Integrand size = 28, antiderivative size = 211 \[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\sqrt {a-b x^4}} \, dx=-\frac {(7 A b+5 a C) x \sqrt {a-b x^4}}{21 b^2}-\frac {B x^3 \sqrt {a-b x^4}}{5 b}-\frac {C x^5 \sqrt {a-b x^4}}{7 b}+\frac {3 a^{7/4} B \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 b^{7/4} \sqrt {a-b x^4}}+\frac {a^{5/4} \left (35 A b-63 \sqrt {a} \sqrt {b} B+25 a C\right ) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{105 b^{9/4} \sqrt {a-b x^4}} \] Output:
-1/21*(7*A*b+5*C*a)*x*(-b*x^4+a)^(1/2)/b^2-1/5*B*x^3*(-b*x^4+a)^(1/2)/b-1/ 7*C*x^5*(-b*x^4+a)^(1/2)/b+3/5*a^(7/4)*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)+1/105*a^(5/4)*(35*A*b-63*a^(1/2)* b^(1/2)*B+25*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 = 10.10 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.67 \[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\sqrt {a-b x^4}} \, dx=\frac {x \left (-a+b x^4\right ) \left (35 A b+25 a C+21 b B x^2+15 b C x^4\right )+5 a (7 A b+5 a C) x \sqrt {1-\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {b x^4}{a}\right )+21 a b B x^3 \sqrt {1-\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {b x^4}{a}\right )}{105 b^2 \sqrt {a-b x^4}} \] Input:
Integrate[(x^4*(A + B*x^2 + C*x^4))/Sqrt[a - b*x^4],x]
Output:
(x*(-a + b*x^4)*(35*A*b + 25*a*C + 21*b*B*x^2 + 15*b*C*x^4) + 5*a*(7*A*b + 5*a*C)*x*Sqrt[1 - (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, (b*x^4)/a] + 21*a*b*B*x^3*Sqrt[1 - (b*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, (b*x^4 )/a])/(105*b^2*Sqrt[a - b*x^4])
Time = 0.95 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {2375, 25, 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.
| \(\displaystyle \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\sqrt {a-b x^4}} \, dx\) |
| \(\Big \downarrow \) 2375 |
| \(\displaystyle -\frac {\int -\frac {x^4 \left (7 b B x^2+7 A b+5 a C\right )}{\sqrt {a-b x^4}}dx}{7 b}-\frac {C x^5 \sqrt {a-b x^4}}{7 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {x^4 \left (7 b B x^2+7 A b+5 a C\right )}{\sqrt {a-b x^4}}dx}{7 b}-\frac {C x^5 \sqrt {a-b x^4}}{7 b}\) |
| \(\Big \downarrow \) 1603 |
| \(\displaystyle \frac {\frac {\int \frac {b x^2 \left (5 (7 A b+5 a C) x^2+21 a B\right )}{\sqrt {a-b x^4}}dx}{5 b}-\frac {7}{5} B x^3 \sqrt {a-b x^4}}{7 b}-\frac {C x^5 \sqrt {a-b x^4}}{7 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \int \frac {x^2 \left (5 (7 A b+5 a C) x^2+21 a B\right )}{\sqrt {a-b x^4}}dx-\frac {7}{5} B x^3 \sqrt {a-b x^4}}{7 b}-\frac {C x^5 \sqrt {a-b x^4}}{7 b}\) |
| \(\Big \downarrow \) 1603 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {\int \frac {a \left (63 b B x^2+5 (7 A b+5 a C)\right )}{\sqrt {a-b x^4}}dx}{3 b}-\frac {5 x \sqrt {a-b x^4} (5 a C+7 A b)}{3 b}\right )-\frac {7}{5} B x^3 \sqrt {a-b x^4}}{7 b}-\frac {C x^5 \sqrt {a-b x^4}}{7 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {a \int \frac {63 b B x^2+5 (7 A b+5 a C)}{\sqrt {a-b x^4}}dx}{3 b}-\frac {5 x \sqrt {a-b x^4} (5 a C+7 A b)}{3 b}\right )-\frac {7}{5} B x^3 \sqrt {a-b x^4}}{7 b}-\frac {C x^5 \sqrt {a-b x^4}}{7 b}\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {a \left (\left (-63 \sqrt {a} \sqrt {b} B+25 a C+35 A b\right ) \int \frac {1}{\sqrt {a-b x^4}}dx+63 \sqrt {a} \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-b x^4}}dx\right )}{3 b}-\frac {5 x \sqrt {a-b x^4} (5 a C+7 A b)}{3 b}\right )-\frac {7}{5} B x^3 \sqrt {a-b x^4}}{7 b}-\frac {C x^5 \sqrt {a-b x^4}}{7 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {a \left (\left (-63 \sqrt {a} \sqrt {b} B+25 a C+35 A b\right ) \int \frac {1}{\sqrt {a-b x^4}}dx+63 \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx\right )}{3 b}-\frac {5 x \sqrt {a-b x^4} (5 a C+7 A b)}{3 b}\right )-\frac {7}{5} B x^3 \sqrt {a-b x^4}}{7 b}-\frac {C x^5 \sqrt {a-b x^4}}{7 b}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {a \left (\frac {\sqrt {1-\frac {b x^4}{a}} \left (-63 \sqrt {a} \sqrt {b} B+25 a C+35 A b\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {a-b x^4}}+63 \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx\right )}{3 b}-\frac {5 x \sqrt {a-b x^4} (5 a C+7 A b)}{3 b}\right )-\frac {7}{5} B x^3 \sqrt {a-b x^4}}{7 b}-\frac {C x^5 \sqrt {a-b x^4}}{7 b}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {a \left (63 \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (-63 \sqrt {a} \sqrt {b} B+25 a C+35 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )}{3 b}-\frac {5 x \sqrt {a-b x^4} (5 a C+7 A b)}{3 b}\right )-\frac {7}{5} B x^3 \sqrt {a-b x^4}}{7 b}-\frac {C x^5 \sqrt {a-b x^4}}{7 b}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {a \left (\frac {63 \sqrt {b} B \sqrt {1-\frac {b x^4}{a}} \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (-63 \sqrt {a} \sqrt {b} B+25 a C+35 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )}{3 b}-\frac {5 x \sqrt {a-b x^4} (5 a C+7 A b)}{3 b}\right )-\frac {7}{5} B x^3 \sqrt {a-b x^4}}{7 b}-\frac {C x^5 \sqrt {a-b x^4}}{7 b}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {a \left (\frac {63 \sqrt {a} \sqrt {b} B \sqrt {1-\frac {b x^4}{a}} \int \frac {\sqrt {\frac {\sqrt {b} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}}dx}{\sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (-63 \sqrt {a} \sqrt {b} B+25 a C+35 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )}{3 b}-\frac {5 x \sqrt {a-b x^4} (5 a C+7 A b)}{3 b}\right )-\frac {7}{5} B x^3 \sqrt {a-b x^4}}{7 b}-\frac {C x^5 \sqrt {a-b x^4}}{7 b}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {a \left (\frac {63 a^{3/4} \sqrt [4]{b} B \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (-63 \sqrt {a} \sqrt {b} B+25 a C+35 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )}{3 b}-\frac {5 x \sqrt {a-b x^4} (5 a C+7 A b)}{3 b}\right )-\frac {7}{5} B x^3 \sqrt {a-b x^4}}{7 b}-\frac {C x^5 \sqrt {a-b x^4}}{7 b}\) |
Input:
Int[(x^4*(A + B*x^2 + C*x^4))/Sqrt[a - b*x^4],x]
Output:
-1/7*(C*x^5*Sqrt[a - b*x^4])/b + ((-7*B*x^3*Sqrt[a - b*x^4])/5 + ((-5*(7*A *b + 5*a*C)*x*Sqrt[a - b*x^4])/(3*b) + (a*((63*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)*(35*A*b - 63*Sqrt[a]*Sqrt[b]*B + 25*a*C)*Sqrt[1 - (b*x^4)/a]*Ellip ticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(b^(1/4)*Sqrt[a - b*x^4])))/(3*b))/ 5)/(7*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[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 = 3.35 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.09
| method | result | size |
| elliptic | \(-\frac {C \,x^{5} \sqrt {-b \,x^{4}+a}}{7 b}-\frac {B \,x^{3} \sqrt {-b \,x^{4}+a}}{5 b}-\frac {\left (A +\frac {5 a C}{7 b}\right ) x \sqrt {-b \,x^{4}+a}}{3 b}+\frac {\left (A +\frac {5 a C}{7 b}\right ) a \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{3 b \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {3 B \,a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 b^{\frac {3}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\) | \(230\) |
| risch | \(-\frac {x \left (15 C b \,x^{4}+21 B b \,x^{2}+35 A b +25 C a \right ) \sqrt {-b \,x^{4}+a}}{105 b^{2}}+\frac {a \left (\frac {35 A b \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}+\frac {25 C a \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {63 B \sqrt {b}\, \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )}{105 b^{2}}\) | \(268\) |
| default | \(A \left (-\frac {x \sqrt {-b \,x^{4}+a}}{3 b}+\frac {a \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{3 b \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )+B \left (-\frac {x^{3} \sqrt {-b \,x^{4}+a}}{5 b}-\frac {3 a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 b^{\frac {3}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )+C \left (-\frac {x^{5} \sqrt {-b \,x^{4}+a}}{7 b}-\frac {5 a x \sqrt {-b \,x^{4}+a}}{21 b^{2}}+\frac {5 a^{2} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{21 b^{2} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )\) | \(305\) |
