Integrand size = 23, antiderivative size = 334 \[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )} \, dx=\frac {b x}{6 a (b c-a d) \left (a-b x^4\right )^{3/2}}+\frac {b (5 b c-11 a d) x}{12 a^2 (b c-a d)^2 \sqrt {a-b x^4}}+\frac {b^{3/4} (5 b c-11 a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{12 a^{7/4} (b c-a d)^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} d^2 \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c (b c-a d)^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} d^2 \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c (b c-a d)^2 \sqrt {a-b x^4}} \] Output:
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)*(1-b*x^4/a)^(1/2)*Elli pticF(b^(1/4)*x/a^(1/4),I)/a^(7/4)/(-a*d+b*c)^2/(-b*x^4+a)^(1/2)+1/2*a^(1/ 4)*d^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/(-a*d+b*c)^2/(-b*x^4+a)^(1/2)+1/2*a^(1/4)*d^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/(-a*d+b*c)^2/(-b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.76 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )} \, dx=\frac {x \left (\frac {b d (-5 b c+11 a d) x^4 \sqrt {1-\frac {b x^4}{a}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )}{c}-\frac {5 \left (5 a c \left (12 a^3 d^2+a^2 b d \left (-24 c+d x^4\right )+5 b^3 c x^4 \left (-2 c+d x^4\right )+a b^2 \left (12 c^2+15 c d x^4-11 d^2 x^8\right )\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 b x^4 \left (-c+d x^4\right ) \left (13 a^2 d+5 b^2 c x^4-a b \left (7 c+11 d x^4\right )\right ) \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}{\left (a-b x^4\right ) \left (-c+d x^4\right ) \left (5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^4 \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}\right )}{60 a^2 (b c-a d)^2 \sqrt {a-b x^4}} \] Input:
Integrate[1/((a - b*x^4)^(5/2)*(c - d*x^4)),x]
Output:
(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])
Time = 1.06 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {931, 1024, 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.
| \(\displaystyle \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )} \, dx\) |
| \(\Big \downarrow \) 931 |
| \(\displaystyle \frac {\int \frac {-5 b d x^4+5 b c-6 a d}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )}dx}{6 a (b c-a d)}+\frac {b x}{6 a \left (a-b x^4\right )^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle \frac {\frac {\int \frac {-b d (5 b c-11 a d) x^4+5 b^2 c^2+12 a^2 d^2-11 a b c d}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx}{2 a (b c-a d)}+\frac {b x (5 b c-11 a d)}{2 a \sqrt {a-b x^4} (b c-a d)}}{6 a (b c-a d)}+\frac {b x}{6 a \left (a-b x^4\right )^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 1021 |
| \(\displaystyle \frac {\frac {12 a^2 d^2 \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx+b (5 b c-11 a d) \int \frac {1}{\sqrt {a-b x^4}}dx}{2 a (b c-a d)}+\frac {b x (5 b c-11 a d)}{2 a \sqrt {a-b x^4} (b c-a d)}}{6 a (b c-a d)}+\frac {b x}{6 a \left (a-b x^4\right )^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\frac {12 a^2 d^2 \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx+\frac {b \sqrt {1-\frac {b x^4}{a}} (5 b c-11 a d) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {a-b x^4}}}{2 a (b c-a d)}+\frac {b x (5 b c-11 a d)}{2 a \sqrt {a-b x^4} (b c-a d)}}{6 a (b c-a d)}+\frac {b x}{6 a \left (a-b x^4\right )^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\frac {12 a^2 d^2 \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx+\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} (5 b c-11 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {a-b x^4}}}{2 a (b c-a d)}+\frac {b x (5 b c-11 a d)}{2 a \sqrt {a-b x^4} (b c-a d)}}{6 a (b c-a d)}+\frac {b x}{6 a \left (a-b x^4\right )^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle \frac {\frac {12 a^2 d^2 \left (\frac {\int \frac {\sqrt {c}}{\left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {a-b x^4}}dx}{2 c}+\frac {\int \frac {\sqrt {c}}{\left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {a-b x^4}}dx}{2 c}\right )+\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} (5 b c-11 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {a-b x^4}}}{2 a (b c-a d)}+\frac {b x (5 b c-11 a d)}{2 a \sqrt {a-b x^4} (b c-a d)}}{6 a (b c-a d)}+\frac {b x}{6 a \left (a-b x^4\right )^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {12 a^2 d^2 \left (\frac {\int \frac {1}{\left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {a-b x^4}}dx}{2 \sqrt {c}}+\frac {\int \frac {1}{\left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {a-b x^4}}dx}{2 \sqrt {c}}\right )+\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} (5 b c-11 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {a-b x^4}}}{2 a (b c-a d)}+\frac {b x (5 b c-11 a d)}{2 a \sqrt {a-b x^4} (b c-a d)}}{6 a (b c-a d)}+\frac {b x}{6 a \left (a-b x^4\right )^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle \frac {\frac {12 a^2 d^2 \left (\frac {\sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {1-\frac {b x^4}{a}}}dx}{2 \sqrt {c} \sqrt {a-b x^4}}+\frac {\sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {1-\frac {b x^4}{a}}}dx}{2 \sqrt {c} \sqrt {a-b x^4}}\right )+\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} (5 b c-11 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {a-b x^4}}}{2 a (b c-a d)}+\frac {b x (5 b c-11 a d)}{2 a \sqrt {a-b x^4} (b c-a d)}}{6 a (b c-a d)}+\frac {b x}{6 a \left (a-b x^4\right )^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle \frac {\frac {12 a^2 d^2 \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4}}\right )+\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} (5 b c-11 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {a-b x^4}}}{2 a (b c-a d)}+\frac {b x (5 b c-11 a d)}{2 a \sqrt {a-b x^4} (b c-a d)}}{6 a (b c-a d)}+\frac {b x}{6 a \left (a-b x^4\right )^{3/2} (b c-a d)}\) |
