3.184 \(\int \frac{(a-b x^4)^{7/2}}{(c-d x^4)^2} \, dx\)

Optimal. Leaf size=426 \[ \frac{\sqrt [4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} \left (349 a^2 b c d^2+21 a^3 d^3-553 a b^2 c^2 d+231 b^3 c^3\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{84 c d^4 \sqrt{a-b x^4}}-\frac{b x \sqrt{a-b x^4} \left (21 a^2 d^2-122 a b c d+77 b^2 c^2\right )}{84 c d^3}-\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (3 a d+11 b c) (b c-a d)^3 \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^4 \sqrt{a-b x^4}}-\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (3 a d+11 b c) (b c-a d)^3 \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^4 \sqrt{a-b x^4}}+\frac{b x \left (a-b x^4\right )^{3/2} (11 b c-7 a d)}{28 c d^2}-\frac{x \left (a-b x^4\right )^{5/2} (b c-a d)}{4 c d \left (c-d x^4\right )} \]

[Out]

-(b*(77*b^2*c^2 - 122*a*b*c*d + 21*a^2*d^2)*x*Sqrt[a - b*x^4])/(84*c*d^3) + (b*(11*b*c - 7*a*d)*x*(a - b*x^4)^
(3/2))/(28*c*d^2) - ((b*c - a*d)*x*(a - b*x^4)^(5/2))/(4*c*d*(c - d*x^4)) + (a^(1/4)*b^(3/4)*(231*b^3*c^3 - 55
3*a*b^2*c^2*d + 349*a^2*b*c*d^2 + 21*a^3*d^3)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/
(84*c*d^4*Sqrt[a - b*x^4]) - (a^(1/4)*(b*c - a*d)^3*(11*b*c + 3*a*d)*Sqrt[1 - (b*x^4)/a]*EllipticPi[-((Sqrt[a]
*Sqrt[d])/(Sqrt[b]*Sqrt[c])), ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(8*b^(1/4)*c^2*d^4*Sqrt[a - b*x^4]) - (a^(1/4)
*(b*c - a*d)^3*(11*b*c + 3*a*d)*Sqrt[1 - (b*x^4)/a]*EllipticPi[(Sqrt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c]), ArcSin[(b^
(1/4)*x)/a^(1/4)], -1])/(8*b^(1/4)*c^2*d^4*Sqrt[a - b*x^4])

________________________________________________________________________________________

Rubi [A]  time = 0.535672, antiderivative size = 426, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {413, 528, 523, 224, 221, 409, 1219, 1218} \[ -\frac{b x \sqrt{a-b x^4} \left (21 a^2 d^2-122 a b c d+77 b^2 c^2\right )}{84 c d^3}+\frac{\sqrt [4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} \left (349 a^2 b c d^2+21 a^3 d^3-553 a b^2 c^2 d+231 b^3 c^3\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{84 c d^4 \sqrt{a-b x^4}}-\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (3 a d+11 b c) (b c-a d)^3 \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^4 \sqrt{a-b x^4}}-\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (3 a d+11 b c) (b c-a d)^3 \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^4 \sqrt{a-b x^4}}+\frac{b x \left (a-b x^4\right )^{3/2} (11 b c-7 a d)}{28 c d^2}-\frac{x \left (a-b x^4\right )^{5/2} (b c-a d)}{4 c d \left (c-d x^4\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x^4)^(7/2)/(c - d*x^4)^2,x]

[Out]

-(b*(77*b^2*c^2 - 122*a*b*c*d + 21*a^2*d^2)*x*Sqrt[a - b*x^4])/(84*c*d^3) + (b*(11*b*c - 7*a*d)*x*(a - b*x^4)^
(3/2))/(28*c*d^2) - ((b*c - a*d)*x*(a - b*x^4)^(5/2))/(4*c*d*(c - d*x^4)) + (a^(1/4)*b^(3/4)*(231*b^3*c^3 - 55
3*a*b^2*c^2*d + 349*a^2*b*c*d^2 + 21*a^3*d^3)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/
(84*c*d^4*Sqrt[a - b*x^4]) - (a^(1/4)*(b*c - a*d)^3*(11*b*c + 3*a*d)*Sqrt[1 - (b*x^4)/a]*EllipticPi[-((Sqrt[a]
*Sqrt[d])/(Sqrt[b]*Sqrt[c])), ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(8*b^(1/4)*c^2*d^4*Sqrt[a - b*x^4]) - (a^(1/4)
*(b*c - a*d)^3*(11*b*c + 3*a*d)*Sqrt[1 - (b*x^4)/a]*EllipticPi[(Sqrt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c]), ArcSin[(b^
(1/4)*x)/a^(1/4)], -1])/(8*b^(1/4)*c^2*d^4*Sqrt[a - b*x^4])

