\(\int \frac {1}{(a+b x^4)^{7/4} (c+d x^4)} \, dx\) [113]

Optimal result

Integrand size = 21, antiderivative size = 304 \[ \int \frac {1}{\left (a+b x^4\right )^{7/4} \left (c+d x^4\right )} \, dx=\frac {b x}{3 a (b c-a d) \left (a+b x^4\right )^{3/4}}-\frac {b^{3/2} (2 b c-5 a d) \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{3 a^{3/2} (b c-a d)^2 \left (a+b x^4\right )^{3/4}}+\frac {d^2 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \operatorname {EllipticPi}\left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right ),-1\right )}{2 \sqrt [4]{b} c (b c-a d)^2}+\frac {d^2 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \operatorname {EllipticPi}\left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right ),-1\right )}{2 \sqrt [4]{b} c (b c-a d)^2} \] Output:

1/3*b*x/a/(-a*d+b*c)/(b*x^4+a)^(3/4)-1/3*b^(3/2)*(-5*a*d+2*b*c)*(1+a/b/x^4 
)^(3/4)*x^3*InverseJacobiAM(1/2*arccot(b^(1/2)*x^2/a^(1/2)),2^(1/2))/a^(3/ 
2)/(-a*d+b*c)^2/(b*x^4+a)^(3/4)+1/2*d^2*(a/(b*x^4+a))^(1/2)*(b*x^4+a)^(1/2 
)*EllipticPi(b^(1/4)*x/(b*x^4+a)^(1/4),-(-a*d+b*c)^(1/2)/b^(1/2)/c^(1/2),I 
)/b^(1/4)/c/(-a*d+b*c)^2+1/2*d^2*(a/(b*x^4+a))^(1/2)*(b*x^4+a)^(1/2)*Ellip 
ticPi(b^(1/4)*x/(b*x^4+a)^(1/4),(-a*d+b*c)^(1/2)/b^(1/2)/c^(1/2),I)/b^(1/4 
)/c/(-a*d+b*c)^2
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 10.29 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (a+b x^4\right )^{7/4} \left (c+d x^4\right )} \, dx=\frac {x \left (-\frac {2 b d x^4 \left (1+\frac {b x^4}{a}\right )^{3/4} \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{c}+\frac {5 \left (5 a c \left (3 a d-b \left (3 c+d x^4\right )\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {3}{4},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+b x^4 \left (c+d x^4\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {7}{4},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )\right )}{\left (c+d x^4\right ) \left (5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {3}{4},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )-x^4 \left (4 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {7}{4},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )\right )}\right )}{15 a (-b c+a d) \left (a+b x^4\right )^{3/4}} \] Input:

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

Output:

(x*((-2*b*d*x^4*(1 + (b*x^4)/a)^(3/4)*AppellF1[5/4, 3/4, 1, 9/4, -((b*x^4) 
/a), -((d*x^4)/c)])/c + (5*(5*a*c*(3*a*d - b*(3*c + d*x^4))*AppellF1[1/4, 
3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + b*x^4*(c + d*x^4)*(4*a*d*Appell 
F1[5/4, 3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4, 7/4 
, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))/((c + d*x^4)*(5*a*c*AppellF1[1/4, 
 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] - x^4*(4*a*d*AppellF1[5/4, 3/4, 
2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*AppellF1[5/4, 7/4, 1, 9/4, -(( 
b*x^4)/a), -((d*x^4)/c)])))))/(15*a*(-(b*c) + a*d)*(a + b*x^4)^(3/4))
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.94, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {931, 25, 404, 768, 858, 807, 229, 923, 925, 27, 1542}

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.

