Integrand size = 23, antiderivative size = 426 \[ \int \frac {\left (a-b x^4\right )^{7/2}}{\left (c-d x^4\right )^2} \, dx=-\frac {b \left (77 b^2 c^2-122 a b c d+21 a^2 d^2\right ) x \sqrt {a-b x^4}}{84 c d^3}+\frac {b (11 b c-7 a d) x \left (a-b x^4\right )^{3/2}}{28 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{5/2}}{4 c d \left (c-d x^4\right )}+\frac {\sqrt [4]{a} b^{3/4} \left (231 b^3 c^3-553 a b^2 c^2 d+349 a^2 b c d^2+21 a^3 d^3\right ) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{84 c d^4 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} (b c-a d)^3 (11 b c+3 a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d^4 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} (b c-a d)^3 (11 b c+3 a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d^4 \sqrt {a-b x^4}} \]
1/28*b*(-7*a*d+11*b*c)*x*(-b*x^4+a)^(3/2)/c/d^2-1/4*(-a*d+b*c)*x*(-b*x^4+a )^(5/2)/c/d/(-d*x^4+c)-1/84*b*(21*a^2*d^2-122*a*b*c*d+77*b^2*c^2)*x*(-b*x^ 4+a)^(1/2)/c/d^3+1/84*a^(1/4)*b^(3/4)*(21*a^3*d^3+349*a^2*b*c*d^2-553*a*b^ 2*c^2*d+231*b^3*c^3)*EllipticF(b^(1/4)*x/a^(1/4),I)*(1-b*x^4/a)^(1/2)/c/d^ 4/(-b*x^4+a)^(1/2)-1/8*a^(1/4)*(-a*d+b*c)^3*(3*a*d+11*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^4/(-b*x^4+a)^(1/2)-1/8*a^(1/4)*(-a*d+b*c)^3*(3*a*d+11*b*c)*Ellipti cPi(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^4/(-b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.82 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a-b x^4\right )^{7/2}}{\left (c-d x^4\right )^2} \, dx=-\frac {b \left (231 b^3 c^3-553 a b^2 c^2 d+349 a^2 b c d^2+21 a^3 d^3\right ) x^5 \sqrt {1-\frac {b x^4}{a}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+\frac {5 c \left (5 a c x \left (-84 a^4 d^3+29 a^2 b^2 c d^2 x^4+21 a^3 b d^3 x^4+a b^3 c d x^4 \left (111 c-104 d x^4\right )+b^4 c x^4 \left (-77 c^2+44 c d x^4+12 d^2 x^8\right )\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^5 \left (-a+b x^4\right ) \left (-63 a^2 b c d^2+21 a^3 d^3+a b^2 c d \left (155 c-92 d x^4\right )+b^3 c \left (-77 c^2+44 c d x^4+12 d^2 x^8\right )\right ) \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}{\left (c-d x^4\right ) \left (5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^4 \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}}{420 c^2 d^3 \sqrt {a-b x^4}} \]
-1/420*(b*(231*b^3*c^3 - 553*a*b^2*c^2*d + 349*a^2*b*c*d^2 + 21*a^3*d^3)*x ^5*Sqrt[1 - (b*x^4)/a]*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c] + (5*c*(5*a*c*x*(-84*a^4*d^3 + 29*a^2*b^2*c*d^2*x^4 + 21*a^3*b*d^3*x^4 + a*b ^3*c*d*x^4*(111*c - 104*d*x^4) + b^4*c*x^4*(-77*c^2 + 44*c*d*x^4 + 12*d^2* x^8))*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*x^5*(-a + b*x^4 )*(-63*a^2*b*c*d^2 + 21*a^3*d^3 + a*b^2*c*d*(155*c - 92*d*x^4) + b^3*c*(-7 7*c^2 + 44*c*d*x^4 + 12*d^2*x^8))*(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]))))/(c^2*d^3*Sqrt[a - b*x^4])
Time = 0.82 (sec) , antiderivative size = 422, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {930, 25, 1025, 1025, 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.
