Integrand size = 28, antiderivative size = 221 \[ \int \frac {A+B x^2+C x^4}{x^6 \sqrt {a-b x^4}} \, dx=-\frac {A \sqrt {a-b x^4}}{5 a x^5}-\frac {B \sqrt {a-b x^4}}{3 a x^3}-\frac {(3 A b+5 a C) \sqrt {a-b x^4}}{5 a^2 x}-\frac {\sqrt [4]{b} (3 A b+5 a C) \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 a^{5/4} \sqrt {a-b x^4}}+\frac {\sqrt [4]{b} \left (9 A b+5 \sqrt {a} \sqrt {b} B+15 a C\right ) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{15 a^{5/4} \sqrt {a-b x^4}} \] Output:
-1/5*A*(-b*x^4+a)^(1/2)/a/x^5-1/3*B*(-b*x^4+a)^(1/2)/a/x^3-1/5*(3*A*b+5*C* a)*(-b*x^4+a)^(1/2)/a^2/x-1/5*b^(1/4)*(3*A*b+5*C*a)*(1-b*x^4/a)^(1/2)*Elli pticE(b^(1/4)*x/a^(1/4),I)/a^(5/4)/(-b*x^4+a)^(1/2)+1/15*b^(1/4)*(9*A*b+5* a^(1/2)*b^(1/2)*B+15*C*a)*(1-b*x^4/a)^(1/2)*EllipticF(b^(1/4)*x/a^(1/4),I) /a^(5/4)/(-b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.06 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.47 \[ \int \frac {A+B x^2+C x^4}{x^6 \sqrt {a-b x^4}} \, dx=-\frac {\sqrt {1-\frac {b x^4}{a}} \left (3 A \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {1}{2},-\frac {1}{4},\frac {b x^4}{a}\right )+5 B x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {1}{4},\frac {b x^4}{a}\right )+15 C x^4 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},\frac {b x^4}{a}\right )\right )}{15 x^5 \sqrt {a-b x^4}} \] Input:
Integrate[(A + B*x^2 + C*x^4)/(x^6*Sqrt[a - b*x^4]),x]
Output:
-1/15*(Sqrt[1 - (b*x^4)/a]*(3*A*Hypergeometric2F1[-5/4, 1/2, -1/4, (b*x^4) /a] + 5*B*x^2*Hypergeometric2F1[-3/4, 1/2, 1/4, (b*x^4)/a] + 15*C*x^4*Hype rgeometric2F1[-1/4, 1/2, 3/4, (b*x^4)/a]))/(x^5*Sqrt[a - b*x^4])
Time = 1.00 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.04, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {2374, 9, 27, 1605, 25, 27, 1605, 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.
| \(\displaystyle \int \frac {A+B x^2+C x^4}{x^6 \sqrt {a-b x^4}} \, dx\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle -\frac {\int -\frac {2 \left ((3 A b+5 a C) x^3+5 a B x\right )}{x^5 \sqrt {a-b x^4}}dx}{10 a}-\frac {A \sqrt {a-b x^4}}{5 a x^5}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle -\frac {\int -\frac {2 \left ((3 A b+5 a C) x^2+5 a B\right )}{x^4 \sqrt {a-b x^4}}dx}{10 a}-\frac {A \sqrt {a-b x^4}}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(3 A b+5 a C) x^2+5 a B}{x^4 \sqrt {a-b x^4}}dx}{5 a}-\frac {A \sqrt {a-b x^4}}{5 a x^5}\) |
| \(\Big \downarrow \) 1605 |
| \(\displaystyle \frac {-\frac {\int -\frac {a \left (5 b B x^2+3 (3 A b+5 a C)\right )}{x^2 \sqrt {a-b x^4}}dx}{3 a}-\frac {5 B \sqrt {a-b x^4}}{3 x^3}}{5 a}-\frac {A \sqrt {a-b x^4}}{5 a x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {a \left (5 b B x^2+3 (3 A b+5 a C)\right )}{x^2 \sqrt {a-b x^4}}dx}{3 a}-\frac {5 B \sqrt {a-b x^4}}{3 x^3}}{5 a}-\frac {A \sqrt {a-b x^4}}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \int \frac {5 b B x^2+3 (3 A b+5 a C)}{x^2 \sqrt {a-b x^4}}dx-\frac {5 B \sqrt {a-b x^4}}{3 x^3}}{5 a}-\frac {A \sqrt {a-b x^4}}{5 a x^5}\) |
| \(\Big \downarrow \) 1605 |
