Integrand size = 23, antiderivative size = 365 \[ \int \frac {\left (a-b x^4\right )^{5/2}}{\left (c-d x^4\right )^2} \, dx=\frac {b (7 b c-3 a d) x \sqrt {a-b x^4}}{12 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{3/2}}{4 c d \left (c-d x^4\right )}-\frac {\sqrt [4]{a} b^{3/4} \left (21 b^2 c^2-26 a b c d-3 a^2 d^2\right ) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{12 c d^3 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} (b c-a d)^2 (7 b c+3 a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d^3 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} (b c-a d)^2 (7 b c+3 a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d^3 \sqrt {a-b x^4}} \] Output:
1/12*b*(-3*a*d+7*b*c)*x*(-b*x^4+a)^(1/2)/c/d^2-1/4*(-a*d+b*c)*x*(-b*x^4+a) ^(3/2)/c/d/(-d*x^4+c)-1/12*a^(1/4)*b^(3/4)*(-3*a^2*d^2-26*a*b*c*d+21*b^2*c ^2)*(1-b*x^4/a)^(1/2)*EllipticF(b^(1/4)*x/a^(1/4),I)/c/d^3/(-b*x^4+a)^(1/2 )+1/8*a^(1/4)*(-a*d+b*c)^2*(3*a*d+7*b*c)*(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^2/d^3/(-b*x^4+ a)^(1/2)+1/8*a^(1/4)*(-a*d+b*c)^2*(3*a*d+7*b*c)*(1-b*x^4/a)^(1/2)*Elliptic Pi(b^(1/4)*x/a^(1/4),a^(1/2)*d^(1/2)/b^(1/2)/c^(1/2),I)/b^(1/4)/c^2/d^3/(- b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.64 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a-b x^4\right )^{5/2}}{\left (c-d x^4\right )^2} \, dx=-\frac {b \left (-21 b^2 c^2+26 a b c d+3 a^2 d^2\right ) x^5 \sqrt {1-\frac {b x^4}{a}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+\frac {5 c \left (5 a c x \left (12 a^3 d^2+2 a b^2 c d x^4-3 a^2 b d^2 x^4+b^3 c x^4 \left (-7 c+4 d x^4\right )\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^5 \left (a-b x^4\right ) \left (-6 a b c d+3 a^2 d^2+b^2 c \left (7 c-4 d x^4\right )\right ) \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}{\left (-c+d x^4\right ) \left (5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^4 \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}}{60 c^2 d^2 \sqrt {a-b x^4}} \] Input:
Integrate[(a - b*x^4)^(5/2)/(c - d*x^4)^2,x]
Output:
-1/60*(b*(-21*b^2*c^2 + 26*a*b*c*d + 3*a^2*d^2)*x^5*Sqrt[1 - (b*x^4)/a]*Ap pellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c] + (5*c*(5*a*c*x*(12*a^3*d^2 + 2*a*b^2*c*d*x^4 - 3*a^2*b*d^2*x^4 + b^3*c*x^4*(-7*c + 4*d*x^4))*AppellF 1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c] + 2*x^5*(a - b*x^4)*(-6*a*b*c*d + 3*a^2*d^2 + b^2*c*(7*c - 4*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*Appel lF1[5/4, 3/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c]))))/(c^2*d^2*Sqrt[a - b*x^4])
Time = 1.10 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {930, 25, 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.
