Integrand size = 28, antiderivative size = 251 \[ \int x^4 \sqrt {a-b x^4} \left (A+B x^2+C x^4\right ) \, dx=\frac {a x \left (5 (11 A b+5 a C)+77 b B x^2\right ) \sqrt {a-b x^4}}{1155 b^2}-\frac {(11 A b+5 a C) x \left (a-b x^4\right )^{3/2}}{77 b^2}-\frac {B x^3 \left (a-b x^4\right )^{3/2}}{9 b}-\frac {C x^5 \left (a-b x^4\right )^{3/2}}{11 b}+\frac {2 a^{11/4} B \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{15 b^{7/4} \sqrt {a-b x^4}}+\frac {2 a^{9/4} \left (55 A b-77 \sqrt {a} \sqrt {b} B+25 a C\right ) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{1155 b^{9/4} \sqrt {a-b x^4}} \] Output:
1/1155*a*x*(77*B*b*x^2+55*A*b+25*C*a)*(-b*x^4+a)^(1/2)/b^2-1/77*(11*A*b+5* C*a)*x*(-b*x^4+a)^(3/2)/b^2-1/9*B*x^3*(-b*x^4+a)^(3/2)/b-1/11*C*x^5*(-b*x^ 4+a)^(3/2)/b+2/15*a^(11/4)*B*(1-b*x^4/a)^(1/2)*EllipticE(b^(1/4)*x/a^(1/4) ,I)/b^(7/4)/(-b*x^4+a)^(1/2)+2/1155*a^(9/4)*(55*A*b-77*a^(1/2)*b^(1/2)*B+2 5*C*a)*(1-b*x^4/a)^(1/2)*EllipticF(b^(1/4)*x/a^(1/4),I)/b^(9/4)/(-b*x^4+a) ^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.13 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.56 \[ \int x^4 \sqrt {a-b x^4} \left (A+B x^2+C x^4\right ) \, dx=\frac {x \sqrt {a-b x^4} \left (-\left (\left (a-b x^4\right ) \sqrt {1-\frac {b x^4}{a}} \left (99 A b+45 a C+77 b B x^2+63 b C x^4\right )\right )+9 a (11 A b+5 a C) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {b x^4}{a}\right )+77 a b B x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {b x^4}{a}\right )\right )}{693 b^2 \sqrt {1-\frac {b x^4}{a}}} \] Input:
Integrate[x^4*Sqrt[a - b*x^4]*(A + B*x^2 + C*x^4),x]
Output:
(x*Sqrt[a - b*x^4]*(-((a - b*x^4)*Sqrt[1 - (b*x^4)/a]*(99*A*b + 45*a*C + 7 7*b*B*x^2 + 63*b*C*x^4)) + 9*a*(11*A*b + 5*a*C)*Hypergeometric2F1[-1/2, 1/ 4, 5/4, (b*x^4)/a] + 77*a*b*B*x^2*Hypergeometric2F1[-1/2, 3/4, 7/4, (b*x^4 )/a]))/(693*b^2*Sqrt[1 - (b*x^4)/a])
Time = 1.12 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2375, 25, 1597, 1603, 27, 1603, 27, 1513, 27, 765, 762, 1390, 1389, 327}
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.