Input:
int(x^4*(C*x^4+B*x^2+A)/(-b*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/7*C*x^5*(-b*x^4+a)^(1/2)/b-1/5*B*x^3*(-b*x^4+a)^(1/2)/b-1/3*(A+5/7*a*C/ b)/b*x*(-b*x^4+a)^(1/2)+1/3*(A+5/7*a*C/b)/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)-3/5*B*a^(3/2)/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 = 0.08 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.60 \[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\sqrt {a-b x^4}} \, dx=-\frac {63 \, B a \sqrt {-b} x \left (\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left ({\left (63 \, B + 25 \, C\right )} a + 35 \, A b\right )} \sqrt {-b} x \left (\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (15 \, C b x^{6} + 21 \, B b x^{4} + 5 \, {\left (5 \, C a + 7 \, A b\right )} x^{2} + 63 \, B a\right )} \sqrt {-b x^{4} + a}}{105 \, b^{2} x} \] Input:
integrate(x^4*(C*x^4+B*x^2+A)/(-b*x^4+a)^(1/2),x, algorithm="fricas")
Output:
-1/105*(63*B*a*sqrt(-b)*x*(a/b)^(3/4)*elliptic_e(arcsin((a/b)^(1/4)/x), -1 ) - ((63*B + 25*C)*a + 35*A*b)*sqrt(-b)*x*(a/b)^(3/4)*elliptic_f(arcsin((a /b)^(1/4)/x), -1) + (15*C*b*x^6 + 21*B*b*x^4 + 5*(5*C*a + 7*A*b)*x^2 + 63* B*a)*sqrt(-b*x^4 + a))/(b^2*x)
Time = 1.78 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.60 \[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\sqrt {a-b x^4}} \, dx=\frac {A x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {9}{4}\right )} + \frac {B x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {11}{4}\right )} + \frac {C x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {13}{4}\right )} \] Input:
integrate(x**4*(C*x**4+B*x**2+A)/(-b*x**4+a)**(1/2),x)
Output:
A*x**5*gamma(5/4)*hyper((1/2, 5/4), (9/4,), b*x**4*exp_polar(2*I*pi)/a)/(4 *sqrt(a)*gamma(9/4)) + B*x**7*gamma(7/4)*hyper((1/2, 7/4), (11/4,), b*x**4 *exp_polar(2*I*pi)/a)/(4*sqrt(a)*gamma(11/4)) + C*x**9*gamma(9/4)*hyper((1 /2, 9/4), (13/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(a)*gamma(13/4))
\[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\sqrt {a-b x^4}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} x^{4}}{\sqrt {-b x^{4} + a}} \,d x } \] Input:
integrate(x^4*(C*x^4+B*x^2+A)/(-b*x^4+a)^(1/2),x, algorithm="maxima")
Output:
integrate((C*x^4 + B*x^2 + A)*x^4/sqrt(-b*x^4 + a), x)
\[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\sqrt {a-b x^4}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} x^{4}}{\sqrt {-b x^{4} + a}} \,d x } \] Input:
integrate(x^4*(C*x^4+B*x^2+A)/(-b*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate((C*x^4 + B*x^2 + A)*x^4/sqrt(-b*x^4 + a), x)
Timed out. \[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\sqrt {a-b x^4}} \, dx=\int \frac {x^4\,\left (C\,x^4+B\,x^2+A\right )}{\sqrt {a-b\,x^4}} \,d x \] Input:
int((x^4*(A + B*x^2 + C*x^4))/(a - b*x^4)^(1/2),x)
Output:
int((x^4*(A + B*x^2 + C*x^4))/(a - b*x^4)^(1/2), x)
\[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\sqrt {a-b x^4}} \, dx=\frac {-35 \sqrt {-b \,x^{4}+a}\, a b x -25 \sqrt {-b \,x^{4}+a}\, a c x -21 \sqrt {-b \,x^{4}+a}\, b^{2} x^{3}-15 \sqrt {-b \,x^{4}+a}\, b c \,x^{5}+35 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,x^{4}+a}d x \right ) a^{2} b +25 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,x^{4}+a}d x \right ) a^{2} c +63 \left (\int \frac {\sqrt {-b \,x^{4}+a}\, x^{2}}{-b \,x^{4}+a}d x \right ) a \,b^{2}}{105 b^{2}} \] Input:
int(x^4*(C*x^4+B*x^2+A)/(-b*x^4+a)^(1/2),x)
Output:
( - 35*sqrt(a - b*x**4)*a*b*x - 25*sqrt(a - b*x**4)*a*c*x - 21*sqrt(a - b* x**4)*b**2*x**3 - 15*sqrt(a - b*x**4)*b*c*x**5 + 35*int(sqrt(a - b*x**4)/( a - b*x**4),x)*a**2*b + 25*int(sqrt(a - b*x**4)/(a - b*x**4),x)*a**2*c + 6 3*int((sqrt(a - b*x**4)*x**2)/(a - b*x**4),x)*a*b**2)/(105*b**2)