Integrand size = 23, antiderivative size = 439 \[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )^2} \, dx=\frac {b (2 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a-b x^4\right )^{3/2}}+\frac {b \left (5 b^2 c^2-17 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \left (a-b x^4\right )^{3/2} \left (c-d x^4\right )}+\frac {b^{3/4} \left (5 b^2 c^2-17 a b c d-3 a^2 d^2\right ) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{12 a^{7/4} c (b c-a d)^3 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} d^2 (13 b c-3 a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)^3 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} d^2 (13 b c-3 a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)^3 \sqrt {a-b x^4}} \]
1/12*b*(3*a*d+2*b*c)*x/a/c/(-a*d+b*c)^2/(-b*x^4+a)^(3/2)-1/4*d*x/c/(-a*d+b *c)/(-b*x^4+a)^(3/2)/(-d*x^4+c)+1/12*b*(-3*a^2*d^2-17*a*b*c*d+5*b^2*c^2)*x /a^2/c/(-a*d+b*c)^3/(-b*x^4+a)^(1/2)+1/12*b^(3/4)*(-3*a^2*d^2-17*a*b*c*d+5 *b^2*c^2)*EllipticF(b^(1/4)*x/a^(1/4),I)*(1-b*x^4/a)^(1/2)/a^(7/4)/c/(-a*d +b*c)^3/(-b*x^4+a)^(1/2)+1/8*a^(1/4)*d^2*(-3*a*d+13*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/(-a*d+b*c)^3/(-b*x^4+a)^(1/2)+1/8*a^(1/4)*d^2*(-3*a*d+13*b*c)*Elliptic Pi(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/(-a*d+b*c)^3/(-b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.88 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )^2} \, dx=\frac {x \left (\frac {b d \left (-5 b^2 c^2+17 a b c d+3 a^2 d^2\right ) x^4 \sqrt {1-\frac {b x^4}{a}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )}{a^2 c^2}+5 \left (\frac {5 b^3 c}{a^2}-\frac {17 b^2 d}{a}-\frac {2 b^2 d}{a-b x^4}+\frac {2 b^3 c}{a^2-a b x^4}-\frac {3 a d^3}{c^2-c d x^4}+\frac {3 b d^3 x^4}{c^2-c d x^4}+\frac {5 \left (5 b^3 c^3-17 a b^2 c^2 d+36 a^2 b c d^2-9 a^3 d^3\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )}{a \left (c-d x^4\right ) \left (5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^4 \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}\right )\right )}{60 (b c-a d)^3 \sqrt {a-b x^4}} \]
(x*((b*d*(-5*b^2*c^2 + 17*a*b*c*d + 3*a^2*d^2)*x^4*Sqrt[1 - (b*x^4)/a]*App ellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])/(a^2*c^2) + 5*((5*b^3*c)/a^ 2 - (17*b^2*d)/a - (2*b^2*d)/(a - b*x^4) + (2*b^3*c)/(a^2 - a*b*x^4) - (3* a*d^3)/(c^2 - c*d*x^4) + (3*b*d^3*x^4)/(c^2 - c*d*x^4) + (5*(5*b^3*c^3 - 1 7*a*b^2*c^2*d + 36*a^2*b*c*d^2 - 9*a^3*d^3)*AppellF1[1/4, 1/2, 1, 5/4, (b* x^4)/a, (d*x^4)/c])/(a*(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*(b *c - a*d)^3*Sqrt[a - b*x^4])
Time = 0.87 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {931, 25, 1024, 27, 1024, 27, 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.