| \(\displaystyle \int x^4 \sqrt {a-b x^4} \left (A+B x^2+C x^4\right ) \, dx\) |
| \(\Big \downarrow \) 2375 |
| \(\displaystyle -\frac {\int -x^4 \left (11 b B x^2+11 A b+5 a C\right ) \sqrt {a-b x^4}dx}{11 b}-\frac {C x^5 \left (a-b x^4\right )^{3/2}}{11 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int x^4 \left (11 b B x^2+11 A b+5 a C\right ) \sqrt {a-b x^4}dx}{11 b}-\frac {C x^5 \left (a-b x^4\right )^{3/2}}{11 b}\) |
| \(\Big \downarrow \) 1597 |
| \(\displaystyle \frac {\frac {2}{63} a \int \frac {x^4 \left (77 b B x^2+9 (11 A b+5 a C)\right )}{\sqrt {a-b x^4}}dx+\frac {1}{63} x^5 \sqrt {a-b x^4} \left (9 (5 a C+11 A b)+77 b B x^2\right )}{11 b}-\frac {C x^5 \left (a-b x^4\right )^{3/2}}{11 b}\) |
| \(\Big \downarrow \) 1603 |
| \(\displaystyle \frac {\frac {2}{63} a \left (\frac {\int \frac {3 b x^2 \left (15 (11 A b+5 a C) x^2+77 a B\right )}{\sqrt {a-b x^4}}dx}{5 b}-\frac {77}{5} B x^3 \sqrt {a-b x^4}\right )+\frac {1}{63} x^5 \sqrt {a-b x^4} \left (9 (5 a C+11 A b)+77 b B x^2\right )}{11 b}-\frac {C x^5 \left (a-b x^4\right )^{3/2}}{11 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{63} a \left (\frac {3}{5} \int \frac {x^2 \left (15 (11 A b+5 a C) x^2+77 a B\right )}{\sqrt {a-b x^4}}dx-\frac {77}{5} B x^3 \sqrt {a-b x^4}\right )+\frac {1}{63} x^5 \sqrt {a-b x^4} \left (9 (5 a C+11 A b)+77 b B x^2\right )}{11 b}-\frac {C x^5 \left (a-b x^4\right )^{3/2}}{11 b}\) |
| \(\Big \downarrow \) 1603 |
| \(\displaystyle \frac {\frac {2}{63} a \left (\frac {3}{5} \left (\frac {\int \frac {3 a \left (77 b B x^2+5 (11 A b+5 a C)\right )}{\sqrt {a-b x^4}}dx}{3 b}-\frac {5 x \sqrt {a-b x^4} (5 a C+11 A b)}{b}\right )-\frac {77}{5} B x^3 \sqrt {a-b x^4}\right )+\frac {1}{63} x^5 \sqrt {a-b x^4} \left (9 (5 a C+11 A b)+77 b B x^2\right )}{11 b}-\frac {C x^5 \left (a-b x^4\right )^{3/2}}{11 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{63} a \left (\frac {3}{5} \left (\frac {a \int \frac {77 b B x^2+5 (11 A b+5 a C)}{\sqrt {a-b x^4}}dx}{b}-\frac {5 x \sqrt {a-b x^4} (5 a C+11 A b)}{b}\right )-\frac {77}{5} B x^3 \sqrt {a-b x^4}\right )+\frac {1}{63} x^5 \sqrt {a-b x^4} \left (9 (5 a C+11 A b)+77 b B x^2\right )}{11 b}-\frac {C x^5 \left (a-b x^4\right )^{3/2}}{11 b}\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \frac {\frac {2}{63} a \left (\frac {3}{5} \left (\frac {a \left (\left (-77 \sqrt {a} \sqrt {b} B+25 a C+55 A b\right ) \int \frac {1}{\sqrt {a-b x^4}}dx+77 \sqrt {a} \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-b x^4}}dx\right )}{b}-\frac {5 x \sqrt {a-b x^4} (5 a C+11 A b)}{b}\right )-\frac {77}{5} B x^3 \sqrt {a-b x^4}\right )+\frac {1}{63} x^5 \sqrt {a-b x^4} \left (9 (5 a C+11 A b)+77 b B x^2\right )}{11 b}-\frac {C x^5 \left (a-b x^4\right )^{3/2}}{11 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{63} a \left (\frac {3}{5} \left (\frac {a \left (\left (-77 \sqrt {a} \sqrt {b} B+25 a C+55 A b\right ) \int \frac {1}{\sqrt {a-b