Rule 413

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

Rule 528

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[
(f*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*(n*(p + q + 1) + 1)), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a +
b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*
d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1
, 0]

Rule 523

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + (b*x^4)/a]/Sqrt[a + b*x^4], Int[1/Sqrt[1 + (b*x^4)
/a], x], x] /; FreeQ[{a, b}, x] && NegQ[b/a] &&  !GtQ[a, 0]

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 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 1219

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[Sqrt[1 + (c*x^4)/a]/Sqrt[a + c*x^4]
, Int[1/((d + e*x^2)*Sqrt[1 + (c*x^4)/a]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] &&  !GtQ[a, 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{\left (a-b x^4\right )^{7/2}}{\left (c-d x^4\right )^2} \, dx &=-\frac{(b c-a d) x \left (a-b x^4\right )^{5/2}}{4 c d \left (c-d x^4\right )}-\frac{\int \frac{\left (a-b x^4\right )^{3/2} \left (-a (b c+3 a d)+b (11 b c-7 a d) x^4\right )}{c-d x^4} \, dx}{4 c d}\\ &=\frac{b (11 b c-7 a d) x \left (a-b x^4\right )^{3/2}}{28 c d^2}-\frac{(b c-a d) x \left (a-b x^4\right )^{5/2}}{4 c d \left (c-d x^4\right )}+\frac{\int \frac{\sqrt{a-b x^4} \left (-a \left (11 b^2 c^2-14 a b c d-21 a^2 d^2\right )+b \left (77 b^2 c^2-122 a b c d+21 a^2 d^2\right ) x^4\right )}{c-d x^4} \, dx}{28 c d^2}\\ &=-\frac{b \left (77 b^2 c^2-122 a b c d+21 a^2 d^2\right ) x \sqrt{a-b x^4}}{84 c d^3}+\frac{b (11 b c-7 a d) x \left (a-b x^4\right )^{3/2}}{28 c d^2}-\frac{(b c-a d) x \left (a-b x^4\right )^{5/2}}{4 c d \left (c-d x^4\right )}-\frac{\int \frac{-a \left (77 b^3 c^3-155 a b^2 c^2 d+63 a^2 b c d^2+63 a^3 d^3\right )+b \left (231 b^3 c^3-553 a b^2 c^2 d+349 a^2 b c d^2+21 a^3 d^3\right ) x^4}{\sqrt{a-b x^4} \left (c-d x^4\right )} \, dx}{84 c d^3}\\ &=-\frac{b \left (77 b^2 c^2-122 a b c d+21 a^2 d^2\right ) x \sqrt{a-b x^4}}{84 c d^3}+\frac{b (11 b c-7 a d) x \left (a-b x^4\right )^{3/2}}{28 c d^2}-\frac{(b c-a d) x \left (a-b x^4\right )^{5/2}}{4 c d \left (c-d x^4\right )}-\frac{\left ((b c-a d)^3 (11 b c+3 a d)\right ) \int \frac{1}{\sqrt{a-b x^4} \left (c-d x^4\right )} \, dx}{4 c d^4}+\frac{\left (b \left (231 b^3 c^3-553 a b^2 c^2 d+349 a^2 b c d^2+21 a^3 d^3\right )\right ) \int \frac{1}{\sqrt{a-b x^4}} \, dx}{84 c d^4}\\ &=-\frac{b \left (77 b^2 c^2-122 a b c d+21 a^2 d^2\right ) x \sqrt{a-b x^4}}{84 c d^3}+\frac{b (11 b c-7 a d) x \left (a-b x^4\right )^{3/2}}{28 c d^2}-\frac{(b c-a d) x \left (a-b x^4\right )^{5/2}}{4 c d \left (c-d x^4\right )}-\frac{\left ((b c-a d)^3 (11 b c+3 a d)\right ) \int \frac{1}{\left (1-\frac{\sqrt{d} x^2}{\sqrt{c}}\right ) \sqrt{a-b x^4}} \, dx}{8 c^2 d^4}-\frac{\left ((b c-a d)^3 (11 b c+3 a d)\right ) \int \frac{1}{\left (1+\frac{\sqrt{d} x^2}{\sqrt{c}}\right ) \sqrt{a-b x^4}} \, dx}{8 c^2 d^4}+\frac{\left (b \left (231 b^3 c^3-553 a b^2 c^2 d+349 a^2 b c d^2+21 a^3 d^3\right ) \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1}{\sqrt{1-\frac{b x^4}{a}}} \, dx}{84 c d^4 \sqrt{a-b x^4}}\\ &=-\frac{b \left (77 b^2 c^2-122 a b c d+21 a^2 d^2\right ) x \sqrt{a-b x^4}}{84 c d^3}+\frac{b (11 b c-7 a d) x \left (a-b x^4\right )^{3/2}}{28 c d^2}-\frac{(b c-a d) x \left (a-b x^4\right )^{5/2}}{4 c d \left (c-d x^4\right )}+\frac{\sqrt [4]{a} b^{3/4} \left (231 b^3 c^3-553 a b^2 c^2 d+349 a^2 b c d^2+21 a^3 d^3\right ) \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{84 c d^4 \sqrt{a-b x^4}}-\frac{\left ((b c-a d)^3 (11 b c+3 a d) \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1}{\left (1-\frac{\sqrt{d} x^2}{\sqrt{c}}\right ) \sqrt{1-\frac{b x^4}{a}}} \, dx}{8 c^2 d^4 \sqrt{a-b x^4}}-\frac{\left ((b c-a d)^3 (11 b c+3 a d) \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1}{\left (1+\frac{\sqrt{d} x^2}{\sqrt{c}}\right ) \sqrt{1-\frac{b x^4}{a}}} \, dx}{8 c^2 d^4 \sqrt{a-b x^4}}\\ &=-\frac{b \left (77 b^2 c^2-122 a b c d+21 a^2 d^2\right ) x \sqrt{a-b x^4}}{84 c d^3}+\frac{b (11 b c-7 a d) x \left (a-b x^4\right )^{3/2}}{28 c d^2}-\frac{(b c-a d) x \left (a-b x^4\right )^{5/2}}{4 c d \left (c-d x^4\right )}+\frac{\sqrt [4]{a} b^{3/4} \left (231 b^3 c^3-553 a b^2 c^2 d+349 a^2 b c d^2+21 a^3 d^3\right ) \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{84 c d^4 \sqrt{a-b x^4}}-\frac{\sqrt [4]{a} (b c-a d)^3 (11 b c+3 a d) \sqrt{1-\frac{b x^4}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^4 \sqrt{a-b x^4}}-\frac{\sqrt [4]{a} (b c-a d)^3 (11 b c+3 a d) \sqrt{1-\frac{b x^4}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d^4 \sqrt{a-b x^4}}\\ \end{align*}

