Integrand size = 28, antiderivative size = 217 \[ \int \frac {A+B x^2+C x^4}{x^4 \left (a-b x^4\right )^{3/2}} \, dx=\frac {x \left (a \left (\frac {A b}{a}+C\right )+b B x^2\right )}{2 a^2 \sqrt {a-b x^4}}-\frac {A \sqrt {a-b x^4}}{3 a^2 x^3}-\frac {B \sqrt {a-b x^4}}{a^2 x}-\frac {3 \sqrt [4]{b} B \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 a^{5/4} \sqrt {a-b x^4}}+\frac {\left (5 A b+9 \sqrt {a} \sqrt {b} B+3 a C\right ) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{6 a^{7/4} \sqrt [4]{b} \sqrt {a-b x^4}} \] Output:
1/2*x*(a*(A*b/a+C)+B*b*x^2)/a^2/(-b*x^4+a)^(1/2)-1/3*A*(-b*x^4+a)^(1/2)/a^ 2/x^3-B*(-b*x^4+a)^(1/2)/a^2/x-3/2*b^(1/4)*B*(1-b*x^4/a)^(1/2)*EllipticE(b ^(1/4)*x/a^(1/4),I)/a^(5/4)/(-b*x^4+a)^(1/2)+1/6*(5*A*b+9*a^(1/2)*b^(1/2)* B+3*C*a)*(1-b*x^4/a)^(1/2)*EllipticF(b^(1/4)*x/a^(1/4),I)/a^(7/4)/b^(1/4)/ (-b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.09 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.65 \[ \int \frac {A+B x^2+C x^4}{x^4 \left (a-b x^4\right )^{3/2}} \, dx=\frac {-2 A \sqrt {1-\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {3}{2},\frac {1}{4},\frac {b x^4}{a}\right )-6 B x^2 \sqrt {1-\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {3}{2},\frac {3}{4},\frac {b x^4}{a}\right )+3 C x^4 \left (1+\sqrt {1-\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {b x^4}{a}\right )\right )}{6 a x^3 \sqrt {a-b x^4}} \] Input:
Integrate[(A + B*x^2 + C*x^4)/(x^4*(a - b*x^4)^(3/2)),x]
Output:
(-2*A*Sqrt[1 - (b*x^4)/a]*Hypergeometric2F1[-3/4, 3/2, 1/4, (b*x^4)/a] - 6 *B*x^2*Sqrt[1 - (b*x^4)/a]*Hypergeometric2F1[-1/4, 3/2, 3/4, (b*x^4)/a] + 3*C*x^4*(1 + Sqrt[1 - (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, (b*x^4)/ a]))/(6*a*x^3*Sqrt[a - b*x^4])
Time = 1.32 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.06, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2368, 2374, 9, 27, 2374, 9, 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.
| \(\displaystyle \int \frac {A+B x^2+C x^4}{x^4 \left (a-b x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle \frac {\int \frac {-\frac {b^2 B x^6}{a}+b \left (\frac {A b}{a}+C\right ) x^4+2 b B x^2+2 A b}{x^4 \sqrt {a-b x^4}}dx}{2 a b}+\frac {x \left (a C+A b+b B x^2\right )}{2 a^2 \sqrt {a-b x^4}}\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle \frac {-\frac {\int -\frac {2 \left (-3 b^2 B x^5+b (5 A b+3 a C) x^3+6 a b B x\right )}{x^3 \sqrt {a-b x^4}}dx}{6 a}-\frac {2 A b \sqrt {a-b x^4}}{3 a x^3}}{2 a b}+\frac {x \left (a C+A b+b B x^2\right )}{2 a^2 \sqrt {a-b x^4}}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {-\frac {\int -\frac {2 \left (-3 b^2 B x^4+b (5 A b+3 a C) x^2+6 a b B\right )}{x^2 \sqrt {a-b x^4}}dx}{6 a}-\frac {2 A b \sqrt {a-b x^4}}{3 a x^3}}{2 a b}+\frac {x \left (a C+A b+b B x^2\right )}{2 a^2 \sqrt {a-b x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {-3 b^2 B x^4+b (5 A b+3 a C) x^2+6 a b B}{x^2 \sqrt {a-b x^4}}dx}{3 a}-\frac {2 A b \sqrt {a-b x^4}}{3 a x^3}}{2 a b}+\frac {x \left (a C+A b+b B x^2\right )}{2 a^2 \sqrt {a-b x^4}}\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {2 \left (a b (5 A b+3 a C) x-9 a b^2 B x^3\right )}{x \sqrt {a-b x^4}}dx}{2 a}-\frac {6 b B \sqrt {a-b x^4}}{x}}{3 a}-\frac {2 A b \sqrt {a-b x^4}}{3 a x^3}}{2 a b}+\frac {x \left (a C+A b+b B x^2\right )}{2 a^2 \sqrt {a-b x^4}}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {2 a b \left (-9 b B x^2+5 A b+3 a C\right )}{\sqrt {a-b x^4}}dx}{2 a}-\frac {6 b B \sqrt {a-b x^4}}{x}}{3 a}-\frac {2 A b \sqrt {a-b x^4}}{3 a x^3}}{2 a b}+\frac {x \left (a C+A b+b B x^2\right )}{2 a^2 \sqrt {a-b x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {b \int \frac {-9 b B x^2+5 A b+3 a C}{\sqrt {a-b x^4}}dx-\frac {6 b B \sqrt {a-b x^4}}{x}}{3 a}-\frac {2 A b \sqrt {a-b x^4}}{3 a x^3}}{2 a b}+\frac {x \left (a C+A b+b B x^2\right )}{2 a^2 \sqrt {a-b x^4}}\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \frac {\frac {b \left (\left (9 \sqrt {a} \sqrt {b} B+3 a C+5 A b\right ) \int \frac {1}{\sqrt {a-b x^4}}dx-9 \sqrt {a} \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-b x^4}}dx\right )-\frac {6 b B \sqrt {a-b x^4}}{x}}{3 a}-\frac {2 A b \sqrt {a-b x^4}}{3 a x^3}}{2 a b}+\frac {x \left (a C+A b+b B x^2\right )}{2 a^2 \sqrt {a-b x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {b \left (\left (9 \sqrt {a} \sqrt {b} B+3 a C+5 A b\right ) \int \frac {1}{\sqrt {a-b x^4}}dx-9 \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx\right )-\frac {6 b B \sqrt {a-b x^4}}{x}}{3 a}-\frac {2 A b \sqrt {a-b x^4}}{3 a x^3}}{2 a b}+\frac {x \left (a C+A b+b B x^2\right )}{2 a^2 \sqrt {a-b x^4}}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\frac {b \left (\frac {\sqrt {1-\frac {b x^4}{a}} \left (9 \sqrt {a} \sqrt {b} B+3 a C+5 A b\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {a-b x^4}}-9 \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx\right )-\frac {6 b B \sqrt {a-b x^4}}{x}}{3 a}-\frac {2 A b \sqrt {a-b x^4}}{3 a x^3}}{2 a b}+\frac {x \left (a C+A b+b B x^2\right )}{2 a^2 \sqrt {a-b x^4}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\frac {b \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (9 \sqrt {a} \sqrt {b} B+3 a C+5 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}-9 \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx\right )-\frac {6 b B \sqrt {a-b x^4}}{x}}{3 a}-\frac {2 A b \sqrt {a-b x^4}}{3 a x^3}}{2 a b}+\frac {x \left (a C+A b+b B x^2\right )}{2 a^2 \sqrt {a-b x^4}}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {\frac {b \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (9 \sqrt {a} \sqrt {b} B+3 a C+5 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}-\frac {9 \sqrt {b} B \sqrt {1-\frac {b x^4}{a}} \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {a-b x^4}}\right )-\frac {6 b B \sqrt {a-b x^4}}{x}}{3 a}-\frac {2 A b \sqrt {a-b x^4}}{3 a x^3}}{2 a b}+\frac {x \left (a C+A b+b B x^2\right )}{2 a^2 \sqrt {a-b x^4}}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {\frac {b \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (9 \sqrt {a} \sqrt {b} B+3 a C+5 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}-\frac {9 \sqrt {a} \sqrt {b} B \sqrt {1-\frac {b x^4}{a}} \int \frac {\sqrt {\frac {\sqrt {b} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}}dx}{\sqrt {a-b x^4}}\right )-\frac {6 b B \sqrt {a-b x^4}}{x}}{3 a}-\frac {2 A b \sqrt {a-b x^4}}{3 a x^3}}{2 a b}+\frac {x \left (a C+A b+b B x^2\right )}{2 a^2 \sqrt {a-b x^4}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {b \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (9 \sqrt {a} \sqrt {b} B+3 a C+5 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}-\frac {9 a^{3/4} \sqrt [4]{b} B \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt {a-b x^4}}\right )-\frac {6 b B \sqrt {a-b x^4}}{x}}{3 a}-\frac {2 A b \sqrt {a-b x^4}}{3 a x^3}}{2 a b}+\frac {x \left (a C+A b+b B x^2\right )}{2 a^2 \sqrt {a-b x^4}}\) |