| \(\displaystyle \int \frac {\left (a-b x^4\right )^{5/2}}{\left (c-d x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 930 |
| \(\displaystyle -\frac {\int -\frac {\sqrt {a-b x^4} \left (a (b c+3 a d)-b (7 b c-3 a d) x^4\right )}{c-d x^4}dx}{4 c d}-\frac {x \left (a-b x^4\right )^{3/2} (b c-a d)}{4 c d \left (c-d x^4\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\sqrt {a-b x^4} \left (a (b c+3 a d)-b (7 b c-3 a d) x^4\right )}{c-d x^4}dx}{4 c d}-\frac {x \left (a-b x^4\right )^{3/2} (b c-a d)}{4 c d \left (c-d x^4\right )}\) |
| \(\Big \downarrow \) 1025 |
| \(\displaystyle \frac {\frac {b x \sqrt {a-b x^4} (7 b c-3 a d)}{3 d}-\frac {\int \frac {a \left (7 b^2 c^2-6 a b d c-9 a^2 d^2\right )-b \left (21 b^2 c^2-26 a b d c-3 a^2 d^2\right ) x^4}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx}{3 d}}{4 c d}-\frac {x \left (a-b x^4\right )^{3/2} (b c-a d)}{4 c d \left (c-d x^4\right )}\) |
| \(\Big \downarrow \) 1021 |
| \(\displaystyle \frac {\frac {b x \sqrt {a-b x^4} (7 b c-3 a d)}{3 d}-\frac {\frac {b \left (-3 a^2 d^2-26 a b c d+21 b^2 c^2\right ) \int \frac {1}{\sqrt {a-b x^4}}dx}{d}-\frac {3 (b c-a d)^2 (3 a d+7 b c) \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx}{d}}{3 d}}{4 c d}-\frac {x \left (a-b x^4\right )^{3/2} (b c-a d)}{4 c d \left (c-d x^4\right )}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\frac {b x \sqrt {a-b x^4} (7 b c-3 a d)}{3 d}-\frac {\frac {b \sqrt {1-\frac {b x^4}{a}} \left (-3 a^2 d^2-26 a b c d+21 b^2 c^2\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{d \sqrt {a-b x^4}}-\frac {3 (b c-a d)^2 (3 a d+7 b c) \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx}{d}}{3 d}}{4 c d}-\frac {x \left (a-b x^4\right )^{3/2} (b c-a d)}{4 c d \left (c-d x^4\right )}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\frac {b x \sqrt {a-b x^4} (7 b c-3 a d)}{3 d}-\frac {\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (-3 a^2 d^2-26 a b c d+21 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{d \sqrt {a-b x^4}}-\frac {3 (b c-a d)^2 (3 a d+7 b c) \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )}dx}{d}}{3 d}}{4 c d}-\frac {x \left (a-b x^4\right )^{3/2} (b c-a d)}{4 c d \left (c-d x^4\right )}\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle \frac {\frac {b x \sqrt {a-b x^4} (7 b c-3 a d)}{3 d}-\frac {\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (-3 a^2 d^2-26 a b c d+21 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{d \sqrt {a-b x^4}}-\frac {3 (b c-a d)^2 (3 a d+7 b c) \left (\frac {\int \frac {\sqrt {c}}{\left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {a-b x^4}}dx}{2 c}+\frac {\int \frac {\sqrt {c}}{\left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {a-b x^4}}dx}{2 c}\right )}{d}}{3 d}}{4 c d}-\frac {x \left (a-b x^4\right )^{3/2} (b c-a d)}{4 c d \left (c-d x^4\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {b x \sqrt {a-b x^4} (7 b c-3 a d)}{3 d}-\frac {\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (-3 a^2 d^2-26 a b c d+21 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{d \sqrt {a-b x^4}}-\frac {3 (b c-a d)^2 (3 a d+7 b c) \left (\frac {\int \frac {1}{\left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {a-b x^4}}dx}{2 \sqrt {c}}+\frac {\int \frac {1}{\left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {a-b x^4}}dx}{2 \sqrt {c}}\right )}{d}}{3 d}}{4 c d}-\frac {x \left (a-b x^4\right )^{3/2} (b c-a d)}{4 c d \left (c-d x^4\right )}\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle \frac {\frac {b x \sqrt {a-b x^4} (7 b c-3 a d)}{3 d}-\frac {\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (-3 a^2 d^2-26 a b c d+21 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{d \sqrt {a-b x^4}}-\frac {3 (b c-a d)^2 (3 a d+7 b c) \left (\frac {\sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\left (\sqrt {c}-\sqrt {d} x^2\right ) \sqrt {1-\frac {b x^4}{a}}}dx}{2 \sqrt {c} \sqrt {a-b x^4}}+\frac {\sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {1-\frac {b x^4}{a}}}dx}{2 \sqrt {c} \sqrt {a-b x^4}}\right )}{d}}{3 d}}{4 c d}-\frac {x \left (a-b x^4\right )^{3/2} (b c-a d)}{4 c d \left (c-d x^4\right )}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle \frac {\frac {b x \sqrt {a-b x^4} (7 b c-3 a d)}{3 d}-\frac {\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (-3 a^2 d^2-26 a b c d+21 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{d \sqrt {a-b x^4}}-\frac {3 (b c-a d)^2 (3 a d+7 b c) \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4}}\right )}{d}}{3 d}}{4 c d}-\frac {x \left (a-b x^4\right )^{3/2} (b c-a d)}{4 c d \left (c-d x^4\right )}\) |
Input:
Int[(a - b*x^4)^(5/2)/(c - d*x^4)^2,x]
Output:
-1/4*((b*c - a*d)*x*(a - b*x^4)^(3/2))/(c*d*(c - d*x^4)) + ((b*(7*b*c - 3* a*d)*x*Sqrt[a - b*x^4])/(3*d) - ((a^(1/4)*b^(3/4)*(21*b^2*c^2 - 26*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])/(d*Sqrt[a - b*x^4]) - (3*(b*c - a*d)^2*(7*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))/(4*c*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[(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 = 7.70 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.13
| method | result | size |
| default | \(\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x \sqrt {-b \,x^{4}+a}}{4 c \,d^{2} \left (-d \,x^{4}+c \right )}+\frac {b^{2} x \sqrt {-b \,x^{4}+a}}{3 d^{2}}+\frac {\left (\frac {b^{2} \left (3 a d -2 b c \right )}{d^{3}}+\frac {b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{4 d^{3} c}-\frac {b^{2} a}{3 d^{2}}\right ) \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} d -c \right )}{\sum }\frac {\left (3 a^{3} d^{3}+a^{2} b c \,d^{2}-11 a \,b^{2} c^{2} d +7 b^{3} c^{3}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{32 c \,d^{4}}\) | \(411\) |
| elliptic | \(\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x \sqrt {-b \,x^{4}+a}}{4 c \,d^{2} \left (-d \,x^{4}+c \right )}+\frac {b^{2} x \sqrt {-b \,x^{4}+a}}{3 d^{2}}+\frac {\left (\frac {b^{2} \left (3 a d -2 b c \right )}{d^{3}}+\frac {b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{4 d^{3} c}-\frac {b^{2} a}{3 d^{2}}\right ) \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} d -c \right )}{\sum }\frac {\left (3 a^{3} d^{3}+a^{2} b c \,d^{2}-11 a \,b^{2} c^{2} d +7 b^{3} c^{3}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{32 c \,d^{4}}\) | \(411\) |
| risch | \(\frac {b^{2} x \sqrt {-b \,x^{4}+a}}{3 d^{2}}+\frac {\frac {3 \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 {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 {\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 )}{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 (\textit {\_Z}^{4} d -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 {\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 )}{\left (a d -b c \right ) \underline {\hspace {1.25 ex}}\alpha ^{3}}}{32 c d}\right )}{d}+\frac {2 b^{2} \left (4 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 {9 b \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^{2}}\) | \(672\) |