| \(\displaystyle \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 931 |
| \(\displaystyle -\frac {\int -\frac {9 b d x^4+4 b c-3 a d}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )}dx}{4 c (b c-a d)}-\frac {d x}{4 c \left (a-b x^4\right )^{3/2} \left (c-d x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {9 b d x^4+4 b c-3 a d}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )}dx}{4 c (b c-a d)}-\frac {d x}{4 c \left (a-b x^4\right )^{3/2} \left (c-d x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle \frac {\frac {\int \frac {2 \left (-5 b d (2 b c+3 a d) x^4+10 b^2 c^2+9 a^2 d^2-24 a b c d\right )}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )}dx}{6 a (b c-a d)}+\frac {b x (3 a d+2 b c)}{3 a \left (a-b x^4\right )^{3/2} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{4 c \left (a-b x^4\right )^{3/2} \left (c-d x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {-5 b d (2 b c+3 a d) x^4+10 b^2 c^2+9 a^2 d^2-24 a b c d}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )}dx}{3 a (b c-a d)}+\frac {b x (3 a d+2 b c)}{3 a \left (a-b x^4\right )^{3/2} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{4 c \left (a-b x^4\right )^{3/2} \left (c-d x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {2 \left (-b d \left (5 b^2 c^2-17 a b d c-3 a^2 d^2\right ) x^4+5 b^3 c^3-9 a^3 d^3+36 a^2 b c d^2-17 a b^2 c^2 d\right )}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx}{2 a (b c-a d)}+\frac {b x \left (-3 a^2 d^2-17 a b c d+5 b^2 c^2\right )}{a \sqrt {a-b x^4} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (3 a d+2 b c)}{3 a \left (a-b x^4\right )^{3/2} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{4 c \left (a-b x^4\right )^{3/2} \left (c-d x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {-b d \left (5 b^2 c^2-17 a b d c-3 a^2 d^2\right ) x^4+5 b^3 c^3-9 a^3 d^3+36 a^2 b c d^2-17 a b^2 c^2 d}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx}{a (b c-a d)}+\frac {b x \left (-3 a^2 d^2-17 a b c d+5 b^2 c^2\right )}{a \sqrt {a-b x^4} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (3 a d+2 b c)}{3 a \left (a-b x^4\right )^{3/2} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{4 c \left (a-b x^4\right )^{3/2} \left (c-d x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 1021 |
| \(\displaystyle \frac {\frac {\frac {b \left (-3 a^2 d^2-17 a b c d+5 b^2 c^2\right ) \int \frac {1}{\sqrt {a-b x^4}}dx+3 a^2 d^2 (13 b c-3 a d) \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx}{a (b c-a d)}+\frac {b x \left (-3 a^2 d^2-17 a b c d+5 b^2 c^2\right )}{a \sqrt {a-b x^4} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (3 a d+2 b c)}{3 a \left (a-b x^4\right )^{3/2} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{4 c \left (a-b x^4\right )^{3/2} \left (c-d x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\frac {\frac {\frac {b \sqrt {1-\frac {b x^4}{a}} \left (-3 a^2 d^2-17 a b c d+5 b^2 c^2\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {a-b x^4}}+3 a^2 d^2 (13 b c-3 a d) \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx}{a (b c-a d)}+\frac {b x \left (-3 a^2 d^2-17 a b c d+5 b^2 c^2\right )}{a \sqrt {a-b x^4} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (3 a d+2 b c)}{3 a \left (a-b x^4\right )^{3/2} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{4 c \left (a-b x^4\right )^{3/2} \left (c-d x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\frac {\frac {3 a^2 d^2 (13 b c-3 a d) \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx+\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (-3 a^2 d^2-17 a b c d+5 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {a-b x^4}}}{a (b c-a d)}+\frac {b x \left (-3 a^2 d^2-17 a b c d+5 b^2 c^2\right )}{a \sqrt {a-b x^4} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (3 a d+2 b c)}{3 a \left (a-b x^4\right )^{3/2} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{4 c \left (a-b x^4\right )^{3/2} \left (c-d x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle \frac {\frac {\frac {3 a^2 d^2 (13 b c-3 a d) \left (\frac {\int \frac {\sqrt {c}}{\left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {a-b x^4}}dx}{2 c}+\frac {\int \frac {\sqrt {c}}{\left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {a-b x^4}}dx}{2 c}\right )+\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (-3 a^2 d^2-17 a b c d+5 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {a-b x^4}}}{a (b c-a d)}+\frac {b x \left (-3 a^2 d^2-17 a b c d+5 b^2 c^2\right )}{a \sqrt {a-b x^4} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (3 a d+2 b c)}{3 a \left (a-b x^4\right )^{3/2} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{4 c \left (a-b x^4\right )^{3/2} \left (c-d x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 a^2 d^2 (13 b c-3 a d) \left (\frac {\int \frac {1}{\left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {a-b x^4}}dx}{2 \sqrt {c}}+\frac {\int \frac {1}{\left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {a-b x^4}}dx}{2 \sqrt {c}}\right )+\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (-3 a^2 d^2-17 a b c d+5 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {a-b x^4}}}{a (b c-a d)}+\frac {b x \left (-3 a^2 d^2-17 a b c d+5 b^2 c^2\right )}{a \sqrt {a-b x^4} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (3 a d+2 b c)}{3 a \left (a-b x^4\right )^{3/2} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{4 c \left (a-b x^4\right )^{3/2} \left (c-d x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle \frac {\frac {\frac {3 a^2 d^2 (13 b c-3 a d) \left (\frac {\sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {1-\frac {b x^4}{a}}}dx}{2 \sqrt {c} \sqrt {a-b x^4}}+\frac {\sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {1-\frac {b x^4}{a}}}dx}{2 \sqrt {c} \sqrt {a-b x^4}}\right )+\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (-3 a^2 d^2-17 a b c d+5 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {a-b x^4}}}{a (b c-a d)}+\frac {b x \left (-3 a^2 d^2-17 a b c d+5 b^2 c^2\right )}{a \sqrt {a-b x^4} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (3 a d+2 b c)}{3 a \left (a-b x^4\right )^{3/2} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{4 c \left (a-b x^4\right )^{3/2} \left (c-d x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle \frac {\frac {\frac {\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (-3 a^2 d^2-17 a b c d+5 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {a-b x^4}}+3 a^2 d^2 (13 b c-3 a d) \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4}}\right )}{a (b c-a d)}+\frac {b x \left (-3 a^2 d^2-17 a b c d+5 b^2 c^2\right )}{a \sqrt {a-b x^4} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (3 a d+2 b c)}{3 a \left (a-b x^4\right )^{3/2} (b c-a d)}}{4 c (b c-a d)}-\frac {d x}{4 c \left (a-b x^4\right )^{3/2} \left (c-d x^4\right ) (b c-a d)}\) |
-1/4*(d*x)/(c*(b*c - a*d)*(a - b*x^4)^(3/2)*(c - d*x^4)) + ((b*(2*b*c + 3* a*d)*x)/(3*a*(b*c - a*d)*(a - b*x^4)^(3/2)) + ((b*(5*b^2*c^2 - 17*a*b*c*d - 3*a^2*d^2)*x)/(a*(b*c - a*d)*Sqrt[a - b*x^4]) + ((a^(1/4)*b^(3/4)*(5*b^2 *c^2 - 17*a*b*c*d - 3*a^2*d^2)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/ 4)*x)/a^(1/4)], -1])/Sqrt[a - b*x^4] + 3*a^2*d^2*(13*b*c - 3*a*d)*((a^(1/4 )*Sqrt[1 - (b*x^4)/a]*EllipticPi[-((Sqrt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c])), A rcSin[(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])))/(a*(b*c - a*d) ))/(3*a*(b*c - a*d)))/(4*c*(b*c - a*d))
3.2.90.3.1 Defintions of rubi rules used
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 = 4.43 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.10
| method | result | size |
| default | \(-\frac {b \,d^{3} x \sqrt {-b \,x^{4}+a}}{4 \left (a d -b c \right ) c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (b d \,x^{4}-b c \right )}+\frac {x \sqrt {-b \,x^{4}+a}}{6 \left (a d -b c \right )^{2} a \left (x^{4}-\frac {a}{b}\right )^{2}}+\frac {b^{2} x \left (17 a d -5 b c \right )}{12 a^{2} \left (a d -b c \right )^{3} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {\left (\frac {b \,d^{2}}{4 \left (a d -b c \right ) c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {b^{2} \left (17 a d -5 b c \right )}{12 a^{2} \left (a d -b c \right )^{3}}\right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {d \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (3 a d -13 b c \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\left (a d -b c \right )^{3} \underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{32 c}\) | \(484\) |
| elliptic | \(-\frac {b \,d^{3} x \sqrt {-b \,x^{4}+a}}{4 \left (a d -b c \right ) c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (b d \,x^{4}-b c \right )}+\frac {x \sqrt {-b \,x^{4}+a}}{6 \left (a d -b c \right )^{2} a \left (x^{4}-\frac {a}{b}\right )^{2}}+\frac {b^{2} x \left (17 a d -5 b c \right )}{12 a^{2} \left (a d -b c \right )^{3} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {\left (\frac {b \,d^{2}}{4 \left (a d -b c \right ) c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {b^{2} \left (17 a d -5 b c \right )}{12 a^{2} \left (a d -b c \right )^{3}}\right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {d \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (3 a d -13 b c \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\left (a d -b c \right )^{3} \underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{32 c}\) | \(484\) |
-1/4*b*d^3/(a*d-b*c)/c/(a^2*d^2-2*a*b*c*d+b^2*c^2)*x*(-b*x^4+a)^(1/2)/(b*d *x^4-b*c)+1/6/(a*d-b*c)^2/a*x*(-b*x^4+a)^(1/2)/(x^4-a/b)^2+1/12*b^2*x/a^2* (17*a*d-5*b*c)/(a*d-b*c)^3/(-(x^4-a/b)*b)^(1/2)+(1/4*b*d^2/(a*d-b*c)/c/(a^ 2*d^2-2*a*b*c*d+b^2*c^2)+1/12*b^2/a^2*(17*a*d-5*b*c)/(a*d-b*c)^3)/(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)-1/32*d/c*sum ((3*a*d-13*b*c)/(a*d-b*c)^3/_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))
Timed out. \[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )^2} \, dx=\int \frac {1}{\left (a - b x^{4}\right )^{\frac {5}{2}} \left (- c + d x^{4}\right )^{2}}\, dx \]
\[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )^2} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {5}{2}} {\left (d x^{4} - c\right )}^{2}} \,d x } \]
\[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )^2} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {5}{2}} {\left (d x^{4} - c\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )^2} \, dx=\int \frac {1}{{\left (a-b\,x^4\right )}^{5/2}\,{\left (c-d\,x^4\right )}^2} \,d x \]