Input:
Int[(A + B*x^2 + C*x^4)/(x^4*(a - b*x^4)^(3/2)),x]
Output:
(x*(A*b + a*C + b*B*x^2))/(2*a^2*Sqrt[a - b*x^4]) + ((-2*A*b*Sqrt[a - b*x^ 4])/(3*a*x^3) + ((-6*b*B*Sqrt[a - b*x^4])/x + b*((-9*a^(3/4)*b^(1/4)*B*Sqr t[1 - (b*x^4)/a]*EllipticE[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/Sqrt[a - b*x^ 4] + (a^(1/4)*(5*A*b + 9*Sqrt[a]*Sqrt[b]*B + 3*a*C)*Sqrt[1 - (b*x^4)/a]*El lipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(b^(1/4)*Sqrt[a - b*x^4])))/(3*a ))/(2*a*b)
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[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^ m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1)*x^m *Pq, a + b*x^n, x], i}, Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a^2*n*(p + 1)*b^( Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) Int[x^m*(a + b*x^n)^(p + 1)*ExpandToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)/a)*Coeff[R, x, i]*x^(i - m), {i, 0, n - 1}], x], x], x]]] /; F reeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]
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 = 3.46 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.18
| method | result | size |
| elliptic | \(-\frac {A \sqrt {-b \,x^{4}+a}}{3 a^{2} x^{3}}-\frac {B \sqrt {-b \,x^{4}+a}}{a^{2} x}+\frac {2 b \left (\frac {B \,x^{3}}{4 a^{2}}+\frac {\left (A b +C a \right ) x}{4 b \,a^{2}}\right )}{\sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {\left (\frac {A b}{3 a^{2}}+\frac {A b +C a}{2 a^{2}}\right ) \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}+\frac {3 B \sqrt {b}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 a^{\frac {3}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\) | \(255\) |
| default | \(C \left (\frac {x}{2 a \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {\sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )+A \left (-\frac {\sqrt {-b \,x^{4}+a}}{3 a^{2} x^{3}}+\frac {b x}{2 a^{2} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {5 b \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{6 a^{2} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )+B \left (-\frac {\sqrt {-b \,x^{4}+a}}{a^{2} x}+\frac {b \,x^{3}}{2 a^{2} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {3 \sqrt {b}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 a^{\frac {3}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )\) | \(336\) |
| risch | \(-\frac {\sqrt {-b \,x^{4}+a}\, \left (3 B \,x^{2}+A \right )}{3 a^{2} x^{3}}+\frac {\frac {A b \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}+\frac {3 B \sqrt {b}\, \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-3 a \left (\frac {2 b \left (-\frac {x^{3} B}{4 a}-\frac {\left (A b +C a \right ) x}{4 b a}\right )}{\sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}-\frac {\left (A b +C a \right ) \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\sqrt {b}\, B \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )}{3 a^{2}}\) | \(397\) |
Input:
