3.2.89 \(\int \frac {1}{(a-b x^4)^{3/2} (c-d x^4)^2} \, dx\) [189]

Optimal. Leaf size=362 \[ \frac {b (2 b c+a d) x}{4 a c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \sqrt {a-b x^4} \left (c-d x^4\right )}+\frac {b^{3/4} (2 b c+a d) \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 a^{3/4} c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {3 \sqrt [4]{a} d (3 b c-a d) \sqrt {1-\frac {b x^4}{a}} \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)^2 \sqrt {a-b x^4}}-\frac {3 \sqrt [4]{a} d (3 b c-a d) \sqrt {1-\frac {b x^4}{a}} \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)^2 \sqrt {a-b x^4}} \]

[Out]

1/4*b*(a*d+2*b*c)*x/a/c/(-a*d+b*c)^2/(-b*x^4+a)^(1/2)-1/4*d*x/c/(-a*d+b*c)/(-d*x^4+c)/(-b*x^4+a)^(1/2)+1/4*b^(
3/4)*(a*d+2*b*c)*EllipticF(b^(1/4)*x/a^(1/4),I)*(1-b*x^4/a)^(1/2)/a^(3/4)/c/(-a*d+b*c)^2/(-b*x^4+a)^(1/2)-3/8*
a^(1/4)*d*(-a*d+3*b*c)*EllipticPi(b^(1/4)*x/a^(1/4),-a^(1/2)*d^(1/2)/b^(1/2)/c^(1/2),I)*(1-b*x^4/a)^(1/2)/b^(1
/4)/c^2/(-a*d+b*c)^2/(-b*x^4+a)^(1/2)-3/8*a^(1/4)*d*(-a*d+3*b*c)*EllipticPi(b^(1/4)*x/a^(1/4),a^(1/2)*d^(1/2)/
b^(1/2)/c^(1/2),I)*(1-b*x^4/a)^(1/2)/b^(1/4)/c^2/(-a*d+b*c)^2/(-b*x^4+a)^(1/2)

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Rubi [A]
time = 0.44, antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {425, 541, 537, 230, 227, 418, 1233, 1232} \begin {gather*} \frac {b^{3/4} \sqrt {1-\frac {b x^4}{a}} (a d+2 b c) F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 a^{3/4} c \sqrt {a-b x^4} (b c-a d)^2}-\frac {3 \sqrt [4]{a} d \sqrt {1-\frac {b x^4}{a}} (3 b c-a d) \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt {a-b x^4} (b c-a d)^2}-\frac {3 \sqrt [4]{a} d \sqrt {1-\frac {b x^4}{a}} (3 b c-a d) \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt {a-b x^4} (b c-a d)^2}+\frac {b x (a d+2 b c)}{4 a c \sqrt {a-b x^4} (b c-a d)^2}-\frac {d x}{4 c \sqrt {a-b x^4} \left (c-d x^4\right ) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a - b*x^4)^(3/2)*(c - d*x^4)^2),x]

[Out]

(b*(2*b*c + a*d)*x)/(4*a*c*(b*c - a*d)^2*Sqrt[a - b*x^4]) - (d*x)/(4*c*(b*c - a*d)*Sqrt[a - b*x^4]*(c - d*x^4)
) + (b^(3/4)*(2*b*c + a*d)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(4*a^(3/4)*c*(b*c -
a*d)^2*Sqrt[a - b*x^4]) - (3*a^(1/4)*d*(3*b*c - 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*(b*c - a*d)^2*Sqrt[a - b*x^4]) - (3*a^(1/4)*d*
(3*b*c - 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*(b*c - a*d)^2*Sqrt[a - b*x^4])

Rule 227

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 230

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 418

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 425

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]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 537

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 541

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 1232

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]

Rule 1233

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]

