3.204 \(\int \frac{1}{(a+b x^4)^{11/4} (c+d x^4)} \, dx\)

Optimal. Leaf size=357 \[ -\frac{b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} \left (47 a^2 d^2-38 a b c d+12 b^2 c^2\right ) \text{EllipticF}\left (\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{21 a^{5/2} \left (a+b x^4\right )^{3/4} (b c-a d)^3}+\frac{b x (6 b c-13 a d)}{21 a^2 \left (a+b x^4\right )^{3/4} (b c-a d)^2}-\frac{d^3 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c (b c-a d)^3}-\frac{d^3 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c (b c-a d)^3}+\frac{b x}{7 a \left (a+b x^4\right )^{7/4} (b c-a d)} \]

[Out]

(b*x)/(7*a*(b*c - a*d)*(a + b*x^4)^(7/4)) + (b*(6*b*c - 13*a*d)*x)/(21*a^2*(b*c - a*d)^2*(a + b*x^4)^(3/4)) -
(b^(3/2)*(12*b^2*c^2 - 38*a*b*c*d + 47*a^2*d^2)*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[
a]]/2, 2])/(21*a^(5/2)*(b*c - a*d)^3*(a + b*x^4)^(3/4)) - (d^3*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*(b*c - a*d)^3)
- (d^3*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*(b*c - a*d)^3)

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Rubi [A]  time = 0.400936, antiderivative size = 357, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {414, 527, 529, 237, 335, 275, 231, 407, 409, 1218} \[ -\frac{b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} \left (47 a^2 d^2-38 a b c d+12 b^2 c^2\right ) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{21 a^{5/2} \left (a+b x^4\right )^{3/4} (b c-a d)^3}+\frac{b x (6 b c-13 a d)}{21 a^2 \left (a+b x^4\right )^{3/4} (b c-a d)^2}-\frac{d^3 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c (b c-a d)^3}-\frac{d^3 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c (b c-a d)^3}+\frac{b x}{7 a \left (a+b x^4\right )^{7/4} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^4)^(11/4)*(c + d*x^4)),x]

[Out]

(b*x)/(7*a*(b*c - a*d)*(a + b*x^4)^(7/4)) + (b*(6*b*c - 13*a*d)*x)/(21*a^2*(b*c - a*d)^2*(a + b*x^4)^(3/4)) -
(b^(3/2)*(12*b^2*c^2 - 38*a*b*c*d + 47*a^2*d^2)*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[
a]]/2, 2])/(21*a^(5/2)*(b*c - a*d)^3*(a + b*x^4)^(3/4)) - (d^3*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*(b*c - a*d)^3)
- (d^3*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*(b*c - a*d)^3)

Rule 414

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] + Dist[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]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 527

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] + Dist[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] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 529

Int[((e_) + (f_.)*(x_)^4)/(((a_) + (b_.)*(x_)^4)^(3/4)*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Dist[(b*e - a*f)/(
b*c - a*d), Int[1/(a + b*x^4)^(3/4), x], x] - Dist[(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 237

Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Dist[(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 335

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] && IntegerQ[m]

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[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 231

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

Rule 407

Int[((a_) + (b_.)*(x_)^4)^(1/4)/((c_) + (d_.)*(x_)^4), x_Symbol] :> Dist[Sqrt[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 409

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1
- Rt[-(d/c), 2]*x^2)), x], x] + Dist[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 1218