| \(\displaystyle \frac {\frac {1}{3} \left (-\frac {\int -\frac {b \left (5 a B-3 (3 A b+5 a C) x^2\right )}{\sqrt {a-b x^4}}dx}{a}-\frac {3 \sqrt {a-b x^4} (5 a C+3 A b)}{a x}\right )-\frac {5 B \sqrt {a-b x^4}}{3 x^3}}{5 a}-\frac {A \sqrt {a-b x^4}}{5 a x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {\int \frac {b \left (5 a B-3 (3 A b+5 a C) x^2\right )}{\sqrt {a-b x^4}}dx}{a}-\frac {3 \sqrt {a-b x^4} (5 a C+3 A b)}{a x}\right )-\frac {5 B \sqrt {a-b x^4}}{3 x^3}}{5 a}-\frac {A \sqrt {a-b x^4}}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {b \int \frac {5 a B-3 (3 A b+5 a C) x^2}{\sqrt {a-b x^4}}dx}{a}-\frac {3 \sqrt {a-b x^4} (5 a C+3 A b)}{a x}\right )-\frac {5 B \sqrt {a-b x^4}}{3 x^3}}{5 a}-\frac {A \sqrt {a-b x^4}}{5 a x^5}\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {b \left (\frac {\sqrt {a} \left (5 \sqrt {a} \sqrt {b} B+15 a C+9 A b\right ) \int \frac {1}{\sqrt {a-b x^4}}dx}{\sqrt {b}}-\frac {3 \sqrt {a} (5 a C+3 A b) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-b x^4}}dx}{\sqrt {b}}\right )}{a}-\frac {3 \sqrt {a-b x^4} (5 a C+3 A b)}{a x}\right )-\frac {5 B \sqrt {a-b x^4}}{3 x^3}}{5 a}-\frac {A \sqrt {a-b x^4}}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {b \left (\frac {\sqrt {a} \left (5 \sqrt {a} \sqrt {b} B+15 a C+9 A b\right ) \int \frac {1}{\sqrt {a-b x^4}}dx}{\sqrt {b}}-\frac {3 (5 a C+3 A b) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx}{\sqrt {b}}\right )}{a}-\frac {3 \sqrt {a-b x^4} (5 a C+3 A b)}{a x}\right )-\frac {5 B \sqrt {a-b x^4}}{3 x^3}}{5 a}-\frac {A \sqrt {a-b x^4}}{5 a x^5}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {b \left (\frac {\sqrt {a} \sqrt {1-\frac {b x^4}{a}} \left (5 \sqrt {a} \sqrt {b} B+15 a C+9 A b\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {b} \sqrt {a-b x^4}}-\frac {3 (5 a C+3 A b) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx}{\sqrt {b}}\right )}{a}-\frac {3 \sqrt {a-b x^4} (5 a C+3 A b)}{a x}\right )-\frac {5 B \sqrt {a-b x^4}}{3 x^3}}{5 a}-\frac {A \sqrt {a-b x^4}}{5 a x^5}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {b \left (\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (5 \sqrt {a} \sqrt {b} B+15 a C+9 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^4}}-\frac {3 (5 a C+3 A b) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx}{\sqrt {b}}\right )}{a}-\frac {3 \sqrt {a-b x^4} (5 a C+3 A b)}{a x}\right )-\frac {5 B \sqrt {a-b x^4}}{3 x^3}}{5 a}-\frac {A \sqrt {a-b x^4}}{5 a x^5}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {b \left (\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (5 \sqrt {a} \sqrt {b} B+15 a C+9 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^4}}-\frac {3 \sqrt {1-\frac {b x^4}{a}} (5 a C+3 A b) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {b} \sqrt {a-b x^4}}\right )}{a}-\frac {3 \sqrt {a-b x^4} (5 a C+3 A b)}{a x}\right )-\frac {5 B \sqrt {a-b x^4}}{3 x^3}}{5 a}-\frac {A \sqrt {a-b x^4}}{5 a x^5}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {b \left (\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (5 \sqrt {a} \sqrt {b} B+15 a C+9 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^4}}-\frac {3 \sqrt {a} \sqrt {1-\frac {b x^4}{a}} (5 a C+3 A b) \int \frac {\sqrt {\frac {\sqrt {b} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}}dx}{\sqrt {b} \sqrt {a-b x^4}}\right )}{a}-\frac {3 \sqrt {a-b x^4} (5 a C+3 A b)}{a x}\right )-\frac {5 B \sqrt {a-b x^4}}{3 x^3}}{5 a}-\frac {A \sqrt {a-b x^4}}{5 a x^5}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {b \left (\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (5 \sqrt {a} \sqrt {b} B+15 a C+9 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^4}}-\frac {3 a^{3/4} \sqrt {1-\frac {b x^4}{a}} (5 a C+3 A b) E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{b^{3/4} \sqrt {a-b x^4}}\right )}{a}-\frac {3 \sqrt {a-b x^4} (5 a C+3 A b)}{a x}\right )-\frac {5 B \sqrt {a-b x^4}}{3 x^3}}{5 a}-\frac {A \sqrt {a-b x^4}}{5 a x^5}\) |