x^4}}dx+77 \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx\right )}{b}-\frac {5 x \sqrt {a-b x^4} (5 a C+11 A b)}{b}\right )-\frac {77}{5} B x^3 \sqrt {a-b x^4}\right )+\frac {1}{63} x^5 \sqrt {a-b x^4} \left (9 (5 a C+11 A b)+77 b B x^2\right )}{11 b}-\frac {C x^5 \left (a-b x^4\right )^{3/2}}{11 b}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\frac {2}{63} a \left (\frac {3}{5} \left (\frac {a \left (\frac {\sqrt {1-\frac {b x^4}{a}} \left (-77 \sqrt {a} \sqrt {b} B+25 a C+55 A b\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {a-b x^4}}+77 \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx\right )}{b}-\frac {5 x \sqrt {a-b x^4} (5 a C+11 A b)}{b}\right )-\frac {77}{5} B x^3 \sqrt {a-b x^4}\right )+\frac {1}{63} x^5 \sqrt {a-b x^4} \left (9 (5 a C+11 A b)+77 b B x^2\right )}{11 b}-\frac {C x^5 \left (a-b x^4\right )^{3/2}}{11 b}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\frac {2}{63} a \left (\frac {3}{5} \left (\frac {a \left (77 \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (-77 \sqrt {a} \sqrt {b} B+25 a C+55 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )}{b}-\frac {5 x \sqrt {a-b x^4} (5 a C+11 A b)}{b}\right )-\frac {77}{5} B x^3 \sqrt {a-b x^4}\right )+\frac {1}{63} x^5 \sqrt {a-b x^4} \left (9 (5 a C+11 A b)+77 b B x^2\right )}{11 b}-\frac {C x^5 \left (a-b x^4\right )^{3/2}}{11 b}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {\frac {2}{63} a \left (\frac {3}{5} \left (\frac {a \left (\frac {77 \sqrt {b} B \sqrt {1-\frac {b x^4}{a}} \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (-77 \sqrt {a} \sqrt {b} B+25 a C+55 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )}{b}-\frac {5 x \sqrt {a-b x^4} (5 a C+11 A b)}{b}\right )-\frac {77}{5} B x^3 \sqrt {a-b x^4}\right )+\frac {1}{63} x^5 \sqrt {a-b x^4} \left (9 (5 a C+11 A b)+77 b B x^2\right )}{11 b}-\frac {C x^5 \left (a-b x^4\right )^{3/2}}{11 b}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {\frac {2}{63} a \left (\frac {3}{5} \left (\frac {a \left (\frac {77 \sqrt {a} \sqrt {b} B \sqrt {1-\frac {b x^4}{a}} \int \frac {\sqrt {\frac {\sqrt {b} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}}dx}{\sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (-77 \sqrt {a} \sqrt {b} B+25 a C+55 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )}{b}-\frac {5 x \sqrt {a-b x^4} (5 a C+11 A b)}{b}\right )-\frac {77}{5} B x^3 \sqrt {a-b x^4}\right )+\frac {1}{63} x^5 \sqrt {a-b x^4} \left (9 (5 a C+11 A b)+77 b B x^2\right )}{11 b}-\frac {C x^5 \left (a-b x^4\right )^{3/2}}{11 b}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {2}{63} a \left (\frac {3}{5} \left (\frac {a \left (\frac {77 a^{3/4} \sqrt [4]{b} B \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (-77 \sqrt {a} \sqrt {b} B+25 a C+55 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )}{b}-\frac {5 x \sqrt {a-b x^4} (5 a C+11 A b)}{b}\right )-\frac {77}{5} B x^3 \sqrt {a-b x^4}\right )+\frac {1}{63} x^5 \sqrt {a-b x^4} \left (9 (5 a C+11 A b)+77 b B x^2\right )}{11 b}-\frac {C x^5 \left (a-b x^4\right )^{3/2}}{11 b}\) |
