Integrand size = 28, antiderivative size = 190 \[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\frac {x \left (A b+a C+b B x^2\right )}{2 b^2 \sqrt {a-b x^4}}+\frac {C x \sqrt {a-b x^4}}{3 b^2}-\frac {3 a^{3/4} B \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 b^{7/4} \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} \left (3 A b-9 \sqrt {a} \sqrt {b} B+5 a C\right ) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{6 b^{9/4} \sqrt {a-b x^4}} \] Output:
1/2*x*(B*b*x^2+A*b+C*a)/b^2/(-b*x^4+a)^(1/2)+1/3*C*x*(-b*x^4+a)^(1/2)/b^2- 3/2*a^(3/4)*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)-1/6*a^(1/4)*(3*A*b-9*a^(1/2)*b^(1/2)*B+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 = 10.07 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.68 \[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\frac {3 A b x+5 a C x-6 b B x^3-2 b C x^5-(3 A b+5 a C) x \sqrt {1-\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {b x^4}{a}\right )+6 b B x^3 \sqrt {1-\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},\frac {b x^4}{a}\right )}{6 b^2 \sqrt {a-b x^4}} \] Input:
Integrate[(x^4*(A + B*x^2 + C*x^4))/(a - b*x^4)^(3/2),x]
Output:
(3*A*b*x + 5*a*C*x - 6*b*B*x^3 - 2*b*C*x^5 - (3*A*b + 5*a*C)*x*Sqrt[1 - (b *x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, (b*x^4)/a] + 6*b*B*x^3*Sqrt[1 - (b*x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, (b*x^4)/a])/(6*b^2*Sqrt[a - b* x^4])
Time = 0.93 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {2367, 2427, 25, 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.
| \(\displaystyle \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2367 |
| \(\displaystyle \frac {x \left (a C+A b+b B x^2\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {\int \frac {2 a b C x^4+3 a b B x^2+a (A b+a C)}{\sqrt {a-b x^4}}dx}{2 a b^2}\) |
| \(\Big \downarrow \) 2427 |
| \(\displaystyle \frac {x \left (a C+A b+b B x^2\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {-\frac {\int -\frac {a b \left (9 b B x^2+3 A b+5 a C\right )}{\sqrt {a-b x^4}}dx}{3 b}-\frac {2}{3} a C x \sqrt {a-b x^4}}{2 a b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x \left (a C+A b+b B x^2\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {\frac {\int \frac {a b \left (9 b B x^2+3 A b+5 a C\right )}{\sqrt {a-b x^4}}dx}{3 b}-\frac {2}{3} a C x \sqrt {a-b x^4}}{2 a b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \left (a C+A b+b B x^2\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {\frac {1}{3} a \int \frac {9 b B x^2+3 A b+5 a C}{\sqrt {a-b x^4}}dx-\frac {2}{3} a C x \sqrt {a-b x^4}}{2 a b^2}\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \frac {x \left (a C+A b+b B x^2\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {\frac {1}{3} a \left (\left (-9 \sqrt {a} \sqrt {b} B+5 a C+3 A b\right ) \int \frac {1}{\sqrt {a-b x^4}}dx+9 \sqrt {a} \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-b x^4}}dx\right )-\frac {2}{3} a C x \sqrt {a-b x^4}}{2 a b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \left (a C+A b+b B x^2\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {\frac {1}{3} a \left (\left (-9 \sqrt {a} \sqrt {b} B+5 a C+3 A b\right ) \int \frac {1}{\sqrt {a-b x^4}}dx+9 \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx\right )-\frac {2}{3} a C x \sqrt {a-b x^4}}{2 a b^2}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {x \left (a C+A b+b B x^2\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {\frac {1}{3} a \left (\frac {\sqrt {1-\frac {b x^4}{a}} \left (-9 \sqrt {a} \sqrt {b} B+5 a C+3 A b\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {a-b x^4}}+9 \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx\right )-\frac {2}{3} a C x \sqrt {a-b x^4}}{2 a b^2}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {x \left (a C+A b+b B x^2\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {\frac {1}{3} a \left (9 \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (-9 \sqrt {a} \sqrt {b} B+5 a C+3 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )-\frac {2}{3} a C x \sqrt {a-b x^4}}{2 a b^2}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {x \left (a C+A b+b B x^2\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {\frac {1}{3} a \left (\frac {9 \sqrt {b} B \sqrt {1-\frac {b x^4}{a}} \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (-9 \sqrt {a} \sqrt {b} B+5 a C+3 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )-\frac {2}{3} a C x \sqrt {a-b x^4}}{2 a b^2}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {x \left (a C+A b+b B x^2\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {\frac {1}{3} a \left (\frac {9 \sqrt {a} \sqrt {b} B \sqrt {1-\frac {b x^4}{a}} \int \frac {\sqrt {\frac {\sqrt {b} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}}dx}{\sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (-9 \sqrt {a} \sqrt {b} B+5 a C+3 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )-\frac {2}{3} a C x \sqrt {a-b x^4}}{2 a b^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {x \left (a C+A b+b B x^2\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {\frac {1}{3} a \left (\frac {9 a^{3/4} \sqrt [4]{b} B \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (-9 \sqrt {a} \sqrt {b} B+5 a C+3 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )-\frac {2}{3} a C x \sqrt {a-b x^4}}{2 a b^2}\) |
Input:
Int[(x^4*(A + B*x^2 + C*x^4))/(a - b*x^4)^(3/2),x]
Output:
(x*(A*b + a*C + b*B*x^2))/(2*b^2*Sqrt[a - b*x^4]) - ((-2*a*C*x*Sqrt[a - b* x^4])/3 + (a*((9*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)*(3*A*b - 9*Sqrt[a]*Sqr t[b]*B + 5*a*C)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(b^(1/4)*Sqrt[a - b*x^4])))/3)/(2*a*b^2)
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[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = m + Expon[Pq, x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1) *x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*x^ m*Pq, a + b*x^n, x]}, Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floo r[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) I nt[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0 ] && LtQ[p, -1] && IGtQ[m, 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x ]}, With[{Pqq = Coeff[Pq, x, q]}, Simp[Pqq*x^(q - n + 1)*((a + b*x^n)^(p + 1)/(b*(q + n*p + 1))), x] + Simp[1/(b*(q + n*p + 1)) Int[ExpandToSum[b*(q + n*p + 1)*(Pq - Pqq*x^q) - a*Pqq*(q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x]] /; NeQ[q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || IntegerQ [p + (q + 1)/(2*n)])] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]
Time = 4.57 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.22
| method | result | size |
| elliptic | \(\frac {2 b \left (\frac {B \,x^{3}}{4 b^{2}}+\frac {\left (A b +C a \right ) x}{4 b^{3}}\right )}{\sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {C x \sqrt {-b \,x^{4}+a}}{3 b^{2}}+\frac {\left (-\frac {A b +C a}{2 b^{2}}-\frac {C a}{3 b^{2}}\right ) \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}+\frac {3 B \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 b^{\frac {3}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\) | \(231\) |
| default | \(A \left (\frac {x}{2 b \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}-\frac {\sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{2 b \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )+B \left (\frac {x^{3}}{2 b \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {3 \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 b^{\frac {3}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )+C \left (\frac {a x}{2 b^{2} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {x \sqrt {-b \,x^{4}+a}}{3 b^{2}}-\frac {5 a \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{6 b^{2} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )\) | \(315\) |
| risch | \(\frac {C x \sqrt {-b \,x^{4}+a}}{3 b^{2}}-\frac {\frac {3 A b \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}+\frac {4 C a \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {3 B \sqrt {b}\, \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}+3 a \left (\frac {2 b \left (-\frac {x^{3} B}{4 a}-\frac {\left (A b +C a \right ) x}{4 b a}\right )}{\sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}-\frac {\left (A b +C a \right ) \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\sqrt {b}\, B \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )}{3 b^{2}}\) | \(455\) |
Input:
int(x^4*(C*x^4+B*x^2+A)/(-b*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
