\(\int \frac {(a-b x^4)^{5/2}}{c-d x^4} \, dx\) [173]

Optimal result

Integrand size = 23, antiderivative size = 321 \[ \int \frac {\left (a-b x^4\right )^{5/2}}{c-d x^4} \, dx=-\frac {b (7 b c-13 a d) x \sqrt {a-b x^4}}{21 d^2}+\frac {b x \left (a-b x^4\right )^{3/2}}{7 d}+\frac {\sqrt [4]{a} b^{3/4} \left (21 b^2 c^2-56 a b c d+47 a^2 d^2\right ) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{21 d^3 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} (b c-a d)^3 \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c d^3 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} (b c-a d)^3 \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c d^3 \sqrt {a-b x^4}} \]

[Out]

1/7*b*x*(-b*x^4+a)^(3/2)/d-1/21*b*(-13*a*d+7*b*c)*x*(-b*x^4+a)^(1/2)/d^2+1/21*a^(1/4)*b^(3/4)*(47*a^2*d^2-56*a
*b*c*d+21*b^2*c^2)*EllipticF(b^(1/4)*x/a^(1/4),I)*(1-b*x^4/a)^(1/2)/d^3/(-b*x^4+a)^(1/2)-1/2*a^(1/4)*(-a*d+b*c
)^3*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/d^3/(-b*x^4+a
)^(1/2)-1/2*a^(1/4)*(-a*d+b*c)^3*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/d^3/(-b*x^4+a)^(1/2)

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {427, 542, 537, 230, 227, 418, 1233, 1232} \[ \int \frac {\left (a-b x^4\right )^{5/2}}{c-d x^4} \, dx=\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (47 a^2 d^2-56 a b c d+21 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{21 d^3 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (b c-a d)^3 \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c d^3 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (b c-a d)^3 \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c d^3 \sqrt {a-b x^4}}-\frac {b x \sqrt {a-b x^4} (7 b c-13 a d)}{21 d^2}+\frac {b x \left (a-b x^4\right )^{3/2}}{7 d} \]

[In]

Int[(a - b*x^4)^(5/2)/(c - d*x^4),x]

[Out]

-1/21*(b*(7*b*c - 13*a*d)*x*Sqrt[a - b*x^4])/d^2 + (b*x*(a - b*x^4)^(3/2))/(7*d) + (a^(1/4)*b^(3/4)*(21*b^2*c^
2 - 56*a*b*c*d + 47*a^2*d^2)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(21*d^3*Sqrt[a -
b*x^4]) - (a^(1/4)*(b*c - a*d)^3*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*d^3*Sqrt[a - b*x^4]) - (a^(1/4)*(b*c - a*d)^3*Sqrt[1 - (b*x^4)/a]*Ell
ipticPi[(Sqrt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c]), ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(2*b^(1/4)*c*d^3*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 427

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c
+ d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[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 542

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 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 x \left (a-b x^4\right )^{3/2}}{7 d}-\frac {\int \frac {\sqrt {a-b x^4} \left (a (b c-7 a d)-b (7 b c-13 a d) x^4\right )}{c-d x^4} \, dx}{7 d} \\ & = -\frac {b (7 b c-13 a d) x \sqrt {a-b x^4}}{21 d^2}+\frac {b x \left (a-b x^4\right )^{3/2}}{7 d}+\frac {\int \frac {a \left (7 b^2 c^2-16 a b c d+21 a^2 d^2\right )-b \left (21 b^2 c^2-56 a b c d+47 a^2 d^2\right ) x^4}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{21 d^2} \\ & = -\frac {b (7 b c-13 a d) x \sqrt {a-b x^4}}{21 d^2}+\frac {b x \left (a-b x^4\right )^{3/2}}{7 d}-\frac {(b c-a d)^3 \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{d^3}+\frac {\left (b \left (21 b^2 c^2-56 a b c d+47 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a-b x^4}} \, dx}{21 d^3} \\ & = -\frac {b (7 b c-13 a d) x \sqrt {a-b x^4}}{21 d^2}+\frac {b x \left (a-b x^4\right )^{3/2}}{7 d}-\frac {(b c-a d)^3 \int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{2 c d^3}-\frac {(b c-a d)^3 \int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{2 c d^3}+\frac {\left (b \left (21 b^2 c^2-56 a b c d+47 a^2 d^2\right ) \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{21 d^3 \sqrt {a-b x^4}} \\ & = -\frac {b (7 b c-13 a d) x \sqrt {a-b x^4}}{21 d^2}+\frac {b x \left (a-b x^4\right )^{3/2}}{7 d}+\frac {\sqrt [4]{a} b^{3/4} \left (21 b^2 c^2-56 a b c d+47 a^2 d^2\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 )}{21 d^3 \sqrt {a-b x^4}}-\frac {\left ((b c-a d)^3 \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 d^3 \sqrt {a-b x^4}}-\frac {\left ((b c-a d)^3 \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 d^3 \sqrt {a-b x^4}} \\ & = -\frac {b (7 b c-13 a d) x \sqrt {a-b x^4}}{21 d^2}+\frac {b x \left (a-b x^4\right )^{3/2}}{7 d}+\frac {\sqrt [4]{a} b^{3/4} \left (21 b^2 c^2-56 a b c d+47 a^2 d^2\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 )}{21 d^3 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} (b c-a d)^3 \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 d^3 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} (b c-a d)^3 \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 d^3 \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 = 10.70 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a-b x^4\right )^{5/2}}{c-d x^4} \, dx=\frac {x \left (5 b \left (-a+b x^4\right ) \left (7 b c-16 a d+3 b d x^4\right )-\frac {b \left (21 b^2 c^2-56 a b c d+47 a^2 d^2\right ) x^4 \sqrt {1-\frac {b x^4}{a}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )}{c}+\frac {25 a^2 c \left (7 b^2 c^2-16 a b c d+21 a^2 d^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )}{\left (c-d x^4\right ) \left (5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^4 \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}\right )}{105 d^2 \sqrt {a-b x^4}} \]