Input:
Int[x^4*Sqrt[a - b*x^4]*(A + B*x^2 + C*x^4),x]
Output:
-1/11*(C*x^5*(a - b*x^4)^(3/2))/b + ((x^5*(9*(11*A*b + 5*a*C) + 77*b*B*x^2 )*Sqrt[a - b*x^4])/63 + (2*a*((-77*B*x^3*Sqrt[a - b*x^4])/5 + (3*((-5*(11* A*b + 5*a*C)*x*Sqrt[a - b*x^4])/b + (a*((77*a^(3/4)*b^(1/4)*B*Sqrt[1 - (b* x^4)/a]*EllipticE[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/Sqrt[a - b*x^4] + (a^( 1/4)*(55*A*b - 77*Sqrt[a]*Sqrt[b]*B + 25*a*C)*Sqrt[1 - (b*x^4)/a]*Elliptic F[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(b^(1/4)*Sqrt[a - b*x^4])))/b))/5))/63 )/(11*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
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[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq rt[a] Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt [1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && !GtQ [a, 0] && !(LtQ[a, 0] && GtQ[c, 0])
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-c/a, 2]}, Simp[(d*q - e)/q Int[1/Sqrt[a + c*x^4], x], x] + Simp[e/q Int[(1 + q*x^2)/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && Neg Q[c/a] && NeQ[c*d^2 + a*e^2, 0]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_.), x _Symbol] :> Simp[(f*x)^(m + 1)*(a + c*x^4)^p*((c*d*(m + 4*p + 3) + c*e*(4*p + m + 1)*x^2)/(c*f*(4*p + m + 1)*(m + 4*p + 3))), x] + Simp[4*a*(p/((4*p + m + 1)*(m + 4*p + 3))) Int[(f*x)^m*(a + c*x^4)^(p - 1)*Simp[d*(m + 4*p + 3) + e*(4*p + m + 1)*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, m}, x] && Gt Q[p, 0] && NeQ[4*p + m + 1, 0] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && ( IntegerQ[p] || IntegerQ[m])
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_ Symbol] :> Simp[e*f*(f*x)^(m - 1)*((a + c*x^4)^(p + 1)/(c*(m + 4*p + 3))), x] - Simp[f^2/(c*(m + 4*p + 3)) Int[(f*x)^(m - 2)*(a + c*x^4)^p*(a*e*(m - 1) - c*d*(m + 4*p + 3)*x^2), x], x] /; FreeQ[{a, c, d, e, f, p}, x] && GtQ [m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[ m])
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Wi th[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Simp[Pqq*(c*x)^(m + q - n + 1)*((a + b*x^n)^(p + 1)/(b*c^(q - n + 1)*(m + q + n*p + 1))), x] + Si mp[1/(b*(m + q + n*p + 1)) Int[(c*x)^m*ExpandToSum[b*(m + q + n*p + 1)*(P q - Pqq*x^q) - a*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x]] / ; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || IntegerQ[p + ( q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]
Time = 3.38 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.16
| method | result | size |