Rubi steps

\begin {align*} \int \frac {1}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )^2} \, dx &=-\frac {d x}{4 c (b c-a d) \sqrt {a-b x^4} \left (c-d x^4\right )}-\frac {\int \frac {-4 b c+3 a d-5 b d x^4}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx}{4 c (b c-a d)}\\ &=\frac {b (2 b c+a d) x}{4 a c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \sqrt {a-b x^4} \left (c-d x^4\right )}-\frac {\int \frac {-2 \left (2 b^2 c^2-8 a b c d+3 a^2 d^2\right )+2 b d (2 b c+a d) x^4}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{8 a c (b c-a d)^2}\\ &=\frac {b (2 b c+a d) x}{4 a c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \sqrt {a-b x^4} \left (c-d x^4\right )}-\frac {(3 d (3 b c-a d)) \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{4 c (b c-a d)^2}+\frac {(b (2 b c+a d)) \int \frac {1}{\sqrt {a-b x^4}} \, dx}{4 a c (b c-a d)^2}\\ &=\frac {b (2 b c+a d) x}{4 a c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \sqrt {a-b x^4} \left (c-d x^4\right )}-\frac {(3 d (3 b c-a d)) \int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{8 c^2 (b c-a d)^2}-\frac {(3 d (3 b c-a d)) \int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{8 c^2 (b c-a d)^2}+\frac {\left (b (2 b c+a d) \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{4 a c (b c-a d)^2 \sqrt {a-b x^4}}\\ &=\frac {b (2 b c+a d) x}{4 a c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \sqrt {a-b x^4} \left (c-d x^4\right )}+\frac {b^{3/4} (2 b c+a d) \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 )}{4 a^{3/4} c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {\left (3 d (3 b c-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 (b c-a d)^2 \sqrt {a-b x^4}}-\frac {\left (3 d (3 b c-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 (b c-a d)^2 \sqrt {a-b x^4}}\\ &=\frac {b (2 b c+a d) x}{4 a c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \sqrt {a-b x^4} \left (c-d x^4\right )}+\frac {b^{3/4} (2 b c+a d) \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 )}{4 a^{3/4} c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {3 \sqrt [4]{a} d (3 b c-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 (b c-a d)^2 \sqrt {a-b x^4}}-\frac {3 \sqrt [4]{a} d (3 b c-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 (b c-a d)^2 \sqrt {a-b x^4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
time = 10.38, size = 374, normalized size = 1.03 \begin {gather*} \frac {x \left (-b d (2 b c+a d) x^4 \sqrt {1-\frac {b x^4}{a}} 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 {c \left (25 a c \left (4 a^2 d^2+2 b^2 c \left (2 c-d x^4\right )-a b d \left (8 c+d x^4\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 )-10 x^4 \left (-a^2 d^2+a b d^2 x^4-2 b^2 c \left (c-d x^4\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 )\right )}{\left (c-d x^4\right ) \left (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 )+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 )\right )}\right )}{20 a c^2 (b c-a d)^2 \sqrt {a-b x^4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a - b*x^4)^(3/2)*(c - d*x^4)^2),x]

[Out]

(x*(-(b*d*(2*b*c + a*d)*x^4*Sqrt[1 - (b*x^4)/a]*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c]) + (c*(25*a*c
*(4*a^2*d^2 + 2*b^2*c*(2*c - d*x^4) - a*b*d*(8*c + d*x^4))*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] -
10*x^4*(-(a^2*d^2) + a*b*d^2*x^4 - 2*b^2*c*(c - d*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])))/((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])))))/(20*a*c^2*(b*c - a*d)^2*Sqrt[a - b*x^4])

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.27, size = 374, normalized size = 1.03

method	result	size
default	\(\frac {d^{2} x \sqrt {-b \,x^{4}+a}}{4 c \left (a d -b c \right )^{2} \left (-d \,x^{4}+c \right )}+\frac {b^{2} x}{2 a \left (a d -b c \right )^{2} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {\left (\frac {b d}{4 c \left (a d -b c \right )^{2}}+\frac {b^{2}}{2 a \left (a d -b c \right )^{2}}\right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {3 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (a d -3 b c \right ) \left (-\frac {\arctanh \left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \EllipticPi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\left (a d -b c \right )^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{32 c}\)	\(374\)
elliptic	\(\frac {d^{2} x \sqrt {-b \,x^{4}+a}}{4 c \left (a d -b c \right )^{2} \left (-d \,x^{4}+c \right )}+\frac {b^{2} x}{2 a \left (a d -b c \right )^{2} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {\left (\frac {b d}{4 c \left (a d -b c \right )^{2}}+\frac {b^{2}}{2 a \left (a d -b c \right )^{2}}\right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {3 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (a d -3 b c \right ) \left (-\frac {\arctanh \left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \EllipticPi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\left (a d -b c \right )^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{32 c}\)	\(374\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-b*x^4+a)^(3/2)/(-d*x^4+c)^2,x,method=_RETURNVERBOSE)

[Out]

1/4*d^2/c/(a*d-b*c)^2*x*(-b*x^4+a)^(1/2)/(-d*x^4+c)+1/2*b^2/a*x/(a*d-b*c)^2/(-(x^4-a/b)*b)^(1/2)+(1/4*b*d/c/(a
*d-b*c)^2+1/2*b^2/a/(a*d-b*c)^2)/(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)*EllipticF(x*(1/a^(1/2)*b^(1/2))^(1/2),I)-3/32/c*sum((a*d-3*b*c)/(a*d-b*c)^2/_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)*Ell
ipticPi(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))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x^4+a)^(3/2)/(-d*x^4+c)^2,x, algorithm="maxima")

[Out]

integrate(1/((-b*x^4 + a)^(3/2)*(d*x^4 - c)^2), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x^4+a)^(3/2)/(-d*x^4+c)^2,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a - b x^{4}\right )^{\frac {3}{2}} \left (- c + d x^{4}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x**4+a)**(3/2)/(-d*x**4+c)**2,x)

[Out]

Integral(1/((a - b*x**4)**(3/2)*(-c + d*x**4)**2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x^4+a)^(3/2)/(-d*x^4+c)^2,x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(3/2)*(d*x^4 - c)^2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (a-b\,x^4\right )}^{3/2}\,{\left (c-d\,x^4\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a - b*x^4)^(3/2)*(c - d*x^4)^2),x)

[Out]

int(1/((a - b*x^4)^(3/2)*(c - d*x^4)^2), x)

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