Mathematica [C]  time = 0.821353, size = 477, normalized size = 1.12 \[ -\frac{b x^5 \sqrt{1-\frac{b x^4}{a}} \left (349 a^2 b c d^2+21 a^3 d^3-553 a b^2 c^2 d+231 b^3 c^3\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+\frac{5 c \left (2 x^5 \left (b x^4-a\right ) \left (-63 a^2 b c d^2+21 a^3 d^3+a b^2 c d \left (155 c-92 d x^4\right )+b^3 c \left (-77 c^2+44 c d x^4+12 d^2 x^8\right )\right ) \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+5 a c x \left (29 a^2 b^2 c d^2 x^4+21 a^3 b d^3 x^4-84 a^4 d^3+a b^3 c d x^4 \left (111 c-104 d x^4\right )+b^4 c x^4 \left (-77 c^2+44 c d x^4+12 d^2 x^8\right )\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )}{\left (c-d x^4\right ) \left (2 x^4 \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )}}{420 c^2 d^3 \sqrt{a-b x^4}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a - b*x^4)^(7/2)/(c - d*x^4)^2,x]

[Out]

-(b*(231*b^3*c^3 - 553*a*b^2*c^2*d + 349*a^2*b*c*d^2 + 21*a^3*d^3)*x^5*Sqrt[1 - (b*x^4)/a]*AppellF1[5/4, 1/2,
1, 9/4, (b*x^4)/a, (d*x^4)/c] + (5*c*(5*a*c*x*(-84*a^4*d^3 + 29*a^2*b^2*c*d^2*x^4 + 21*a^3*b*d^3*x^4 + a*b^3*c
*d*x^4*(111*c - 104*d*x^4) + b^4*c*x^4*(-77*c^2 + 44*c*d*x^4 + 12*d^2*x^8))*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)
/a, (d*x^4)/c] + 2*x^5*(-a + b*x^4)*(-63*a^2*b*c*d^2 + 21*a^3*d^3 + a*b^2*c*d*(155*c - 92*d*x^4) + b^3*c*(-77*
c^2 + 44*c*d*x^4 + 12*d^2*x^8))*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/
2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])))/((c - d*x^4)*(5*a*c*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*x
^4*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)
/c]))))/(420*c^2*d^3*Sqrt[a - b*x^4])