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

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b x^4\right )^{11/4} \left (c+d x^4\right )} \, dx &=\frac{b x}{7 a (b c-a d) \left (a+b x^4\right )^{7/4}}-\frac{\int \frac{-6 b c+7 a d-6 b d x^4}{\left (a+b x^4\right )^{7/4} \left (c+d x^4\right )} \, dx}{7 a (b c-a d)}\\ &=\frac{b x}{7 a (b c-a d) \left (a+b x^4\right )^{7/4}}+\frac{b (6 b c-13 a d) x}{21 a^2 (b c-a d)^2 \left (a+b x^4\right )^{3/4}}+\frac{\int \frac{12 b^2 c^2-26 a b c d+21 a^2 d^2+2 b d (6 b c-13 a d) x^4}{\left (a+b x^4\right )^{3/4} \left (c+d x^4\right )} \, dx}{21 a^2 (b c-a d)^2}\\ &=\frac{b x}{7 a (b c-a d) \left (a+b x^4\right )^{7/4}}+\frac{b (6 b c-13 a d) x}{21 a^2 (b c-a d)^2 \left (a+b x^4\right )^{3/4}}-\frac{d^3 \int \frac{\sqrt [4]{a+b x^4}}{c+d x^4} \, dx}{(b c-a d)^3}+\frac{\left (b \left (12 b^2 c^2-38 a b c d+47 a^2 d^2\right )\right ) \int \frac{1}{\left (a+b x^4\right )^{3/4}} \, dx}{21 a^2 (b c-a d)^3}\\ &=\frac{b x}{7 a (b c-a d) \left (a+b x^4\right )^{7/4}}+\frac{b (6 b c-13 a d) x}{21 a^2 (b c-a d)^2 \left (a+b x^4\right )^{3/4}}+\frac{\left (b \left (12 b^2 c^2-38 a b c d+47 a^2 d^2\right ) \left (1+\frac{a}{b x^4}\right )^{3/4} x^3\right ) \int \frac{1}{\left (1+\frac{a}{b x^4}\right )^{3/4} x^3} \, dx}{21 a^2 (b c-a d)^3 \left (a+b x^4\right )^{3/4}}-\frac{\left (d^3 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-b x^4} \left (c-(b c-a d) x^4\right )} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{(b c-a d)^3}\\ &=\frac{b x}{7 a (b c-a d) \left (a+b x^4\right )^{7/4}}+\frac{b (6 b c-13 a d) x}{21 a^2 (b c-a d)^2 \left (a+b x^4\right )^{3/4}}-\frac{\left (b \left (12 b^2 c^2-38 a b c d+47 a^2 d^2\right ) \left (1+\frac{a}{b x^4}\right )^{3/4} x^3\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1+\frac{a x^4}{b}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )}{21 a^2 (b c-a d)^3 \left (a+b x^4\right )^{3/4}}-\frac{\left (d^3 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{\sqrt{b c-a d} x^2}{\sqrt{c}}\right ) \sqrt{1-b x^4}} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{2 c (b c-a d)^3}-\frac{\left (d^3 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{\sqrt{b c-a d} x^2}{\sqrt{c}}\right ) \sqrt{1-b x^4}} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{2 c (b c-a d)^3}\\ &=\frac{b x}{7 a (b c-a d) \left (a+b x^4\right )^{7/4}}+\frac{b (6 b c-13 a d) x}{21 a^2 (b c-a d)^2 \left (a+b x^4\right )^{3/4}}-\frac{d^3 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{2 \sqrt [4]{b} c (b c-a d)^3}-\frac{d^3 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{2 \sqrt [4]{b} c (b c-a d)^3}-\frac{\left (b \left (12 b^2 c^2-38 a b c d+47 a^2 d^2\right ) \left (1+\frac{a}{b x^4}\right )^{3/4} x^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{a x^2}{b}\right )^{3/4}} \, dx,x,\frac{1}{x^2}\right )}{42 a^2 (b c-a d)^3 \left (a+b x^4\right )^{3/4}}\\ &=\frac{b x}{7 a (b c-a d) \left (a+b x^4\right )^{7/4}}+\frac{b (6 b c-13 a d) x}{21 a^2 (b c-a d)^2 \left (a+b x^4\right )^{3/4}}-\frac{b^{3/2} \left (12 b^2 c^2-38 a b c d+47 a^2 d^2\right ) \left (1+\frac{a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{21 a^{5/2} (b c-a d)^3 \left (a+b x^4\right )^{3/4}}-\frac{d^3 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{2 \sqrt [4]{b} c (b c-a d)^3}-\frac{d^3 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{2 \sqrt [4]{b} c (b c-a d)^3}\\ \end{align*}

Mathematica [C]  time = 0.724857, size = 430, normalized size = 1.2 \[ \frac{x \left (-\frac{5 \left (5 a c \left (a^2 b d \left (5 d x^4-42 c\right )+21 a^3 d^2+a b^2 \left (21 c^2-30 c d x^4-13 d^2 x^8\right )+6 b^3 c x^4 \left (3 c+d x^4\right )\right ) F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+b x^4 \left (c+d x^4\right ) \left (16 a^2 d+a b \left (13 d x^4-9 c\right )-6 b^2 c x^4\right ) \left (4 a d F_1\left (\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )\right )}{\left (a+b x^4\right ) \left (c+d x^4\right ) \left (x^4 \left (4 a d F_1\left (\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}-\frac{2 b d x^4 \left (\frac{b x^4}{a}+1\right )^{3/4} (13 a d-6 b c) F_1\left (\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{c}\right )}{105 a^2 \left (a+b x^4\right )^{3/4} (b c-a d)^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + b*x^4)^(11/4)*(c + d*x^4)),x]

[Out]

(x*((-2*b*d*(-6*b*c + 13*a*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*(21*a^3*d^2 + 6*b^3*c*x^4*(3*c + d*x^4) + a^2*b*d*(-42*c + 5*d*x^4) + a*b^2*(21*c^2 - 30*c*d*x
^4 - 13*d^2*x^8))*AppellF1[1/4, 3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + b*x^4*(c + d*x^4)*(16*a^2*d - 6*b^2
*c*x^4 + a*b*(-9*c + 13*d*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)])))/((a + b*x^4)*(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*Appell
F1[5/4, 7/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))))/(105*a^2*(b*c - a*d)^2*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.402, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{4}+c} \left ( b{x}^{4}+a \right ) ^{-{\frac{11}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^4+a)^(11/4)/(d*x^4+c),x)

[Out]

int(1/(b*x^4+a)^(11/4)/(d*x^4+c),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{11}{4}}{\left (d x^{4} + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(11/4)/(d*x^4+c),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(11/4)*(d*x^4 + c)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(11/4)/(d*x^4+c),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**4+a)**(11/4)/(d*x**4+c),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{11}{4}}{\left (d x^{4} + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(11/4)/(d*x^4+c),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(11/4)*(d*x^4 + c)), x)