| elliptic | \(\frac {C \,x^{9} \sqrt {-b \,x^{4}+a}}{11}+\frac {B \,x^{7} \sqrt {-b \,x^{4}+a}}{9}-\frac {\left (-A b +\frac {2 C a}{11}\right ) x^{5} \sqrt {-b \,x^{4}+a}}{7 b}-\frac {2 B a \,x^{3} \sqrt {-b \,x^{4}+a}}{45 b}-\frac {\left (A a +\frac {5 \left (-A b +\frac {2 C a}{11}\right ) a}{7 b}\right ) x \sqrt {-b \,x^{4}+a}}{3 b}+\frac {\left (A a +\frac {5 \left (-A b +\frac {2 C a}{11}\right ) a}{7 b}\right ) a \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{3 b \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {2 B \,a^{\frac {5}{2}} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 b^{\frac {3}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\) | \(291\) |
| risch | \(-\frac {x \left (-315 C \,x^{8} b^{2}-385 B \,x^{6} b^{2}-495 A \,b^{2} x^{4}+90 C a b \,x^{4}+154 B a \,x^{2} b +330 a b A +150 a^{2} C \right ) \sqrt {-b \,x^{4}+a}}{3465 b^{2}}+\frac {2 a^{2} \left (\frac {55 A b \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}+\frac {25 C a \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {77 B \sqrt {b}\, \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )}{1155 b^{2}}\) | \(302\) |
| default | \(A \left (\frac {x^{5} \sqrt {-b \,x^{4}+a}}{7}-\frac {2 a x \sqrt {-b \,x^{4}+a}}{21 b}+\frac {2 a^{2} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{21 b \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )+B \left (\frac {x^{7} \sqrt {-b \,x^{4}+a}}{9}-\frac {2 a \,x^{3} \sqrt {-b \,x^{4}+a}}{45 b}-\frac {2 a^{\frac {5}{2}} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 b^{\frac {3}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )+C \left (\frac {x^{9} \sqrt {-b \,x^{4}+a}}{11}-\frac {2 a \,x^{5} \sqrt {-b \,x^{4}+a}}{77 b}-\frac {10 a^{2} x \sqrt {-b \,x^{4}+a}}{231 b^{2}}+\frac {10 a^{3} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{231 b^{2} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )\) | \(357\) |
Input:
int(x^4*(-b*x^4+a)^(1/2)*(C*x^4+B*x^2+A),x,method=_RETURNVERBOSE)
Output:
1/11*C*x^9*(-b*x^4+a)^(1/2)+1/9*B*x^7*(-b*x^4+a)^(1/2)-1/7*(-A*b+2/11*C*a) /b*x^5*(-b*x^4+a)^(1/2)-2/45*B*a/b*x^3*(-b*x^4+a)^(1/2)-1/3*(A*a+5/7*(-A*b +2/11*C*a)/b*a)/b*x*(-b*x^4+a)^(1/2)+1/3*(A*a+5/7*(-A*b+2/11*C*a)/b*a)/b*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)-2/1 5*B*a^(5/2)/b^(3/2)/(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)-EllipticE(x*(1/a^(1/2)*b^(1/2))^(1/2),I))
Time = 0.14 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.67 \[ \int x^4 \sqrt {a-b x^4} \left (A+B x^2+C x^4\right ) \, dx=-\frac {462 \, B a^{2} \sqrt {-b} x \left (\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 6 \, {\left ({\left (77 \, B + 25 \, C\right )} a^{2} + 55 \, A a b\right )} \sqrt {-b} x \left (\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (315 \, C b^{2} x^{10} + 385 \, B b^{2} x^{8} - 154 \, B a b x^{4} - 45 \, {\left (2 \, C a b - 11 \, A b^{2}\right )} x^{6} - 462 \, B a^{2} - 30 \, {\left (5 \, C a^{2} + 11 \, A a b\right )} x^{2}\right )} \sqrt {-b x^{4} + a}}{3465 \, b^{2} x} \] Input:
integrate(x^4*(-b*x^4+a)^(1/2)*(C*x^4+B*x^2+A),x, algorithm="fricas")
Output:
