c-a d)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {4 b d (4 b c+5 a d) x^4+16 b^2 c^2+15 a^2 d^2-40 a b c d}{\left (b x^4+a\right )^{5/4} \left (d x^4+c\right )}dx}{5 a (b c-a d)}+\frac {b x (5 a d+4 b c)}{5 a \left (a+b x^4\right )^{5/4} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{4 c \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) (b c-a d)}\)

\(\Big \downarrow \) 1024

\(\displaystyle \frac {\frac {\frac {b x \left (-5 a^2 d^2-56 a b c d+16 b^2 c^2\right )}{a \sqrt [4]{a+b x^4} (b c-a d)}-\frac {\int -\frac {15 a^2 d^2 (4 b c-a d)}{\sqrt [4]{b x^4+a} \left (d x^4+c\right )}dx}{a (b c-a d)}}{5 a (b c-a d)}+\frac {b x (5 a d+4 b c)}{5 a \left (a+b x^4\right )^{5/4} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{4 c \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {15 a d^2 (4 b c-a d) \int \frac {1}{\sqrt [4]{b x^4+a} \left (d x^4+c\right )}dx}{b c-a d}+\frac {b x \left (-5 a^2 d^2-56 a b c d+16 b^2 c^2\right )}{a \sqrt [4]{a+b x^4} (b c-a d)}}{5 a (b c-a d)}+\frac {b x (5 a d+4 b c)}{5 a \left (a+b x^4\right )^{5/4} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{4 c \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) (b c-a d)}\)

\(\Big \downarrow \) 902

\(\displaystyle \frac {\frac {\frac {15 a d^2 (4 b c-a d) \int \frac {1}{c-\frac {(b c-a d) x^4}{b x^4+a}}d\frac {x}{\sqrt [4]{b x^4+a}}}{b c-a d}+\frac {b x \left (-5 a^2 d^2-56 a b c d+16 b^2 c^2\right )}{a \sqrt [4]{a+b x^4} (b c-a d)}}{5 a (b c-a d)}+\frac {b x (5 a d+4 b c)}{5 a \left (a+b x^4\right )^{5/4} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{4 c \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) (b c-a d)}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {\frac {\frac {15 a d^2 (4 b c-a d) \left (\frac {\int \frac {1}{\sqrt {c}-\frac {\sqrt {b c-a d} x^2}{\sqrt {b x^4+a}}}d\frac {x}{\sqrt [4]{b x^4+a}}}{2 \sqrt {c}}+\frac {\int \frac {1}{\frac {\sqrt {b c-a d} x^2}{\sqrt {b x^4+a}}+\sqrt {c}}d\frac {x}{\sqrt [4]{b x^4+a}}}{2 \sqrt {c}}\right )}{b c-a d}+\frac {b x \left (-5 a^2 d^2-56 a b c d+16 b^2 c^2\right )}{a \sqrt [4]{a+b x^4} (b c-a d)}}{5 a (b c-a d)}+\frac {b x (5 a d+4 b c)}{5 a \left (a+b x^4\right )^{5/4} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{4 c \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) (b c-a d)}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {15 a d^2 (4 b c-a d) \left (\frac {\int \frac {1}{\sqrt {c}-\frac {\sqrt {b c-a d} x^2}{\sqrt {b x^4+a}}}d\frac {x}{\sqrt [4]{b x^4+a}}}{2 \sqrt {c}}+\frac {\arctan \left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} \sqrt [4]{b c-a d}}\right )}{b c-a d}+\frac {b x \left (-5 a^2 d^2-56 a b c d+16 b^2 c^2\right )}{a \sqrt [4]{a+b x^4} (b c-a d)}}{5 a (b c-a d)}+\frac {b x (5 a d+4 b c)}{5 a \left (a+b x^4\right )^{5/4} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{4 c \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {b x \left (-5 a^2 d^2-56 a b c d+16 b^2 c^2\right )}{a \sqrt [4]{a+b x^4} (b c-a d)}+\frac {15 a d^2 (4 b c-a d) \left (\frac {\arctan \left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} \sqrt [4]{b c-a d}}+\frac {\text {arctanh}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} \sqrt [4]{b c-a d}}\right )}{b c-a d}}{5 a (b c-a d)}+\frac {b x (5 a d+4 b c)}{5 a \left (a+b x^4\right )^{5/4} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{4 c \left (a+b x^4\right )^{5/4} \left (c+d x^4\right ) (b c-a d)}\)

