\(\int \frac {(a-b x^4)^{3/2}}{(c-d x^4)^2} \, dx\) [186]

Optimal result

Integrand size = 23, antiderivative size = 309 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{\left (c-d x^4\right )^2} \, dx=-\frac {(b c-a d) x \sqrt {a-b x^4}}{4 c d \left (c-d x^4\right )}+\frac {\sqrt [4]{a} b^{3/4} (3 b c+a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{4 c d^2 \sqrt {a-b x^4}}-\frac {3 \sqrt [4]{a} (b c-a d) (b c+a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d^2 \sqrt {a-b x^4}}-\frac {3 \sqrt [4]{a} (b c-a d) (b c+a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d^2 \sqrt {a-b x^4}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {424, 537, 230, 227, 418, 1233, 1232} \[ \int \frac {\left (a-b x^4\right )^{3/2}}{\left (c-d x^4\right )^2} \, dx=\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} (a d+3 b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{4 c d^2 \sqrt {a-b x^4}}-\frac {3 \sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (b c-a d) (a d+b c) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d^2 \sqrt {a-b x^4}}-\frac {3 \sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (b c-a d) (a d+b c) \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d^2 \sqrt {a-b x^4}}-\frac {x \sqrt {a-b x^4} (b c-a d)}{4 c d \left (c-d x^4\right )} \]

[In]

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

[Out]

-1/4*((b*c - a*d)*x*Sqrt[a - b*x^4])/(c*d*(c - d*x^4)) + (a^(1/4)*b^(3/4)*(3*b*c + a*d)*Sqrt[1 - (b*x^4)/a]*El
lipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(4*c*d^2*Sqrt[a - b*x^4]) - (3*a^(1/4)*(b*c - a*d)*(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*d^2*Sqrt[a - b*x^4]) - (3*a^(1/4)*(b*c - a*d)*(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*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 424

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

Mathematica [C] (warning: unable to verify)

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

Time = 10.33 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{\left (c-d x^4\right )^2} \, dx=\frac {x \left (-b (3 b c+a d) x^4 \sqrt {1-\frac {b x^4}{a}} \left (-c+d x^4\right ) \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+\frac {5 c \left (-5 a c \left (4 a^2 d+b^2 c x^4-a b d x^4\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )-2 (-b c+a d) x^4 \left (a-b x^4\right ) \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}{5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^4 \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )}\right )}{20 c^2 d \sqrt {a-b x^4} \left (-c+d x^4\right )} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 4.03 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.06

method	result	size
default	\(\frac {\left (a d -b c \right ) x \sqrt {-b \,x^{4}+a}}{4 d c \left (-d \,x^{4}+c \right )}+\frac {\left (\frac {b^{2}}{d^{2}}+\frac {b \left (a d -b c \right )}{4 d^{2} c}\right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {3 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (a^{2} d^{2}-b^{2} c^{2}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{32 c \,d^{3}}\)	\(328\)
elliptic	\(\frac {\left (a d -b c \right ) x \sqrt {-b \,x^{4}+a}}{4 d c \left (-d \,x^{4}+c \right )}+\frac {\left (\frac {b^{2}}{d^{2}}+\frac {b \left (a d -b c \right )}{4 d^{2} c}\right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {3 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (a^{2} d^{2}-b^{2} c^{2}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{32 c \,d^{3}}\)	\(328\)

[In]

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

[Out]

1/4/d*(a*d-b*c)/c*x*(-b*x^4+a)^(1/2)/(-d*x^4+c)+(b^2/d^2+1/4*b/d^2*(a*d-b*c)/c)/(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/d^3*sum((a^2*d^2-b^2*c^2)/_alpha^3*(-1/(1/d*(a*d-b*c))^(1/2)*arctanh(1/2*(-2*_alpha^2*b*x^2+2*a)/(1
/d*(a*d-b*c))^(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))

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^4\right )^{3/2}}{\left (c-d x^4\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^4\right )^{3/2}}{\left (c-d x^4\right )^2} \, dx=\int \frac {{\left (a-b\,x^4\right )}^{3/2}}{{\left (c-d\,x^4\right )}^2} \,d x \]

[In]

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

[Out]

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