Integrand size = 23, antiderivative size = 277 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{c-d x^4} \, dx=\frac {b x \sqrt {a-b x^4}}{3 d}-\frac {\sqrt [4]{a} b^{3/4} (3 b c-5 a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{3 d^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} (b c-a d)^2 \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c d^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} (b c-a d)^2 \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c d^2 \sqrt {a-b x^4}} \] Output:
1/3*b*x*(-b*x^4+a)^(1/2)/d-1/3*a^(1/4)*b^(3/4)*(-5*a*d+3*b*c)*(1-b*x^4/a)^ (1/2)*EllipticF(b^(1/4)*x/a^(1/4),I)/d^2/(-b*x^4+a)^(1/2)+1/2*a^(1/4)*(-a* d+b*c)^2*(1-b*x^4/a)^(1/2)*EllipticPi(b^(1/4)*x/a^(1/4),-a^(1/2)*d^(1/2)/b ^(1/2)/c^(1/2),I)/b^(1/4)/c/d^2/(-b*x^4+a)^(1/2)+1/2*a^(1/4)*(-a*d+b*c)^2* (1-b*x^4/a)^(1/2)*EllipticPi(b^(1/4)*x/a^(1/4),a^(1/2)*d^(1/2)/b^(1/2)/c^( 1/2),I)/b^(1/4)/c/d^2/(-b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.36 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{c-d x^4} \, dx=-\frac {x \left (\frac {b (-3 b c+5 a d) x^4 \sqrt {1-\frac {b x^4}{a}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )}{c}+\frac {5 \left (5 a c \left (3 a^2 d-a b d x^4+b^2 x^4 \left (-c+d x^4\right )\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 b x^4 \left (a-b x^4\right ) \left (c-d x^4\right ) \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}{\left (-c+d x^4\right ) \left (5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^4 \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}\right )}{15 d \sqrt {a-b x^4}} \] Input:
Integrate[(a - b*x^4)^(3/2)/(c - d*x^4),x]
Output:
-1/15*(x*((b*(-3*b*c + 5*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*(3*a^2*d - a*b*d*x^4 + b^2*x^4 *(-c + d*x^4))*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*b*x^4* (a - b*x^4)*(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])))))/(d*Sqrt[a - b*x^4])
Time = 0.82 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {933, 1021, 765, 762, 925, 27, 1543, 1542}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a-b x^4\right )^{3/2}}{c-d x^4} \, dx\) |
| \(\Big \downarrow \) 933 |
| \(\displaystyle \frac {b x \sqrt {a-b x^4}}{3 d}-\frac {\int \frac {a (b c-3 a d)-b (3 b c-5 a d) x^4}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx}{3 d}\) |
| \(\Big \downarrow \) 1021 |
| \(\displaystyle \frac {b x \sqrt {a-b x^4}}{3 d}-\frac {\frac {b (3 b c-5 a d) \int \frac {1}{\sqrt {a-b x^4}}dx}{d}-\frac {3 (b c-a d)^2 \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx}{d}}{3 d}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {b x \sqrt {a-b x^4}}{3 d}-\frac {\frac {b \sqrt {1-\frac {b x^4}{a}} (3 b c-5 a d) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{d \sqrt {a-b x^4}}-\frac {3 (b c-a d)^2 \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx}{d}}{3 d}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {b x \sqrt {a-b x^4}}{3 d}-\frac {\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} (3 b c-5 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{d \sqrt {a-b x^4}}-\frac {3 (b c-a d)^2 \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx}{d}}{3 d}\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle \frac {b x \sqrt {a-b x^4}}{3 d}-\frac {\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} (3 b c-5 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{d \sqrt {a-b x^4}}-\frac {3 (b c-a d)^2 \left (\frac {\int \frac {\sqrt {c}}{\left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {a-b x^4}}dx}{2 c}+\frac {\int \frac {\sqrt {c}}{\left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {a-b x^4}}dx}{2 c}\right )}{d}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b x \sqrt {a-b x^4}}{3 d}-\frac {\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} (3 b c-5 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{d \sqrt {a-b x^4}}-\frac {3 (b c-a d)^2 \left (\frac {\int \frac {1}{\left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {a-b x^4}}dx}{2 \sqrt {c}}+\frac {\int \frac {1}{\left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {a-b x^4}}dx}{2 \sqrt {c}}\right )}{d}}{3 d}\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle \frac {b x \sqrt {a-b x^4}}{3 d}-\frac {\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} (3 b c-5 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{d \sqrt {a-b x^4}}-\frac {3 (b c-a d)^2 \left (\frac {\sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {1-\frac {b x^4}{a}}}dx}{2 \sqrt {c} \sqrt {a-b x^4}}+\frac {\sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {1-\frac {b x^4}{a}}}dx}{2 \sqrt {c} \sqrt {a-b x^4}}\right )}{d}}{3 d}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle \frac {b x \sqrt {a-b x^4}}{3 d}-\frac {\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} (3 b c-5 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{d \sqrt {a-b x^4}}-\frac {3 (b c-a d)^2 \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4}}\right )}{d}}{3 d}\) |
Input:
Int[(a - b*x^4)^(3/2)/(c - d*x^4),x]
Output:
(b*x*Sqrt[a - b*x^4])/(3*d) - ((a^(1/4)*b^(3/4)*(3*b*c - 5*a*d)*Sqrt[1 - ( b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(d*Sqrt[a - b*x^4]) - (3*(b*c - a*d)^2*((a^(1/4)*Sqrt[1 - (b*x^4)/a]*EllipticPi[-((Sqrt[a]*Sqr t[d])/(Sqrt[b]*Sqrt[c])), ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(2*b^(1/4)*c*S qrt[a - b*x^4]) + (a^(1/4)*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*Sqrt[a - b*x^4])))/d)/(3*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[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]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Simp[ 1/(2*c) Int[1/(Sqrt[a + b*x^4]*(1 - Rt[-d/c, 2]*x^2)), x], x] + Simp[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]
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] + Simp[1/(b*(n*(p + q) + 1)) Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Sim p[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] /; FreeQ[{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] && IntBinomialQ[ a, b, c, d, n, p, q, x]
Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x _)^(n_)]), x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^n], x], x] + Simp[(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]
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]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ 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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.39 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.10
| method | result | size |
| risch | \(\frac {b x \sqrt {-b \,x^{4}+a}}{3 d}+\frac {\frac {b \left (5 a d -3 b c \right ) \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{d \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {3 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} d -c \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{8 d^{2}}}{3 d}\) | \(304\) |