Input:

Int[1/((a + b*x^4)^(9/4)*(c + d*x^4)^2),x]
 

Output:

-1/4*(d*x)/(c*(b*c - a*d)*(a + b*x^4)^(5/4)*(c + d*x^4)) + ((b*(4*b*c + 5* 
a*d)*x)/(5*a*(b*c - a*d)*(a + b*x^4)^(5/4)) + ((b*(16*b^2*c^2 - 56*a*b*c*d 
 - 5*a^2*d^2)*x)/(a*(b*c - a*d)*(a + b*x^4)^(1/4)) + (15*a*d^2*(4*b*c - a* 
d)*(ArcTan[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))]/(2*c^(3/4)*( 
b*c - a*d)^(1/4)) + ArcTanh[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/ 
4))]/(2*c^(3/4)*(b*c - a*d)^(1/4))))/(b*c - a*d))/(5*a*(b*c - a*d)))/(4*c* 
(b*c - a*d))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 221

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

rule 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 0]
 

rule 902

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Su 
bst[Int[1/(c - (b*c - a*d)*x^n), x], x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b 
, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
 

rule 931

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

rule 1024

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f 
_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c 
+ d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*( 
p + 1))   Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b 
*c - a*d)*(p + 1) + d*(b*e - a*f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; Fr 
eeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 1.86 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.43

method	result	size
pseudoelliptic	\(\frac {-\frac {3 \left (b \,x^{4}+a \right )^{\frac {5}{4}} a^{2} d^{2} \left (d \,x^{4}+c \right ) \left (a d -4 b c \right ) \left (\ln \left (\frac {-\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} \left (b \,x^{4}+a \right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a d -b c}{c}}\, x^{2}+\sqrt {b \,x^{4}+a}}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} \left (b \,x^{4}+a \right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a d -b c}{c}}\, x^{2}+\sqrt {b \,x^{4}+a}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} x}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} x}+1\right )\right ) \sqrt {2}}{8}+c \left (a^{4} d^{3}+2 x^{4} a^{3} b \,d^{3}+12 d \left (\frac {1}{12} d^{2} x^{8}+c d \,x^{4}+c^{2}\right ) b^{2} a^{2}-4 \left (d \,x^{4}+c \right ) c \,b^{3} \left (-\frac {14 d \,x^{4}}{5}+c \right ) a -\frac {16 b^{4} c^{2} x^{4} \left (d \,x^{4}+c \right )}{5}\right ) \left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} x}{4 \left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} \left (b \,x^{4}+a \right )^{\frac {5}{4}} c^{2} \left (d \,x^{4}+c \right ) \left (a d -b c \right )^{3} a^{2}}\)	\(380\)

Input:

int(1/(b*x^4+a)^(9/4)/(d*x^4+c)^2,x,method=_RETURNVERBOSE)
 

Output:

1/4/((a*d-b*c)/c)^(1/4)*(-3/8*(b*x^4+a)^(5/4)*a^2*d^2*(d*x^4+c)*(a*d-4*b*c 
)*(ln((-((a*d-b*c)/c)^(1/4)*(b*x^4+a)^(1/4)*2^(1/2)*x+((a*d-b*c)/c)^(1/2)* 
x^2+(b*x^4+a)^(1/2))/(((a*d-b*c)/c)^(1/4)*(b*x^4+a)^(1/4)*2^(1/2)*x+((a*d- 
b*c)/c)^(1/2)*x^2+(b*x^4+a)^(1/2)))+2*arctan(2^(1/2)/((a*d-b*c)/c)^(1/4)*( 
b*x^4+a)^(1/4)/x+1)-2*arctan(-2^(1/2)/((a*d-b*c)/c)^(1/4)*(b*x^4+a)^(1/4)/ 
x+1))*2^(1/2)+c*(a^4*d^3+2*x^4*a^3*b*d^3+12*d*(1/12*d^2*x^8+c*d*x^4+c^2)*b 
^2*a^2-4*(d*x^4+c)*c*b^3*(-14/5*d*x^4+c)*a-16/5*b^4*c^2*x^4*(d*x^4+c))*((a 
*d-b*c)/c)^(1/4)*x)/(b*x^4+a)^(5/4)/c^2/(d*x^4+c)/(a*d-b*c)^3/a^2
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^4\right )^{9/4} \left (c+d x^4\right )^2} \, dx=\text {Timed out} \] Input:

integrate(1/(b*x^4+a)^(9/4)/(d*x^4+c)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{\left (a+b x^4\right )^{9/4} \left (c+d x^4\right )^2} \, dx=\int \frac {1}{\left (a + b x^{4}\right )^{\frac {9}{4}} \left (c + d x^{4}\right )^{2}}\, dx \] Input:

integrate(1/(b*x**4+a)**(9/4)/(d*x**4+c)**2,x)
 

Output:

Integral(1/((a + b*x**4)**(9/4)*(c + d*x**4)**2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (a+b x^4\right )^{9/4} \left (c+d x^4\right )^2} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {9}{4}} {\left (d x^{4} + c\right )}^{2}} \,d x } \] Input:

integrate(1/(b*x^4+a)^(9/4)/(d*x^4+c)^2,x, algorithm="maxima")
 

Output:

integrate(1/((b*x^4 + a)^(9/4)*(d*x^4 + c)^2), x)
 

Giac [F]

\[ \int \frac {1}{\left (a+b x^4\right )^{9/4} \left (c+d x^4\right )^2} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {9}{4}} {\left (d x^{4} + c\right )}^{2}} \,d x } \] Input:

integrate(1/(b*x^4+a)^(9/4)/(d*x^4+c)^2,x, algorithm="giac")
 

Output:

integrate(1/((b*x^4 + a)^(9/4)*(d*x^4 + c)^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^4\right )^{9/4} \left (c+d x^4\right )^2} \, dx=\int \frac {1}{{\left (b\,x^4+a\right )}^{9/4}\,{\left (d\,x^4+c\right )}^2} \,d x \] Input:

int(1/((a + b*x^4)^(9/4)*(c + d*x^4)^2),x)
                                                                                    
                                                                                    
 

Output:

int(1/((a + b*x^4)^(9/4)*(c + d*x^4)^2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a+b x^4\right )^{9/4} \left (c+d x^4\right )^2} \, dx=\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{2} c^{2}+2 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{2} c d \,x^{4}+\left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{2} d^{2} x^{8}+2 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a b \,c^{2} x^{4}+4 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a b c d \,x^{8}+2 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a b \,d^{2} x^{12}+\left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{2} c^{2} x^{8}+2 \left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{2} c d \,x^{12}+\left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{2} d^{2} x^{16}}d x \] Input:

int(1/(b*x^4+a)^(9/4)/(d*x^4+c)^2,x)
 

Output:

int(1/((a + b*x**4)**(1/4)*a**2*c**2 + 2*(a + b*x**4)**(1/4)*a**2*c*d*x**4 
 + (a + b*x**4)**(1/4)*a**2*d**2*x**8 + 2*(a + b*x**4)**(1/4)*a*b*c**2*x** 
4 + 4*(a + b*x**4)**(1/4)*a*b*c*d*x**8 + 2*(a + b*x**4)**(1/4)*a*b*d**2*x* 
*12 + (a + b*x**4)**(1/4)*b**2*c**2*x**8 + 2*(a + b*x**4)**(1/4)*b**2*c*d* 
x**12 + (a + b*x**4)**(1/4)*b**2*d**2*x**16),x)