3.2.78 \(\int \frac {1}{(a-b x^4)^{5/2} (c-d x^4)} \, dx\) [178]

Optimal. Leaf size=334 \[ \frac {b x}{6 a (b c-a d) \left (a-b x^4\right )^{3/2}}+\frac {b (5 b c-11 a d) x}{12 a^2 (b c-a d)^2 \sqrt {a-b x^4}}+\frac {b^{3/4} (5 b c-11 a d) \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{12 a^{7/4} (b c-a d)^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} d^2 \sqrt {1-\frac {b x^4}{a}} \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c (b c-a d)^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} d^2 \sqrt {1-\frac {b x^4}{a}} \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c (b c-a d)^2 \sqrt {a-b x^4}} \]

[Out]

1/6*b*x/a/(-a*d+b*c)/(-b*x^4+a)^(3/2)+1/12*b*(-11*a*d+5*b*c)*x/a^2/(-a*d+b*c)^2/(-b*x^4+a)^(1/2)+1/12*b^(3/4)*
(-11*a*d+5*b*c)*EllipticF(b^(1/4)*x/a^(1/4),I)*(1-b*x^4/a)^(1/2)/a^(7/4)/(-a*d+b*c)^2/(-b*x^4+a)^(1/2)+1/2*a^(
1/4)*d^2*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/(-a*d+b*
c)^2/(-b*x^4+a)^(1/2)+1/2*a^(1/4)*d^2*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/(-a*d+b*c)^2/(-b*x^4+a)^(1/2)

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Rubi [A]
time = 0.26, antiderivative size = 334, 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}} (5 b c-11 a d) F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{12 a^{7/4} \sqrt {a-b x^4} (b c-a d)^2}+\frac {b x (5 b c-11 a d)}{12 a^2 \sqrt {a-b x^4} (b c-a d)^2}+\frac {\sqrt [4]{a} d^2 \sqrt {1-\frac {b x^4}{a}} \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 )}{2 \sqrt [4]{b} c \sqrt {a-b x^4} (b c-a d)^2}+\frac {\sqrt [4]{a} d^2 \sqrt {1-\frac {b x^4}{a}} \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 )}{2 \sqrt [4]{b} c \sqrt {a-b x^4} (b c-a d)^2}+\frac {b x}{6 a \left (a-b x^4\right )^{3/2} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a - b*x^4)^(5/2)*(c - d*x^4)),x]

[Out]

(b*x)/(6*a*(b*c - a*d)*(a - b*x^4)^(3/2)) + (b*(5*b*c - 11*a*d)*x)/(12*a^2*(b*c - a*d)^2*Sqrt[a - b*x^4]) + (b
^(3/4)*(5*b*c - 11*a*d)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(12*a^(7/4)*(b*c - a*d
)^2*Sqrt[a - b*x^4]) + (a^(1/4)*d^2*Sqrt[1 - (b*x^4)/a]*EllipticPi[-((Sqrt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c])), Arc
Sin[(b^(1/4)*x)/a^(1/4)], -1])/(2*b^(1/4)*c*(b*c - a*d)^2*Sqrt[a - b*x^4]) + (a^(1/4)*d^2*Sqrt[1 - (b*x^4)/a]*
EllipticPi[(Sqrt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c]), ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(2*b^(1/4)*c*(b*c - a*d)^2*S
qrt[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 )^{5/2} \left (c-d x^4\right )} \, dx &=\frac {b x}{6 a (b c-a d) \left (a-b x^4\right )^{3/2}}+\frac {\int \frac {5 b c-6 a d-5 b d x^4}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx}{6 a (b c-a d)}\\ &=\frac {b x}{6 a (b c-a d) \left (a-b x^4\right )^{3/2}}+\frac {b (5 b c-11 a d) x}{12 a^2 (b c-a d)^2 \sqrt {a-b x^4}}+\frac {\int \frac {5 b^2 c^2-11 a b c d+12 a^2 d^2-b d (5 b c-11 a d) x^4}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{12 a^2 (b c-a d)^2}\\ &=\frac {b x}{6 a (b c-a d) \left (a-b x^4\right )^{3/2}}+\frac {b (5 b c-11 a d) x}{12 a^2 (b c-a d)^2 \sqrt {a-b x^4}}+\frac {d^2 \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{(b c-a d)^2}+\frac {(b (5 b c-11 a d)) \int \frac {1}{\sqrt {a-b x^4}} \, dx}{12 a^2 (b c-a d)^2}\\ &=\frac {b x}{6 a (b c-a d) \left (a-b x^4\right )^{3/2}}+\frac {b (5 b c-11 a d) x}{12 a^2 (b c-a d)^2 \sqrt {a-b x^4}}+\frac {d^2 \int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{2 c (b c-a d)^2}+\frac {d^2 \int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{2 c (b c-a d)^2}+\frac {\left (b (5 b c-11 a d) \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{12 a^2 (b c-a d)^2 \sqrt {a-b x^4}}\\ &=\frac {b x}{6 a (b c-a d) \left (a-b x^4\right )^{3/2}}+\frac {b (5 b c-11 a d) x}{12 a^2 (b c-a d)^2 \sqrt {a-b x^4}}+\frac {b^{3/4} (5 b c-11 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 )}{12 a^{7/4} (b c-a d)^2 \sqrt {a-b x^4}}+\frac {\left (d^2 \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}{2 c (b c-a d)^2 \sqrt {a-b x^4}}+\frac {\left (d^2 \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}{2 c (b c-a d)^2 \sqrt {a-b x^4}}\\ &=\frac {b x}{6 a (b c-a d) \left (a-b x^4\right )^{3/2}}+\frac {b (5 b c-11 a d) x}{12 a^2 (b c-a d)^2 \sqrt {a-b x^4}}+\frac {b^{3/4} (5 b c-11 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 )}{12 a^{7/4} (b c-a d)^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} d^2 \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 )}{2 \sqrt [4]{b} c (b c-a d)^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} d^2 \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 )}{2 \sqrt [4]{b} c (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 = 10.51, size = 422, normalized size = 1.26 \begin {gather*} \frac {x \left (\frac {b d (-5 b c+11 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 )}{c}-\frac {5 \left (5 a c \left (12 a^3 d^2+a^2 b d \left (-24 c+d x^4\right )+5 b^3 c x^4 \left (-2 c+d x^4\right )+a b^2 \left (12 c^2+15 c d x^4-11 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 )+2 b x^4 \left (-c+d x^4\right ) \left (13 a^2 d+5 b^2 c x^4-a b \left (7 c+11 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 (a-b x^4\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 )}{60 a^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)^(5/2)*(c - d*x^4)),x]