2*b*(1/4*B*x^3/b^2+1/4/b^3*(A*b+C*a)*x)/(-(x^4-a/b)*b)^(1/2)+1/3*C*x*(-b*x ^4+a)^(1/2)/b^2+(-1/2*(A*b+C*a)/b^2-1/3*C/b^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)+3/2*B/b^(3/2)*a^(1/2)/(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 = 187, normalized size of antiderivative = 0.98 \[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\frac {9 \, {\left (B a b x^{5} - B a^{2} x\right )} \sqrt {-b} \left (\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left ({\left ({\left (9 \, B + 5 \, C\right )} a b + 3 \, A b^{2}\right )} x^{5} - {\left ({\left (9 \, B + 5 \, C\right )} a^{2} + 3 \, A a b\right )} x\right )} \sqrt {-b} \left (\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (2 \, C a b x^{6} + 6 \, B a b x^{4} - 9 \, B a^{2} - {\left (5 \, C a^{2} + 3 \, A a b\right )} x^{2}\right )} \sqrt {-b x^{4} + a}}{6 \, {\left (a b^{3} x^{5} - a^{2} b^{2} x\right )}} \] Input:
integrate(x^4*(C*x^4+B*x^2+A)/(-b*x^4+a)^(3/2),x, algorithm="fricas")
Output:
1/6*(9*(B*a*b*x^5 - B*a^2*x)*sqrt(-b)*(a/b)^(3/4)*elliptic_e(arcsin((a/b)^ (1/4)/x), -1) - (((9*B + 5*C)*a*b + 3*A*b^2)*x^5 - ((9*B + 5*C)*a^2 + 3*A* a*b)*x)*sqrt(-b)*(a/b)^(3/4)*elliptic_f(arcsin((a/b)^(1/4)/x), -1) + (2*C* a*b*x^6 + 6*B*a*b*x^4 - 9*B*a^2 - (5*C*a^2 + 3*A*a*b)*x^2)*sqrt(-b*x^4 + a ))/(a*b^3*x^5 - a^2*b^2*x)
Time = 6.81 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.66 \[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\frac {A x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {3}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {9}{4}\right )} + \frac {B x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {11}{4}\right )} + \frac {C x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {13}{4}\right )} \] Input:
integrate(x**4*(C*x**4+B*x**2+A)/(-b*x**4+a)**(3/2),x)
Output:
A*x**5*gamma(5/4)*hyper((5/4, 3/2), (9/4,), b*x**4*exp_polar(2*I*pi)/a)/(4 *a**(3/2)*gamma(9/4)) + B*x**7*gamma(7/4)*hyper((3/2, 7/4), (11/4,), b*x** 4*exp_polar(2*I*pi)/a)/(4*a**(3/2)*gamma(11/4)) + C*x**9*gamma(9/4)*hyper( (3/2, 9/4), (13/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*a**(3/2)*gamma(13/4))
\[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} x^{4}}{{\left (-b x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^4*(C*x^4+B*x^2+A)/(-b*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((C*x^4 + B*x^2 + A)*x^4/(-b*x^4 + a)^(3/2), x)
\[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} x^{4}}{{\left (-b x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^4*(C*x^4+B*x^2+A)/(-b*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((C*x^4 + B*x^2 + A)*x^4/(-b*x^4 + a)^(3/2), x)
Timed out. \[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\int \frac {x^4\,\left (C\,x^4+B\,x^2+A\right )}{{\left (a-b\,x^4\right )}^{3/2}} \,d x \] Input:
int((x^4*(A + B*x^2 + C*x^4))/(a - b*x^4)^(3/2),x)
Output:
int((x^4*(A + B*x^2 + C*x^4))/(a - b*x^4)^(3/2), x)
\[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\frac {3 \sqrt {-b \,x^{4}+a}\, a b x +5 \sqrt {-b \,x^{4}+a}\, a c x -3 \sqrt {-b \,x^{4}+a}\, b^{2} x^{3}-\sqrt {-b \,x^{4}+a}\, b c \,x^{5}-3 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{b^{2} x^{8}-2 a b \,x^{4}+a^{2}}d x \right ) a^{3} b -5 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{b^{2} x^{8}-2 a b \,x^{4}+a^{2}}d x \right ) a^{3} c +3 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{b^{2} x^{8}-2 a b \,x^{4}+a^{2}}d x \right ) a^{2} b^{2} x^{4}+5 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{b^{2} x^{8}-2 a b \,x^{4}+a^{2}}d x \right ) a^{2} b c \,x^{4}+9 \left (\int \frac {\sqrt {-b \,x^{4}+a}\, x^{2}}{b^{2} x^{8}-2 a b \,x^{4}+a^{2}}d x \right ) a^{2} b^{2}-9 \left (\int \frac {\sqrt {-b \,x^{4}+a}\, x^{2}}{b^{2} x^{8}-2 a b \,x^{4}+a^{2}}d x \right ) a \,b^{3} x^{4}}{3 b^{2} \left (-b \,x^{4}+a \right )} \] Input:
int(x^4*(C*x^4+B*x^2+A)/(-b*x^4+a)^(3/2),x)
Output:
(3*sqrt(a - b*x**4)*a*b*x + 5*sqrt(a - b*x**4)*a*c*x - 3*sqrt(a - b*x**4)* b**2*x**3 - sqrt(a - b*x**4)*b*c*x**5 - 3*int(sqrt(a - b*x**4)/(a**2 - 2*a *b*x**4 + b**2*x**8),x)*a**3*b - 5*int(sqrt(a - b*x**4)/(a**2 - 2*a*b*x**4 + b**2*x**8),x)*a**3*c + 3*int(sqrt(a - b*x**4)/(a**2 - 2*a*b*x**4 + b**2 *x**8),x)*a**2*b**2*x**4 + 5*int(sqrt(a - b*x**4)/(a**2 - 2*a*b*x**4 + b** 2*x**8),x)*a**2*b*c*x**4 + 9*int((sqrt(a - b*x**4)*x**2)/(a**2 - 2*a*b*x** 4 + b**2*x**8),x)*a**2*b**2 - 9*int((sqrt(a - b*x**4)*x**2)/(a**2 - 2*a*b* x**4 + b**2*x**8),x)*a*b**3*x**4)/(3*b**2*(a - b*x**4))