Input:
Int[(A + B*x^2 + C*x^4)/(x^6*Sqrt[a - b*x^4]),x]
Output:
-1/5*(A*Sqrt[a - b*x^4])/(a*x^5) + ((-5*B*Sqrt[a - b*x^4])/(3*x^3) + ((-3* (3*A*b + 5*a*C)*Sqrt[a - b*x^4])/(a*x) + (b*((-3*a^(3/4)*(3*A*b + 5*a*C)*S qrt[1 - (b*x^4)/a]*EllipticE[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(b^(3/4)*Sq rt[a - b*x^4]) + (a^(3/4)*(9*A*b + 5*Sqrt[a]*Sqrt[b]*B + 15*a*C)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(b^(3/4)*Sqrt[a - b *x^4])))/a)/3)/(5*a)
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
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[d*(f*x)^(m + 1)*((a + c*x^4)^(p + 1)/(a*f*(m + 1))), x] + S imp[1/(a*f^2*(m + 1)) Int[(f*x)^(m + 2)*(a + c*x^4)^p*(a*e*(m + 1) - c*d* (m + 4*p + 5)*x^2), x], x] /; FreeQ[{a, c, d, e, f, p}, x] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Wit h[{Pq0 = Coeff[Pq, x, 0]}, Simp[Pq0*(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c *(m + 1))), x] + Simp[1/(2*a*c*(m + 1)) Int[(c*x)^(m + 1)*ExpandToSum[2*a *(m + 1)*((Pq - Pq0)/x) - 2*b*Pq0*(m + n*(p + 1) + 1)*x^(n - 1), x]*(a + b* x^n)^p, x], x] /; NeQ[Pq0, 0]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[m, -1] && LeQ[n - 1, Expon[Pq, x]]
Time = 2.04 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(-\frac {\sqrt {-b \,x^{4}+a}\, \left (9 A b \,x^{4}+15 C a \,x^{4}+5 B a \,x^{2}+3 A a \right )}{15 a^{2} x^{5}}-\frac {b \left (-\frac {\left (9 A b +15 C a \right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, \sqrt {b}}-\frac {5 B a \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )}{15 a^{2}}\) | \(215\) |
| elliptic | \(-\frac {A \sqrt {-b \,x^{4}+a}}{5 a \,x^{5}}-\frac {B \sqrt {-b \,x^{4}+a}}{3 a \,x^{3}}-\frac {\left (3 A b +5 C a \right ) \sqrt {-b \,x^{4}+a}}{5 a^{2} x}+\frac {b B \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{3 a \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}+\frac {\left (3 A b +5 C a \right ) \sqrt {b}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 a^{\frac {3}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\) | \(232\) |
| default | \(A \left (-\frac {\sqrt {-b \,x^{4}+a}}{5 a \,x^{5}}-\frac {3 b \sqrt {-b \,x^{4}+a}}{5 a^{2} x}+\frac {3 b^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 a^{\frac {3}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )+B \left (-\frac {\sqrt {-b \,x^{4}+a}}{3 a \,x^{3}}+\frac {b \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{3 a \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )+C \left (-\frac {\sqrt {-b \,x^{4}+a}}{a x}+\frac {\sqrt {b}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {a}\, \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )\) | \(325\) |
Input:
int((C*x^4+B*x^2+A)/x^6/(-b*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/15*(-b*x^4+a)^(1/2)*(9*A*b*x^4+15*C*a*x^4+5*B*a*x^2+3*A*a)/a^2/x^5-1/15 *b/a^2*(-(9*A*b+15*C*a)*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)/b^(1/2)*(Elli pticF(x*(1/a^(1/2)*b^(1/2))^(1/2),I)-EllipticE(x*(1/a^(1/2)*b^(1/2))^(1/2) ,I))-5*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))
Time = 0.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.55 \[ \int \frac {A+B x^2+C x^4}{x^6 \sqrt {a-b x^4}} \, dx=-\frac {3 \, {\left (5 \, C a + 3 \, A b\right )} \sqrt {a} x^{5} \left (\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - {\left (5 \, {\left (B + 3 \, C\right )} a + 9 \, A b\right )} \sqrt {a} x^{5} \left (\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (3 \, {\left (5 \, C a + 3 \, A b\right )} x^{4} + 5 \, B a x^{2} + 3 \, A a\right )} \sqrt {-b x^{4} + a}}{15 \, a^{2} x^{5}} \] Input:
integrate((C*x^4+B*x^2+A)/x^6/(-b*x^4+a)^(1/2),x, algorithm="fricas")
Output:
-1/15*(3*(5*C*a + 3*A*b)*sqrt(a)*x^5*(b/a)^(3/4)*elliptic_e(arcsin(x*(b/a) ^(1/4)), -1) - (5*(B + 3*C)*a + 9*A*b)*sqrt(a)*x^5*(b/a)^(3/4)*elliptic_f( arcsin(x*(b/a)^(1/4)), -1) + (3*(5*C*a + 3*A*b)*x^4 + 5*B*a*x^2 + 3*A*a)*s qrt(-b*x^4 + a))/(a^2*x^5)
Time = 1.76 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.58 \[ \int \frac {A+B x^2+C x^4}{x^6 \sqrt {a-b x^4}} \, dx=- \frac {i A \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {a}{b x^{4}}} \right )}}{4 \sqrt {b} x^{7} \Gamma \left (- \frac {3}{4}\right )} + \frac {B \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} x^{3} \Gamma \left (\frac {1}{4}\right )} + \frac {C \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} x \Gamma \left (\frac {3}{4}\right )} \] Input:
integrate((C*x**4+B*x**2+A)/x**6/(-b*x**4+a)**(1/2),x)
Output:
-I*A*gamma(-7/4)*hyper((1/2, 7/4), (11/4,), a/(b*x**4))/(4*sqrt(b)*x**7*ga mma(-3/4)) + B*gamma(-3/4)*hyper((-3/4, 1/2), (1/4,), b*x**4*exp_polar(2*I *pi)/a)/(4*sqrt(a)*x**3*gamma(1/4)) + C*gamma(-1/4)*hyper((-1/4, 1/2), (3/ 4,), b*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(a)*x*gamma(3/4))
\[ \int \frac {A+B x^2+C x^4}{x^6 \sqrt {a-b x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {-b x^{4} + a} x^{6}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/x^6/(-b*x^4+a)^(1/2),x, algorithm="maxima")
Output:
integrate((C*x^4 + B*x^2 + A)/(sqrt(-b*x^4 + a)*x^6), x)
\[ \int \frac {A+B x^2+C x^4}{x^6 \sqrt {a-b x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {-b x^{4} + a} x^{6}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/x^6/(-b*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate((C*x^4 + B*x^2 + A)/(sqrt(-b*x^4 + a)*x^6), x)
Timed out. \[ \int \frac {A+B x^2+C x^4}{x^6 \sqrt {a-b x^4}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{x^6\,\sqrt {a-b\,x^4}} \,d x \] Input:
int((A + B*x^2 + C*x^4)/(x^6*(a - b*x^4)^(1/2)),x)
Output:
int((A + B*x^2 + C*x^4)/(x^6*(a - b*x^4)^(1/2)), x)
\[ \int \frac {A+B x^2+C x^4}{x^6 \sqrt {a-b x^4}} \, dx=\left (\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,x^{10}+a \,x^{6}}d x \right ) a +\left (\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,x^{8}+a \,x^{4}}d x \right ) b +\left (\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,x^{6}+a \,x^{2}}d x \right ) c \] Input:
int((C*x^4+B*x^2+A)/x^6/(-b*x^4+a)^(1/2),x)
Output:
int(sqrt(a - b*x**4)/(a*x**6 - b*x**10),x)*a + int(sqrt(a - b*x**4)/(a*x** 4 - b*x**8),x)*b + int(sqrt(a - b*x**4)/(a*x**2 - b*x**6),x)*c