[Out]

(x*((b*d*(-5*b*c + 11*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 - (5*(5
*a*c*(12*a^3*d^2 + a^2*b*d*(-24*c + d*x^4) + 5*b^3*c*x^4*(-2*c + d*x^4) + a*b^2*(12*c^2 + 15*c*d*x^4 - 11*d^2*
x^8))*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*b*x^4*(-c + d*x^4)*(13*a^2*d + 5*b^2*c*x^4 - a*b*(7
*c + 11*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])))/((a - b*x^4)*(-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]
)))))/(60*a^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 = 0.28, size = 361, normalized size = 1.08

method	result	size
default	\(-\frac {x \sqrt {-b \,x^{4}+a}}{6 a b \left (a d -b c \right ) \left (x^{4}-\frac {a}{b}\right )^{2}}-\frac {b x \left (11 a d -5 b c \right )}{12 a^{2} \left (a d -b c \right )^{2} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}-\frac {b \left (11 a d -5 b c \right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{12 a^{2} \left (a d -b c \right )^{2} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {d \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {-\frac {\arctanh \left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \EllipticPi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}}{\left (a d -b c \right )^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{8}\)	\(361\)
elliptic	\(-\frac {x \sqrt {-b \,x^{4}+a}}{6 a b \left (a d -b c \right ) \left (x^{4}-\frac {a}{b}\right )^{2}}-\frac {b x \left (11 a d -5 b c \right )}{12 a^{2} \left (a d -b c \right )^{2} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}-\frac {b \left (11 a d -5 b c \right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{12 a^{2} \left (a d -b c \right )^{2} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {d \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {-\frac {\arctanh \left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \EllipticPi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}}{\left (a d -b c \right )^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{8}\)	\(361\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-b*x^4+a)^(5/2)/(-d*x^4+c),x,method=_RETURNVERBOSE)

[Out]

-1/6/a*x/b/(a*d-b*c)*(-b*x^4+a)^(1/2)/(x^4-a/b)^2-1/12*b/a^2*x*(11*a*d-5*b*c)/(a*d-b*c)^2/(-(x^4-a/b)*b)^(1/2)
-1/12*b/a^2*(11*a*d-5*b*c)/(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)-1/8*d*sum(1/(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)*Elliptic
Pi(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)^(5/2)/(-d*x^4+c),x, algorithm="maxima")

[Out]

-integrate(1/((-b*x^4 + a)^(5/2)*(d*x^4 - c)), 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)^(5/2)/(-d*x^4+c),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}{- a^{2} c \sqrt {a - b x^{4}} + a^{2} d x^{4} \sqrt {a - b x^{4}} + 2 a b c x^{4} \sqrt {a - b x^{4}} - 2 a b d x^{8} \sqrt {a - b x^{4}} - b^{2} c x^{8} \sqrt {a - b x^{4}} + b^{2} d x^{12} \sqrt {a - b x^{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x**4+a)**(5/2)/(-d*x**4+c),x)

[Out]

-Integral(1/(-a**2*c*sqrt(a - b*x**4) + a**2*d*x**4*sqrt(a - b*x**4) + 2*a*b*c*x**4*sqrt(a - b*x**4) - 2*a*b*d
*x**8*sqrt(a - b*x**4) - b**2*c*x**8*sqrt(a - b*x**4) + b**2*d*x**12*sqrt(a - b*x**4)), 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)^(5/2)/(-d*x^4+c),x, algorithm="giac")

[Out]

integrate(-1/((-b*x^4 + a)^(5/2)*(d*x^4 - c)), 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 )}^{5/2}\,\left (c-d\,x^4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a - b*x^4)^(5/2)*(c - d*x^4)),x)

[Out]

int(1/((a - b*x^4)^(5/2)*(c - d*x^4)), x)

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