| default | \(\frac {b x \sqrt {-b \,x^{4}+a}}{3 d}+\frac {\left (\frac {b \left (2 a d -b c \right )}{d^{2}}-\frac {b a}{3 d}\right ) \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} d -c \right )}{\sum }\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{8 d^{3}}\) | \(307\) |
| elliptic | \(\frac {b x \sqrt {-b \,x^{4}+a}}{3 d}+\frac {\left (\frac {b \left (2 a d -b c \right )}{d^{2}}-\frac {b a}{3 d}\right ) \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} d -c \right )}{\sum }\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{8 d^{3}}\) | \(307\) |
Input:
int((-b*x^4+a)^(3/2)/(-d*x^4+c),x,method=_RETURNVERBOSE)
Output:
1/3*b*x*(-b*x^4+a)^(1/2)/d+1/3/d*(b*(5*a*d-3*b*c)/d/(1/a^(1/2)*b^(1/2))^(1 /2)*(1-b^(1/2)*x^2/a^(1/2))^(1/2)*(1+b^(1/2)*x^2/a^(1/2))^(1/2)/(-b*x^4+a) ^(1/2)*EllipticF(x*(1/a^(1/2)*b^(1/2))^(1/2),I)-3/8*(a^2*d^2-2*a*b*c*d+b^2 *c^2)/d^2*sum(1/_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-b^(1/2)*x^2/a^(1/2))^(1/2)*(1+b^(1/2)*x^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)))
Timed out. \[ \int \frac {\left (a-b x^4\right )^{3/2}}{c-d x^4} \, dx=\text {Timed out} \] Input:
integrate((-b*x^4+a)^(3/2)/(-d*x^4+c),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (a-b x^4\right )^{3/2}}{c-d x^4} \, dx=- \int \frac {a \sqrt {a - b x^{4}}}{- c + d x^{4}}\, dx - \int \left (- \frac {b x^{4} \sqrt {a - b x^{4}}}{- c + d x^{4}}\right )\, dx \] Input:
integrate((-b*x**4+a)**(3/2)/(-d*x**4+c),x)
Output:
-Integral(a*sqrt(a - b*x**4)/(-c + d*x**4), x) - Integral(-b*x**4*sqrt(a - b*x**4)/(-c + d*x**4), x)
\[ \int \frac {\left (a-b x^4\right )^{3/2}}{c-d x^4} \, dx=\int { -\frac {{\left (-b x^{4} + a\right )}^{\frac {3}{2}}}{d x^{4} - c} \,d x } \] Input:
integrate((-b*x^4+a)^(3/2)/(-d*x^4+c),x, algorithm="maxima")
Output:
-integrate((-b*x^4 + a)^(3/2)/(d*x^4 - c), x)
\[ \int \frac {\left (a-b x^4\right )^{3/2}}{c-d x^4} \, dx=\int { -\frac {{\left (-b x^{4} + a\right )}^{\frac {3}{2}}}{d x^{4} - c} \,d x } \] Input:
integrate((-b*x^4+a)^(3/2)/(-d*x^4+c),x, algorithm="giac")
Output:
integrate(-(-b*x^4 + a)^(3/2)/(d*x^4 - c), x)
Timed out. \[ \int \frac {\left (a-b x^4\right )^{3/2}}{c-d x^4} \, dx=\int \frac {{\left (a-b\,x^4\right )}^{3/2}}{c-d\,x^4} \,d x \] Input:
int((a - b*x^4)^(3/2)/(c - d*x^4),x)
Output:
int((a - b*x^4)^(3/2)/(c - d*x^4), x)
\[ \int \frac {\left (a-b x^4\right )^{3/2}}{c-d x^4} \, dx=\frac {\sqrt {-b \,x^{4}+a}\, b x +3 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{b d \,x^{8}-a d \,x^{4}-b c \,x^{4}+a c}d x \right ) a^{2} d -\left (\int \frac {\sqrt {-b \,x^{4}+a}}{b d \,x^{8}-a d \,x^{4}-b c \,x^{4}+a c}d x \right ) a b c -5 \left (\int \frac {\sqrt {-b \,x^{4}+a}\, x^{4}}{b d \,x^{8}-a d \,x^{4}-b c \,x^{4}+a c}d x \right ) a b d +3 \left (\int \frac {\sqrt {-b \,x^{4}+a}\, x^{4}}{b d \,x^{8}-a d \,x^{4}-b c \,x^{4}+a c}d x \right ) b^{2} c}{3 d} \] Input:
int((-b*x^4+a)^(3/2)/(-d*x^4+c),x)
Output:
(sqrt(a - b*x**4)*b*x + 3*int(sqrt(a - b*x**4)/(a*c - a*d*x**4 - b*c*x**4 + b*d*x**8),x)*a**2*d - int(sqrt(a - b*x**4)/(a*c - a*d*x**4 - b*c*x**4 + b*d*x**8),x)*a*b*c - 5*int((sqrt(a - b*x**4)*x**4)/(a*c - a*d*x**4 - b*c*x **4 + b*d*x**8),x)*a*b*d + 3*int((sqrt(a - b*x**4)*x**4)/(a*c - a*d*x**4 - b*c*x**4 + b*d*x**8),x)*b**2*c)/(3*d)