Input:
int((-b*x^4+a)^(5/2)/(-d*x^4+c)^2,x,method=_RETURNVERBOSE)
Output:
1/4/c/d^2*(a^2*d^2-2*a*b*c*d+b^2*c^2)*x*(-b*x^4+a)^(1/2)/(-d*x^4+c)+1/3*b^ 2/d^2*x*(-b*x^4+a)^(1/2)+(b^2*(3*a*d-2*b*c)/d^3+1/4*b/d^3*(a^2*d^2-2*a*b*c *d+b^2*c^2)/c-1/3*b^2/d^2*a)/(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)-1/32/c/d^4*sum((3*a^3*d^3+a^2*b*c*d^2-11*a*b^2*c^ 2*d+7*b^3*c^3)/_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. \[ \int \frac {\left (a-b x^4\right )^{5/2}}{\left (c-d x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate((-b*x^4+a)^(5/2)/(-d*x^4+c)^2,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (a-b x^4\right )^{5/2}}{\left (c-d x^4\right )^2} \, dx=\int \frac {\left (a - b x^{4}\right )^{\frac {5}{2}}}{\left (- c + d x^{4}\right )^{2}}\, dx \] Input:
integrate((-b*x**4+a)**(5/2)/(-d*x**4+c)**2,x)
Output:
Integral((a - b*x**4)**(5/2)/(-c + d*x**4)**2, x)
\[ \int \frac {\left (a-b x^4\right )^{5/2}}{\left (c-d x^4\right )^2} \, dx=\int { \frac {{\left (-b x^{4} + a\right )}^{\frac {5}{2}}}{{\left (d x^{4} - c\right )}^{2}} \,d x } \] Input:
integrate((-b*x^4+a)^(5/2)/(-d*x^4+c)^2,x, algorithm="maxima")
Output:
integrate((-b*x^4 + a)^(5/2)/(d*x^4 - c)^2, x)
\[ \int \frac {\left (a-b x^4\right )^{5/2}}{\left (c-d x^4\right )^2} \, dx=\int { \frac {{\left (-b x^{4} + a\right )}^{\frac {5}{2}}}{{\left (d x^{4} - c\right )}^{2}} \,d x } \] Input:
integrate((-b*x^4+a)^(5/2)/(-d*x^4+c)^2,x, algorithm="giac")
Output:
integrate((-b*x^4 + a)^(5/2)/(d*x^4 - c)^2, x)
Timed out. \[ \int \frac {\left (a-b x^4\right )^{5/2}}{\left (c-d x^4\right )^2} \, dx=\int \frac {{\left (a-b\,x^4\right )}^{5/2}}{{\left (c-d\,x^4\right )}^2} \,d x \] Input:
int((a - b*x^4)^(5/2)/(c - d*x^4)^2,x)
Output:
int((a - b*x^4)^(5/2)/(c - d*x^4)^2, x)
\[ \int \frac {\left (a-b x^4\right )^{5/2}}{\left (c-d x^4\right )^2} \, dx=\text {too large to display} \] Input:
int((-b*x^4+a)^(5/2)/(-d*x^4+c)^2,x)
Output:
( - 9*sqrt(a - b*x**4)*a**2*b*d*x + 5*sqrt(a - b*x**4)*a*b**2*c*x - 3*sqrt (a - b*x**4)*a*b**2*d*x**5 + 3*sqrt(a - b*x**4)*b**3*c*x**5 + 9*int(sqrt(a - b*x**4)/(a**2*c**2*d - 2*a**2*c*d**2*x**4 + a**2*d**3*x**8 - a*b*c**3 + a*b*c**2*d*x**4 + a*b*c*d**2*x**8 - a*b*d**3*x**12 + b**2*c**3*x**4 - 2*b **2*c**2*d*x**8 + b**2*c*d**2*x**12),x)*a**5*c*d**3 - 9*int(sqrt(a - b*x** 4)/(a**2*c**2*d - 2*a**2*c*d**2*x**4 + a**2*d**3*x**8 - a*b*c**3 + a*b*c** 2*d*x**4 + a*b*c*d**2*x**8 - a*b*d**3*x**12 + b**2*c**3*x**4 - 2*b**2*c**2 *d*x**8 + b**2*c*d**2*x**12),x)*a**5*d**4*x**4 - 9*int(sqrt(a - b*x**4)/(a **2*c**2*d - 2*a**2*c*d**2*x**4 + a**2*d**3*x**8 - a*b*c**3 + a*b*c**2*d*x **4 + a*b*c*d**2*x**8 - a*b*d**3*x**12 + b**2*c**3*x**4 - 2*b**2*c**2*d*x* *8 + b**2*c*d**2*x**12),x)*a**4*b*c**2*d**2 + 9*int(sqrt(a - b*x**4)/(a**2 *c**2*d - 2*a**2*c*d**2*x**4 + a**2*d**3*x**8 - a*b*c**3 + a*b*c**2*d*x**4 + a*b*c*d**2*x**8 - a*b*d**3*x**12 + b**2*c**3*x**4 - 2*b**2*c**2*d*x**8 + b**2*c*d**2*x**12),x)*a**4*b*c*d**3*x**4 - 5*int(sqrt(a - b*x**4)/(a**2* c**2*d - 2*a**2*c*d**2*x**4 + a**2*d**3*x**8 - a*b*c**3 + a*b*c**2*d*x**4 + a*b*c*d**2*x**8 - a*b*d**3*x**12 + b**2*c**3*x**4 - 2*b**2*c**2*d*x**8 + b**2*c*d**2*x**12),x)*a**3*b**2*c**3*d + 5*int(sqrt(a - b*x**4)/(a**2*c** 2*d - 2*a**2*c*d**2*x**4 + a**2*d**3*x**8 - a*b*c**3 + a*b*c**2*d*x**4 + a *b*c*d**2*x**8 - a*b*d**3*x**12 + b**2*c**3*x**4 - 2*b**2*c**2*d*x**8 + b* *2*c*d**2*x**12),x)*a**3*b**2*c**2*d**2*x**4 + 5*int(sqrt(a - b*x**4)/(...