Input:
Int[1/((a - b*x^4)^(5/2)*(c - d*x^4)),x]
Output:
(b*x)/(6*a*(b*c - a*d)*(a - b*x^4)^(3/2)) + ((b*(5*b*c - 11*a*d)*x)/(2*a*( b*c - a*d)*Sqrt[a - b*x^4]) + ((a^(1/4)*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])/Sqrt[a - b*x^4] + 1 2*a^2*d^2*((a^(1/4)*Sqrt[1 - (b*x^4)/a]*EllipticPi[-((Sqrt[a]*Sqrt[d])/(Sq rt[b]*Sqrt[c])), ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(2*b^(1/4)*c*Sqrt[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] )))/(2*a*(b*c - a*d)))/(6*a*(b*c - a*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[(-b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Simp[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]) && 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[((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] + Simp[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] /; Fr eeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]
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 = 1.66 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.08
| method | result | size |
| default | \(-\frac {x \sqrt {-b \,x^{4}+a}}{6 a \left (a d -b c \right ) b \left (x^{4}-\frac {a}{b}\right )^{2}}-\frac {b x \left (11 a d -5 b c \right )}{12 a^{2} \left (a d -b c \right )^{2} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}-\frac {b \left (11 a d -5 b c \right ) \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{12 a^{2} \left (a d -b c \right )^{2} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {d \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} d -c \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}}{\left (a d -b c \right )^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{8}\) | \(361\) |
| elliptic | \(-\frac {x \sqrt {-b \,x^{4}+a}}{6 a \left (a d -b c \right ) b \left (x^{4}-\frac {a}{b}\right )^{2}}-\frac {b x \left (11 a d -5 b c \right )}{12 a^{2} \left (a d -b c \right )^{2} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}-\frac {b \left (11 a d -5 b c \right ) \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{12 a^{2} \left (a d -b c \right )^{2} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {d \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} d -c \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}}{\left (a d -b c \right )^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{8}\) | \(361\) |
Input:
int(1/(-b*x^4+a)^(5/2)/(-d*x^4+c),x,method=_RETURNVERBOSE)
Output:
-1/6*x/a/(a*d-b*c)/b*(-b*x^4+a)^(1/2)/(x^4-a/b)^2-1/12*b*x/a^2*(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-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)-1/8 *d*sum(1/(a*d-b*c)^2/_alpha^3*(-1/((a*d-b*c)/d)^(1/2)*arctanh(1/2*(-2*_alp ha^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. \[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(-b*x^4+a)^(5/2)/(-d*x^4+c),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )} \, dx=- \int \frac {1}{- a^{2} c \sqrt {a - b x^{4}} + a^{2} d x^{4} \sqrt {a - b x^{4}} + 2 a b c x^{4} \sqrt {a - b x^{4}} - 2 a b d x^{8} \sqrt {a - b x^{4}} - b^{2} c x^{8} \sqrt {a - b x^{4}} + b^{2} d x^{12} \sqrt {a - b x^{4}}}\, dx \] Input:
integrate(1/(-b*x**4+a)**(5/2)/(-d*x**4+c),x)
Output:
-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*s qrt(a - b*x**4) + b**2*d*x**12*sqrt(a - b*x**4)), x)
\[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )} \, dx=\int { -\frac {1}{{\left (-b x^{4} + a\right )}^{\frac {5}{2}} {\left (d x^{4} - c\right )}} \,d x } \] Input:
integrate(1/(-b*x^4+a)^(5/2)/(-d*x^4+c),x, algorithm="maxima")
Output:
-integrate(1/((-b*x^4 + a)^(5/2)*(d*x^4 - c)), x)
\[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )} \, dx=\int { -\frac {1}{{\left (-b x^{4} + a\right )}^{\frac {5}{2}} {\left (d x^{4} - c\right )}} \,d x } \] Input:
integrate(1/(-b*x^4+a)^(5/2)/(-d*x^4+c),x, algorithm="giac")
Output:
integrate(-1/((-b*x^4 + a)^(5/2)*(d*x^4 - c)), x)
Timed out. \[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )} \, dx=\int \frac {1}{{\left (a-b\,x^4\right )}^{5/2}\,\left (c-d\,x^4\right )} \,d x \] Input:
int(1/((a - b*x^4)^(5/2)*(c - d*x^4)),x)
Output:
int(1/((a - b*x^4)^(5/2)*(c - d*x^4)), x)
\[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )} \, dx=\int \frac {\sqrt {-b \,x^{4}+a}}{b^{3} d \,x^{16}-3 a \,b^{2} d \,x^{12}-b^{3} c \,x^{12}+3 a^{2} b d \,x^{8}+3 a \,b^{2} c \,x^{8}-a^{3} d \,x^{4}-3 a^{2} b c \,x^{4}+a^{3} c}d x \] Input:
int(1/(-b*x^4+a)^(5/2)/(-d*x^4+c),x)
Output:
int(sqrt(a - b*x**4)/(a**3*c - a**3*d*x**4 - 3*a**2*b*c*x**4 + 3*a**2*b*d* x**8 + 3*a*b**2*c*x**8 - 3*a*b**2*d*x**12 - b**3*c*x**12 + b**3*d*x**16),x )