[In]

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

[Out]

(x*(5*b*(-a + b*x^4)*(7*b*c - 16*a*d + 3*b*d*x^4) - (b*(21*b^2*c^2 - 56*a*b*c*d + 47*a^2*d^2)*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^2*c*(7*b^2*c^2 - 16*a*b*c*d + 21*a^2*d^2)*
AppellF1[1/4, 1/2, 1, 5/4, (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])))))/(105*d^2*Sqrt[a - b*x^4])

Maple [C] (warning: unable to verify)

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

Time = 6.41 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.09

method	result	size
risch	\(\frac {b x \left (-3 b d \,x^{4}+16 a d -7 b c \right ) \sqrt {-b \,x^{4}+a}}{21 d^{2}}+\frac {\frac {b \left (47 a^{2} d^{2}-56 a b c d +21 b^{2} c^{2}\right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{d \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}+\frac {\left (-21 a^{3} d^{3}+63 a^{2} b c \,d^{2}-63 a \,b^{2} c^{2} d +21 b^{3} c^{3}\right ) \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{8 d^{2}}}{21 d^{2}}\)	\(350\)
default	\(-\frac {b^{2} x^{5} \sqrt {-b \,x^{4}+a}}{7 d}-\frac {\left (-\frac {b^{2} \left (3 a d -b c \right )}{d^{2}}+\frac {5 b^{2} a}{7 d}\right ) x \sqrt {-b \,x^{4}+a}}{3 b}+\frac {\left (\frac {b \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right )}{d^{3}}+\frac {\left (-\frac {b^{2} \left (3 a d -b c \right )}{d^{2}}+\frac {5 b^{2} a}{7 d}\right ) a}{3 b}\right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{8 d^{4}}\)	\(408\)
elliptic	\(-\frac {b^{2} x^{5} \sqrt {-b \,x^{4}+a}}{7 d}-\frac {\left (-\frac {b^{2} \left (3 a d -b c \right )}{d^{2}}+\frac {5 b^{2} a}{7 d}\right ) x \sqrt {-b \,x^{4}+a}}{3 b}+\frac {\left (\frac {b \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right )}{d^{3}}+\frac {\left (-\frac {b^{2} \left (3 a d -b c \right )}{d^{2}}+\frac {5 b^{2} a}{7 d}\right ) a}{3 b}\right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{8 d^{4}}\)	\(408\)

[In]

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

[Out]

1/21*b*x*(-3*b*d*x^4+16*a*d-7*b*c)*(-b*x^4+a)^(1/2)/d^2+1/21/d^2*(b*(47*a^2*d^2-56*a*b*c*d+21*b^2*c^2)/d/(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*(-21*a^3*d^3+63*a^2*b*c*d^2-63*a*b^2*c^2*d+21*b^3*c^3)/d^2*sum(1/_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)*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)))

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^4\right )^{5/2}}{c-d x^4} \, dx=\text {Timed out} \]

[In]

integrate((-b*x^4+a)^(5/2)/(-d*x^4+c),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

-Integral(a**2*sqrt(a - b*x**4)/(-c + d*x**4), x) - Integral(b**2*x**8*sqrt(a - b*x**4)/(-c + d*x**4), x) - In
tegral(-2*a*b*x**4*sqrt(a - b*x**4)/(-c + d*x**4), x)

Maxima [F]

\[ \int \frac {\left (a-b x^4\right )^{5/2}}{c-d x^4} \, dx=\int { -\frac {{\left (-b x^{4} + a\right )}^{\frac {5}{2}}}{d x^{4} - c} \,d x } \]

[In]

integrate((-b*x^4+a)^(5/2)/(-d*x^4+c),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\left (a-b x^4\right )^{5/2}}{c-d x^4} \, dx=\int { -\frac {{\left (-b x^{4} + a\right )}^{\frac {5}{2}}}{d x^{4} - c} \,d x } \]

[In]

integrate((-b*x^4+a)^(5/2)/(-d*x^4+c),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^4\right )^{5/2}}{c-d x^4} \, dx=\int \frac {{\left (a-b\,x^4\right )}^{5/2}}{c-d\,x^4} \,d x \]

[In]

int((a - b*x^4)^(5/2)/(c - d*x^4),x)

[Out]

int((a - b*x^4)^(5/2)/(c - d*x^4), x)