| \(\displaystyle \int \frac {\left (a-b x^4\right )^{7/2}}{\left (c-d x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 930 |
| \(\displaystyle -\frac {\int -\frac {\left (a-b x^4\right )^{3/2} \left (a (b c+3 a d)-b (11 b c-7 a d) x^4\right )}{c-d x^4}dx}{4 c d}-\frac {x \left (a-b x^4\right )^{5/2} (b c-a d)}{4 c d \left (c-d x^4\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\left (a-b x^4\right )^{3/2} \left (a (b c+3 a d)-b (11 b c-7 a d) x^4\right )}{c-d x^4}dx}{4 c d}-\frac {x \left (a-b x^4\right )^{5/2} (b c-a d)}{4 c d \left (c-d x^4\right )}\) |
| \(\Big \downarrow \) 1025 |
| \(\displaystyle \frac {\frac {b x \left (a-b x^4\right )^{3/2} (11 b c-7 a d)}{7 d}-\frac {\int \frac {\sqrt {a-b x^4} \left (a \left (11 b^2 c^2-14 a b d c-21 a^2 d^2\right )-b \left (77 b^2 c^2-122 a b d c+21 a^2 d^2\right ) x^4\right )}{c-d x^4}dx}{7 d}}{4 c d}-\frac {x \left (a-b x^4\right )^{5/2} (b c-a d)}{4 c d \left (c-d x^4\right )}\) |
| \(\Big \downarrow \) 1025 |
| \(\displaystyle \frac {\frac {b x \left (a-b x^4\right )^{3/2} (11 b c-7 a d)}{7 d}-\frac {\frac {b x \sqrt {a-b x^4} \left (21 a^2 d^2-122 a b c d+77 b^2 c^2\right )}{3 d}-\frac {\int \frac {a \left (77 b^3 c^3-155 a b^2 d c^2+63 a^2 b d^2 c+63 a^3 d^3\right )-b \left (231 b^3 c^3-553 a b^2 d c^2+349 a^2 b d^2 c+21 a^3 d^3\right ) x^4}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx}{3 d}}{7 d}}{4 c d}-\frac {x \left (a-b x^4\right )^{5/2} (b c-a d)}{4 c d \left (c-d x^4\right )}\) |
| \(\Big \downarrow \) 1021 |
| \(\displaystyle \frac {\frac {b x \left (a-b x^4\right )^{3/2} (11 b c-7 a d)}{7 d}-\frac {\frac {b x \sqrt {a-b x^4} \left (21 a^2 d^2-122 a b c d+77 b^2 c^2\right )}{3 d}-\frac {\frac {b \left (21 a^3 d^3+349 a^2 b c d^2-553 a b^2 c^2 d+231 b^3 c^3\right ) \int \frac {1}{\sqrt {a-b x^4}}dx}{d}-\frac {21 (b c-a d)^3 (3 a d+11 b c) \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx}{d}}{3 d}}{7 d}}{4 c d}-\frac {x \left (a-b x^4\right )^{5/2} (b c-a d)}{4 c d \left (c-d x^4\right )}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\frac {b x \left (a-b x^4\right )^{3/2} (11 b c-7 a d)}{7 d}-\frac {\frac {b x \sqrt {a-b x^4} \left (21 a^2 d^2-122 a b c d+77 b^2 c^2\right )}{3 d}-\frac {\frac {b \sqrt {1-\frac {b x^4}{a}} \left (21 a^3 d^3+349 a^2 b c d^2-553 a b^2 c^2 d+231 b^3 c^3\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{d \sqrt {a-b x^4}}-\frac {21 (b c-a d)^3 (3 a d+11 b c) \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx}{d}}{3 d}}{7 d}}{4 c d}-\frac {x \left (a-b x^4\right )^{5/2} (b c-a d)}{4 c d \left (c-d x^4\right )}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\frac {b x \left (a-b x^4\right )^{3/2} (11 b c-7 a d)}{7 d}-\frac {\frac {b x \sqrt {a-b x^4} \left (21 a^2 d^2-122 a b c d+77 b^2 c^2\right )}{3 d}-\frac {\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (21 a^3 d^3+349 a^2 b c d^2-553 a b^2 c^2 d+231 b^3 c^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{d \sqrt {a-b x^4}}-\frac {21 (b c-a d)^3 (3 a d+11 b c) \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx}{d}}{3 d}}{7 d}}{4 c d}-\frac {x \left (a-b x^4\right )^{5/2} (b c-a d)}{4 c d \left (c-d x^4\right )}\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle \frac {\frac {b x \left (a-b x^4\right )^{3/2} (11 b c-7 a d)}{7 d}-\frac {\frac {b x \sqrt {a-b x^4} \left (21 a^2 d^2-122 a b c d+77 b^2 c^2\right )}{3 d}-\frac {\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (21 a^3 d^3+349 a^2 b c d^2-553 a b^2 c^2 d+231 b^3 c^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{d \sqrt {a-b x^4}}-\frac {21 (b c-a d)^3 (3 a d+11 b c) \left (\frac {\int \frac {\sqrt {c}}{\left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {a-b x^4}}dx}{2 c}+\frac {\int \frac {\sqrt {c}}{\left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {a-b x^4}}dx}{2 c}\right )}{d}}{3 d}}{7 d}}{4 c d}-\frac {x \left (a-b x^4\right )^{5/2} (b c-a d)}{4 c d \left (c-d x^4\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {b x \left (a-b x^4\right )^{3/2} (11 b c-7 a d)}{7 d}-\frac {\frac {b x \sqrt {a-b x^4} \left (21 a^2 d^2-122 a b c d+77 b^2 c^2\right )}{3 d}-\frac {\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (21 a^3 d^3+349 a^2 b c d^2-553 a b^2 c^2 d+231 b^3 c^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{d \sqrt {a-b x^4}}-\frac {21 (b c-a d)^3 (3 a d+11 b c) \left (\frac {\int \frac {1}{\left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {a-b x^4}}dx}{2 \sqrt {c}}+\frac {\int \frac {1}{\left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {a-b x^4}}dx}{2 \sqrt {c}}\right )}{d}}{3 d}}{7 d}}{4 c d}-\frac {x \left (a-b x^4\right )^{5/2} (b c-a d)}{4 c d \left (c-d x^4\right )}\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle \frac {\frac {b x \left (a-b x^4\right )^{3/2} (11 b c-7 a d)}{7 d}-\frac {\frac {b x \sqrt {a-b x^4} \left (21 a^2 d^2-122 a b c d+77 b^2 c^2\right )}{3 d}-\frac {\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (21 a^3 d^3+349 a^2 b c d^2-553 a b^2 c^2 d+231 b^3 c^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{d \sqrt {a-b x^4}}-\frac {21 (b c-a d)^3 (3 a d+11 b c) \left (\frac {\sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {1-\frac {b x^4}{a}}}dx}{2 \sqrt {c} \sqrt {a-b x^4}}+\frac {\sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {1-\frac {b x^4}{a}}}dx}{2 \sqrt {c} \sqrt {a-b x^4}}\right )}{d}}{3 d}}{7 d}}{4 c d}-\frac {x \left (a-b x^4\right )^{5/2} (b c-a d)}{4 c d \left (c-d x^4\right )}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle \frac {\frac {b x \left (a-b x^4\right )^{3/2} (11 b c-7 a d)}{7 d}-\frac {\frac {b x \sqrt {a-b x^4} \left (21 a^2 d^2-122 a b c d+77 b^2 c^2\right )}{3 d}-\frac {\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (21 a^3 d^3+349 a^2 b c d^2-553 a b^2 c^2 d+231 b^3 c^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{d \sqrt {a-b x^4}}-\frac {21 (b c-a d)^3 (3 a d+11 b c) \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4}}\right )}{d}}{3 d}}{7 d}}{4 c d}-\frac {x \left (a-b x^4\right )^{5/2} (b c-a d)}{4 c d \left (c-d x^4\right )}\) |
-1/4*((b*c - a*d)*x*(a - b*x^4)^(5/2))/(c*d*(c - d*x^4)) + ((b*(11*b*c - 7 *a*d)*x*(a - b*x^4)^(3/2))/(7*d) - ((b*(77*b^2*c^2 - 122*a*b*c*d + 21*a^2* d^2)*x*Sqrt[a - b*x^4])/(3*d) - ((a^(1/4)*b^(3/4)*(231*b^3*c^3 - 553*a*b^2 *c^2*d + 349*a^2*b*c*d^2 + 21*a^3*d^3)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSi n[(b^(1/4)*x)/a^(1/4)], -1])/(d*Sqrt[a - b*x^4]) - (21*(b*c - a*d)^3*(11*b *c + 3*a*d)*((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^(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))/(7*d))/(4*c*d)
3.2.84.3.1 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[(a*d - c*b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*b*n*(p + 1))), x] - Simp[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]
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[f*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/( b*(n*(p + q + 1) + 1))), x] + Simp[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]
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 = 7.94 (sec) , antiderivative size = 539, normalized size of antiderivative = 1.27
| method | result | size |