int((C*x^4+B*x^2+A)/x^4/(-b*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/3*A*(-b*x^4+a)^(1/2)/a^2/x^3-B*(-b*x^4+a)^(1/2)/a^2/x+2*b*(1/4*B/a^2*x^ 3+1/4/b/a^2*(A*b+C*a)*x)/(-(x^4-a/b)*b)^(1/2)+(1/3*A*b/a^2+1/2/a^2*(A*b+C* 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^(1/2)/a^(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 = 0.12 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.85 \[ \int \frac {A+B x^2+C x^4}{x^4 \left (a-b x^4\right )^{3/2}} \, dx=-\frac {9 \, {\left (B b^{2} x^{7} - B a b x^{3}\right )} \sqrt {a} \left (\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - {\left ({\left (3 \, C a b + {\left (5 \, A + 9 \, B\right )} b^{2}\right )} x^{7} - {\left (3 \, C a^{2} + {\left (5 \, A + 9 \, B\right )} a b\right )} x^{3}\right )} \sqrt {a} \left (\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (9 \, B b^{2} x^{6} - 6 \, B a b x^{2} + {\left (3 \, C a b + 5 \, A b^{2}\right )} x^{4} - 2 \, A a b\right )} \sqrt {-b x^{4} + a}}{6 \, {\left (a^{2} b^{2} x^{7} - a^{3} b x^{3}\right )}} \] Input:
integrate((C*x^4+B*x^2+A)/x^4/(-b*x^4+a)^(3/2),x, algorithm="fricas")
Output:
-1/6*(9*(B*b^2*x^7 - B*a*b*x^3)*sqrt(a)*(b/a)^(3/4)*elliptic_e(arcsin(x*(b /a)^(1/4)), -1) - ((3*C*a*b + (5*A + 9*B)*b^2)*x^7 - (3*C*a^2 + (5*A + 9*B )*a*b)*x^3)*sqrt(a)*(b/a)^(3/4)*elliptic_f(arcsin(x*(b/a)^(1/4)), -1) + (9 *B*b^2*x^6 - 6*B*a*b*x^2 + (3*C*a*b + 5*A*b^2)*x^4 - 2*A*a*b)*sqrt(-b*x^4 + a))/(a^2*b^2*x^7 - a^3*b*x^3)
Time = 9.44 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.59 \[ \int \frac {A+B x^2+C x^4}{x^4 \left (a-b x^4\right )^{3/2}} \, dx=\frac {A \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} x^{3} \Gamma \left (\frac {1}{4}\right )} + \frac {B \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} x \Gamma \left (\frac {3}{4}\right )} + \frac {C x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} \] Input:
integrate((C*x**4+B*x**2+A)/x**4/(-b*x**4+a)**(3/2),x)
Output:
A*gamma(-3/4)*hyper((-3/4, 3/2), (1/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*a* *(3/2)*x**3*gamma(1/4)) + B*gamma(-1/4)*hyper((-1/4, 3/2), (3/4,), b*x**4* exp_polar(2*I*pi)/a)/(4*a**(3/2)*x*gamma(3/4)) + C*x*gamma(1/4)*hyper((1/4 , 3/2), (5/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*a**(3/2)*gamma(5/4))
\[ \int \frac {A+B x^2+C x^4}{x^4 \left (a-b x^4\right )^{3/2}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{{\left (-b x^{4} + a\right )}^{\frac {3}{2}} x^{4}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/x^4/(-b*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((C*x^4 + B*x^2 + A)/((-b*x^4 + a)^(3/2)*x^4), x)
\[ \int \frac {A+B x^2+C x^4}{x^4 \left (a-b x^4\right )^{3/2}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{{\left (-b x^{4} + a\right )}^{\frac {3}{2}} x^{4}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/x^4/(-b*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((C*x^4 + B*x^2 + A)/((-b*x^4 + a)^(3/2)*x^4), x)
Timed out. \[ \int \frac {A+B x^2+C x^4}{x^4 \left (a-b x^4\right )^{3/2}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{x^4\,{\left (a-b\,x^4\right )}^{3/2}} \,d x \] Input:
int((A + B*x^2 + C*x^4)/(x^4*(a - b*x^4)^(3/2)),x)
Output:
int((A + B*x^2 + C*x^4)/(x^4*(a - b*x^4)^(3/2)), x)
\[ \int \frac {A+B x^2+C x^4}{x^4 \left (a-b x^4\right )^{3/2}} \, dx=\left (\int \frac {\sqrt {-b \,x^{4}+a}}{b^{2} x^{8}-2 a b \,x^{4}+a^{2}}d x \right ) c +\left (\int \frac {\sqrt {-b \,x^{4}+a}}{b^{2} x^{12}-2 a b \,x^{8}+a^{2} x^{4}}d x \right ) a +\left (\int \frac {\sqrt {-b \,x^{4}+a}}{b^{2} x^{10}-2 a b \,x^{6}+a^{2} x^{2}}d x \right ) b \] Input:
int((C*x^4+B*x^2+A)/x^4/(-b*x^4+a)^(3/2),x)
Output:
int(sqrt(a - b*x**4)/(a**2 - 2*a*b*x**4 + b**2*x**8),x)*c + int(sqrt(a - b *x**4)/(a**2*x**4 - 2*a*b*x**8 + b**2*x**12),x)*a + int(sqrt(a - b*x**4)/( a**2*x**2 - 2*a*b*x**6 + b**2*x**10),x)*b