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

\(\Big \downarrow \) 931

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 404

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

\(\Big \downarrow \) 768

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

\(\Big \downarrow \) 858

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

\(\Big \downarrow \) 807

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

\(\Big \downarrow \) 229

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

\(\Big \downarrow \) 923

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

\(\Big \downarrow \) 925

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1542

\(\displaystyle \frac {\frac {3 a d^2 \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \left (\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right ),-1\right )}{2 \sqrt [4]{b} c}+\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right ),-1\right )}{2 \sqrt [4]{b} c}\right )}{b c-a d}-\frac {b^{3/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} (2 b c-5 a d) \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {b} x^2}\right ),2\right )}{\sqrt {a} \left (a+b x^4\right )^{3/4} (b c-a d)}}{3 a (b c-a d)}+\frac {b x}{3 a \left (a+b x^4\right )^{3/4} (b c-a d)}\)

Input:

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

Output:

(b*x)/(3*a*(b*c - a*d)*(a + b*x^4)^(3/4)) + (-((b^(3/2)*(2*b*c - 5*a*d)*(1 
 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcTan[Sqrt[a]/(Sqrt[b]*x^2)]/2, 2])/(Sq 
rt[a]*(b*c - a*d)*(a + b*x^4)^(3/4))) + (3*a*d^2*Sqrt[a/(a + b*x^4)]*Sqrt[ 
a + b*x^4]*(EllipticPi[-(Sqrt[b*c - a*d]/(Sqrt[b]*Sqrt[c])), ArcSin[(b^(1/ 
4)*x)/(a + b*x^4)^(1/4)], -1]/(2*b^(1/4)*c) + EllipticPi[Sqrt[b*c - a*d]/( 
Sqrt[b]*Sqrt[c]), ArcSin[(b^(1/4)*x)/(a + b*x^4)^(1/4)], -1]/(2*b^(1/4)*c) 
))/(b*c - a*d))/(3*a*(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 229

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]) 
)*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a 
, 0] && PosQ[b/a]
 

rule 404

Int[((e_) + (f_.)*(x_)^4)/(((a_) + (b_.)*(x_)^4)^(3/4)*((c_) + (d_.)*(x_)^4 
)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^4)^(3/4), x] 
, x] - Simp[(d*e - c*f)/(b*c - a*d)   Int[(a + b*x^4)^(1/4)/(c + d*x^4), x] 
, x] /; FreeQ[{a, b, c, d, e, f}, x]
 

rule 768

Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Simp[x^3*((1 + a/(b*x^4))^(3 
/4)/(a + b*x^4)^(3/4))   Int[1/(x^3*(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ 
[{a, b}, x]
 

rule 807

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

rule 858

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + 
b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int 
egerQ[m]
 

rule 923

Int[((a_) + (b_.)*(x_)^4)^(1/4)/((c_) + (d_.)*(x_)^4), x_Symbol] :> Simp[Sq 
rt[a + b*x^4]*Sqrt[a/(a + b*x^4)]   Subst[Int[1/(Sqrt[1 - b*x^4]*(c - (b*c 
- a*d)*x^4)), x], x, x/(a + b*x^4)^(1/4)], x] /; FreeQ[{a, b, c, d}, x] && 
NeQ[b*c - a*d, 0]
 

rule 925

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Simp[ 
1/(2*c)   Int[1/(Sqrt[a + b*x^4]*(1 - Rt[-d/c, 2]*x^2)), x], x] + Simp[1/(2 
*c)   Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], x] /; FreeQ[{a, b, 
 c, d}, x] && NeQ[b*c - a*d, 0]
 

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 1542

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ 
{q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x 
], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (a+b x^4\right )^{7/4} \left (c+d x^4\right )} \, dx=\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {3}{4}} a c +\left (b \,x^{4}+a \right )^{\frac {3}{4}} a d \,x^{4}+\left (b \,x^{4}+a \right )^{\frac {3}{4}} b c \,x^{4}+\left (b \,x^{4}+a \right )^{\frac {3}{4}} b d \,x^{8}}d x \] Input:

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

Output:

int(1/((a + b*x**4)**(3/4)*a*c + (a + b*x**4)**(3/4)*a*d*x**4 + (a + b*x** 
4)**(3/4)*b*c*x**4 + (a + b*x**4)**(3/4)*b*d*x**8),x)