________________________________________________________________________________________

Maple [C]  time = 0.029, size = 540, normalized size = 1.3 \begin{align*} -{\frac{ \left ({a}^{3}{d}^{3}-3\,cb{a}^{2}{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ) x}{4\,c{d}^{3} \left ( d{x}^{4}-c \right ) }\sqrt{-b{x}^{4}+a}}-{\frac{{b}^{3}{x}^{5}}{7\,{d}^{2}}\sqrt{-b{x}^{4}+a}}-{\frac{x}{3\,b} \left ( -2\,{\frac{{b}^{3} \left ( 2\,ad-bc \right ) }{{d}^{3}}}+{\frac{5\,a{b}^{3}}{7\,{d}^{2}}} \right ) \sqrt{-b{x}^{4}+a}}+{ \left ({\frac{{b}^{2} \left ( 6\,{a}^{2}{d}^{2}-8\,cabd+3\,{b}^{2}{c}^{2} \right ) }{{d}^{4}}}+{\frac{b \left ({a}^{3}{d}^{3}-3\,cb{a}^{2}{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ) }{4\,{d}^{4}c}}+{\frac{a}{3\,b} \left ( -2\,{\frac{{b}^{3} \left ( 2\,ad-bc \right ) }{{d}^{3}}}+{\frac{5\,a{b}^{3}}{7\,{d}^{2}}} \right ) } \right ) \sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}}-{\frac{1}{32\,{d}^{5}c}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}d-c \right ) }{\frac{3\,{a}^{4}{d}^{4}+2\,{a}^{3}b{d}^{3}c-24\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}+30\,{c}^{3}a{b}^{3}d-11\,{b}^{4}{c}^{4}}{{{\it \_alpha}}^{3}} \left ( -{{\it Artanh} \left ({\frac{-2\,{{\it \_alpha}}^{2}b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}-2\,{\frac{{{\it \_alpha}}^{3}d}{c\sqrt{-b{x}^{4}+a}}\sqrt{1-{\frac{{x}^{2}\sqrt{b}}{\sqrt{a}}}}\sqrt{1+{\frac{{x}^{2}\sqrt{b}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}},{\frac{\sqrt{a}{{\it \_alpha}}^{2}d}{c\sqrt{b}}},{\sqrt{-{\frac{\sqrt{b}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*x^4+a)^(7/2)/(-d*x^4+c)^2,x)

[Out]

-1/4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/c/d^3*x*(-b*x^4+a)^(1/2)/(d*x^4-c)-1/7*b^3/d^2*x^5*(-b*x^4+
a)^(1/2)-1/3*(-2*b^3/d^3*(2*a*d-b*c)+5/7*b^3/d^2*a)/b*x*(-b*x^4+a)^(1/2)+(b^2*(6*a^2*d^2-8*a*b*c*d+3*b^2*c^2)/
d^4+1/4*b/d^4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/c+1/3*(-2*b^3/d^3*(2*a*d-b*c)+5/7*b^3/d^2*a)/b*a)/
(1/a^(1/2)*b^(1/2))^(1/2)*(1-x^2*b^(1/2)/a^(1/2))^(1/2)*(1+x^2*b^(1/2)/a^(1/2))^(1/2)/(-b*x^4+a)^(1/2)*Ellipti
cF(x*(1/a^(1/2)*b^(1/2))^(1/2),I)-1/32/d^5/c*sum((3*a^4*d^4+2*a^3*b*c*d^3-24*a^2*b^2*c^2*d^2+30*a*b^3*c^3*d-11
*b^4*c^4)/_alpha^3*(-1/((a*d-b*c)/d)^(1/2)*arctanh(1/2*(-2*_alpha^2*b*x^2+2*a)/((a*d-b*c)/d)^(1/2)/(-b*x^4+a)^
(1/2))-2/(1/a^(1/2)*b^(1/2))^(1/2)*_alpha^3*d/c*(1-x^2*b^(1/2)/a^(1/2))^(1/2)*(1+x^2*b^(1/2)/a^(1/2))^(1/2)/(-
b*x^4+a)^(1/2)*EllipticPi(x*(1/a^(1/2)*b^(1/2))^(1/2),a^(1/2)/b^(1/2)*_alpha^2/c*d,(-1/a^(1/2)*b^(1/2))^(1/2)/
(1/a^(1/2)*b^(1/2))^(1/2))),_alpha=RootOf(_Z^4*d-c))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-b x^{4} + a\right )}^{\frac{7}{2}}}{{\left (d x^{4} - c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^4+a)^(7/2)/(-d*x^4+c)^2,x, algorithm="maxima")

[Out]

integrate((-b*x^4 + a)^(7/2)/(d*x^4 - c)^2, x)

________________________________________________________________________________________

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((-b*x^4+a)^(7/2)/(-d*x^4+c)^2,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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((-b*x**4+a)**(7/2)/(-d*x**4+c)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-b x^{4} + a\right )}^{\frac{7}{2}}}{{\left (d x^{4} - c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^4+a)^(7/2)/(-d*x^4+c)^2,x, algorithm="giac")

[Out]

integrate((-b*x^4 + a)^(7/2)/(d*x^4 - c)^2, x)