| default | \(\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x \sqrt {-b \,x^{4}+a}}{4 c \,d^{3} \left (-d \,x^{4}+c \right )}-\frac {b^{3} x^{5} \sqrt {-b \,x^{4}+a}}{7 d^{2}}-\frac {\left (-\frac {2 b^{3} \left (2 a d -b c \right )}{d^{3}}+\frac {5 b^{3} a}{7 d^{2}}\right ) x \sqrt {-b \,x^{4}+a}}{3 b}+\frac {\left (\frac {b^{2} \left (6 a^{2} d^{2}-8 a b c d +3 b^{2} c^{2}\right )}{d^{4}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b}{4 d^{4} c}+\frac {\left (-\frac {2 b^{3} \left (2 a d -b c \right )}{d^{3}}+\frac {5 b^{3} a}{7 d^{2}}\right ) a}{3 b}\right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (3 a^{4} d^{4}+2 a^{3} b c \,d^{3}-24 a^{2} b^{2} c^{2} d^{2}+30 a \,b^{3} c^{3} d -11 b^{4} c^{4}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{32 d^{5} c}\) | \(539\) |
| elliptic | \(\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x \sqrt {-b \,x^{4}+a}}{4 c \,d^{3} \left (-d \,x^{4}+c \right )}-\frac {b^{3} x^{5} \sqrt {-b \,x^{4}+a}}{7 d^{2}}-\frac {\left (-\frac {2 b^{3} \left (2 a d -b c \right )}{d^{3}}+\frac {5 b^{3} a}{7 d^{2}}\right ) x \sqrt {-b \,x^{4}+a}}{3 b}+\frac {\left (\frac {b^{2} \left (6 a^{2} d^{2}-8 a b c d +3 b^{2} c^{2}\right )}{d^{4}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b}{4 d^{4} c}+\frac {\left (-\frac {2 b^{3} \left (2 a d -b c \right )}{d^{3}}+\frac {5 b^{3} a}{7 d^{2}}\right ) a}{3 b}\right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (3 a^{4} d^{4}+2 a^{3} b c \,d^{3}-24 a^{2} b^{2} c^{2} d^{2}+30 a \,b^{3} c^{3} d -11 b^{4} c^{4}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{32 d^{5} c}\) | \(539\) |
| risch | \(\frac {b^{2} x \left (-3 b d \,x^{4}+23 a d -14 b c \right ) \sqrt {-b \,x^{4}+a}}{21 d^{3}}+\frac {\frac {b^{2} \left (103 a^{2} d^{2}-154 a b c d +63 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 {21 b \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 (\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 )}{2 d^{2}}+\frac {\left (21 a^{4} d^{4}-84 a^{3} b c \,d^{3}+126 a^{2} b^{2} c^{2} d^{2}-84 a \,b^{3} c^{3} d +21 b^{4} c^{4}\right ) \left (-\frac {d x \sqrt {-b \,x^{4}+a}}{4 c \left (a d -b c \right ) \left (d \,x^{4}-c \right )}+\frac {b \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 )}{4 c \left (a d -b c \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 (3 a d -5 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 ) \underline {\hspace {1.25 ex}}\alpha ^{3}}}{32 c d}\right )}{d}}{21 d^{3}}\) | \(730\) |
1/4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/c/d^3*x*(-b*x^4+a)^(1/2) /(-d*x^4+c)-1/7*b^3/d^2*x^5*(-b*x^4+a)^(1/2)-1/3*(-2*b^3/d^3*(2*a*d-b*c)+5 /7*b^3/d^2*a)/b*x*(-b*x^4+a)^(1/2)+(b^2*(6*a^2*d^2-8*a*b*c*d+3*b^2*c^2)/d^ 4+1/4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/d^4*b/c+1/3*(-2*b^3/d^ 3*(2*a*d-b*c)+5/7*b^3/d^2*a)/b*a)/(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^5/c*sum((3*a^4*d^4+2*a^3*b*c*d^3-24*a ^2*b^2*c^2*d^2+30*a*b^3*c^3*d-11*b^4*c^4)/_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. \[ \int \frac {\left (a-b x^4\right )^{7/2}}{\left (c-d x^4\right )^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (a-b x^4\right )^{7/2}}{\left (c-d x^4\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a-b x^4\right )^{7/2}}{\left (c-d x^4\right )^2} \, dx=\int { \frac {{\left (-b x^{4} + a\right )}^{\frac {7}{2}}}{{\left (d x^{4} - c\right )}^{2}} \,d x } \]
\[ \int \frac {\left (a-b x^4\right )^{7/2}}{\left (c-d x^4\right )^2} \, dx=\int { \frac {{\left (-b x^{4} + a\right )}^{\frac {7}{2}}}{{\left (d x^{4} - c\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a-b x^4\right )^{7/2}}{\left (c-d x^4\right )^2} \, dx=\int \frac {{\left (a-b\,x^4\right )}^{7/2}}{{\left (c-d\,x^4\right )}^2} \,d x \]