-1/3465*(462*B*a^2*sqrt(-b)*x*(a/b)^(3/4)*elliptic_e(arcsin((a/b)^(1/4)/x) , -1) - 6*((77*B + 25*C)*a^2 + 55*A*a*b)*sqrt(-b)*x*(a/b)^(3/4)*elliptic_f (arcsin((a/b)^(1/4)/x), -1) - (315*C*b^2*x^10 + 385*B*b^2*x^8 - 154*B*a*b* x^4 - 45*(2*C*a*b - 11*A*b^2)*x^6 - 462*B*a^2 - 30*(5*C*a^2 + 11*A*a*b)*x^ 2)*sqrt(-b*x^4 + a))/(b^2*x)
Time = 1.68 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.52 \[ \int x^4 \sqrt {a-b x^4} \left (A+B x^2+C x^4\right ) \, dx=\frac {A \sqrt {a} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {B \sqrt {a} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} + \frac {C \sqrt {a} x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} \] Input:
integrate(x**4*(-b*x**4+a)**(1/2)*(C*x**4+B*x**2+A),x)
Output:
A*sqrt(a)*x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), b*x**4*exp_polar(2*I* pi)/a)/(4*gamma(9/4)) + B*sqrt(a)*x**7*gamma(7/4)*hyper((-1/2, 7/4), (11/4 ,), b*x**4*exp_polar(2*I*pi)/a)/(4*gamma(11/4)) + C*sqrt(a)*x**9*gamma(9/4 )*hyper((-1/2, 9/4), (13/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*gamma(13/4))
\[ \int x^4 \sqrt {a-b x^4} \left (A+B x^2+C x^4\right ) \, dx=\int { {\left (C x^{4} + B x^{2} + A\right )} \sqrt {-b x^{4} + a} x^{4} \,d x } \] Input:
integrate(x^4*(-b*x^4+a)^(1/2)*(C*x^4+B*x^2+A),x, algorithm="maxima")
Output:
integrate((C*x^4 + B*x^2 + A)*sqrt(-b*x^4 + a)*x^4, x)
\[ \int x^4 \sqrt {a-b x^4} \left (A+B x^2+C x^4\right ) \, dx=\int { {\left (C x^{4} + B x^{2} + A\right )} \sqrt {-b x^{4} + a} x^{4} \,d x } \] Input:
integrate(x^4*(-b*x^4+a)^(1/2)*(C*x^4+B*x^2+A),x, algorithm="giac")
Output:
integrate((C*x^4 + B*x^2 + A)*sqrt(-b*x^4 + a)*x^4, x)
Timed out. \[ \int x^4 \sqrt {a-b x^4} \left (A+B x^2+C x^4\right ) \, dx=\int x^4\,\sqrt {a-b\,x^4}\,\left (C\,x^4+B\,x^2+A\right ) \,d x \] Input:
int(x^4*(a - b*x^4)^(1/2)*(A + B*x^2 + C*x^4),x)
Output:
int(x^4*(a - b*x^4)^(1/2)*(A + B*x^2 + C*x^4), x)
\[ \int x^4 \sqrt {a-b x^4} \left (A+B x^2+C x^4\right ) \, dx=\frac {-330 \sqrt {-b \,x^{4}+a}\, a^{2} b x -150 \sqrt {-b \,x^{4}+a}\, a^{2} c x +495 \sqrt {-b \,x^{4}+a}\, a \,b^{2} x^{5}-154 \sqrt {-b \,x^{4}+a}\, a \,b^{2} x^{3}-90 \sqrt {-b \,x^{4}+a}\, a b c \,x^{5}+385 \sqrt {-b \,x^{4}+a}\, b^{3} x^{7}+315 \sqrt {-b \,x^{4}+a}\, b^{2} c \,x^{9}+330 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,x^{4}+a}d x \right ) a^{3} b +150 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,x^{4}+a}d x \right ) a^{3} c +462 \left (\int \frac {\sqrt {-b \,x^{4}+a}\, x^{2}}{-b \,x^{4}+a}d x \right ) a^{2} b^{2}}{3465 b^{2}} \] Input:
int(x^4*(-b*x^4+a)^(1/2)*(C*x^4+B*x^2+A),x)
Output:
( - 330*sqrt(a - b*x**4)*a**2*b*x - 150*sqrt(a - b*x**4)*a**2*c*x + 495*sq rt(a - b*x**4)*a*b**2*x**5 - 154*sqrt(a - b*x**4)*a*b**2*x**3 - 90*sqrt(a - b*x**4)*a*b*c*x**5 + 385*sqrt(a - b*x**4)*b**3*x**7 + 315*sqrt(a - b*x** 4)*b**2*c*x**9 + 330*int(sqrt(a - b*x**4)/(a - b*x**4),x)*a**3*b + 150*int (sqrt(a - b*x**4)/(a - b*x**4),x)*a**3*c + 462*int((sqrt(a - b*x**4)*x**2) /(a - b*x**4),x)*a**2*b**2)/(3465*b**2)