Integrand size = 28, antiderivative size = 195 \[ \int \frac {\sqrt {a-b x^4} \left (A+B x^2+C x^4\right )}{x^4} \, dx=-\frac {\left (3 a B+(A b-a C) x^2\right ) \sqrt {a-b x^4}}{3 a x}-\frac {A \left (a-b x^4\right )^{3/2}}{3 a x^3}-\frac {2 a^{3/4} \sqrt [4]{b} B \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt {a-b x^4}}-\frac {2 \sqrt [4]{a} \left (A b-3 \sqrt {a} \sqrt {b} B-a C\right ) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{3 \sqrt [4]{b} \sqrt {a-b x^4}} \] Output:
-1/3*(3*a*B+(A*b-C*a)*x^2)*(-b*x^4+a)^(1/2)/a/x-1/3*A*(-b*x^4+a)^(3/2)/a/x ^3-2*a^(3/4)*b^(1/4)*B*(1-b*x^4/a)^(1/2)*EllipticE(b^(1/4)*x/a^(1/4),I)/(- b*x^4+a)^(1/2)-2/3*a^(1/4)*(A*b-3*a^(1/2)*b^(1/2)*B-C*a)*(1-b*x^4/a)^(1/2) *EllipticF(b^(1/4)*x/a^(1/4),I)/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 = 7.23 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {a-b x^4} \left (A+B x^2+C x^4\right )}{x^4} \, dx=\frac {\sqrt {a-b x^4} \left (-A \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},-\frac {1}{2},\frac {1}{4},\frac {b x^4}{a}\right )-3 B x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{4},\frac {3}{4},\frac {b x^4}{a}\right )+3 C x^4 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {b x^4}{a}\right )\right )}{3 x^3 \sqrt {1-\frac {b x^4}{a}}} \] Input:
Integrate[(Sqrt[a - b*x^4]*(A + B*x^2 + C*x^4))/x^4,x]
Output:
(Sqrt[a - b*x^4]*(-(A*Hypergeometric2F1[-3/4, -1/2, 1/4, (b*x^4)/a]) - 3*B *x^2*Hypergeometric2F1[-1/2, -1/4, 3/4, (b*x^4)/a] + 3*C*x^4*Hypergeometri c2F1[-1/2, 1/4, 5/4, (b*x^4)/a]))/(3*x^3*Sqrt[1 - (b*x^4)/a])
Time = 0.83 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2374, 9, 27, 1595, 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.
| \(\displaystyle \int \frac {\sqrt {a-b x^4} \left (A+B x^2+C x^4\right )}{x^4} \, dx\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle -\frac {\int -\frac {6 \left (a B x-(A b-a C) x^3\right ) \sqrt {a-b x^4}}{x^3}dx}{6 a}-\frac {A \left (a-b x^4\right )^{3/2}}{3 a x^3}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle -\frac {\int -\frac {6 \left (a B-(A b-a C) x^2\right ) \sqrt {a-b x^4}}{x^2}dx}{6 a}-\frac {A \left (a-b x^4\right )^{3/2}}{3 a x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (a B-(A b-a C) x^2\right ) \sqrt {a-b x^4}}{x^2}dx}{a}-\frac {A \left (a-b x^4\right )^{3/2}}{3 a x^3}\) |
| \(\Big \downarrow \) 1595 |
| \(\displaystyle \frac {-\frac {2}{3} \int \frac {a \left (3 b B x^2+A b-a C\right )}{\sqrt {a-b x^4}}dx-\frac {\sqrt {a-b x^4} \left (x^2 (A b-a C)+3 a B\right )}{3 x}}{a}-\frac {A \left (a-b x^4\right )^{3/2}}{3 a x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {2}{3} a \int \frac {3 b B x^2+A b-a C}{\sqrt {a-b x^4}}dx-\frac {\sqrt {a-b x^4} \left (x^2 (A b-a C)+3 a B\right )}{3 x}}{a}-\frac {A \left (a-b x^4\right )^{3/2}}{3 a x^3}\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \frac {-\frac {2}{3} a \left (\left (-3 \sqrt {a} \sqrt {b} B-a C+A b\right ) \int \frac {1}{\sqrt {a-b x^4}}dx+3 \sqrt {a} \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-b x^4}}dx\right )-\frac {\sqrt {a-b x^4} \left (x^2 (A b-a C)+3 a B\right )}{3 x}}{a}-\frac {A \left (a-b x^4\right )^{3/2}}{3 a x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {2}{3} a \left (\left (-3 \sqrt {a} \sqrt {b} B-a C+A b\right ) \int \frac {1}{\sqrt {a-b x^4}}dx+3 \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx\right )-\frac {\sqrt {a-b x^4} \left (x^2 (A b-a C)+3 a B\right )}{3 x}}{a}-\frac {A \left (a-b x^4\right )^{3/2}}{3 a x^3}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {-\frac {2}{3} a \left (\frac {\sqrt {1-\frac {b x^4}{a}} \left (-3 \sqrt {a} \sqrt {b} B-a C+A b\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {a-b x^4}}+3 \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx\right )-\frac {\sqrt {a-b x^4} \left (x^2 (A b-a C)+3 a B\right )}{3 x}}{a}-\frac {A \left (a-b x^4\right )^{3/2}}{3 a x^3}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {-\frac {2}{3} a \left (3 \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (-3 \sqrt {a} \sqrt {b} B-a C+A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )-\frac {\sqrt {a-b x^4} \left (x^2 (A b-a C)+3 a B\right )}{3 x}}{a}-\frac {A \left (a-b x^4\right )^{3/2}}{3 a x^3}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {-\frac {2}{3} a \left (\frac {3 \sqrt {b} B \sqrt {1-\frac {b x^4}{a}} \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (-3 \sqrt {a} \sqrt {b} B-a C+A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )-\frac {\sqrt {a-b x^4} \left (x^2 (A b-a C)+3 a B\right )}{3 x}}{a}-\frac {A \left (a-b x^4\right )^{3/2}}{3 a x^3}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {-\frac {2}{3} a \left (\frac {3 \sqrt {a} \sqrt {b} B \sqrt {1-\frac {b x^4}{a}} \int \frac {\sqrt {\frac {\sqrt {b} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}}dx}{\sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (-3 \sqrt {a} \sqrt {b} B-a C+A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )-\frac {\sqrt {a-b x^4} \left (x^2 (A b-a C)+3 a B\right )}{3 x}}{a}-\frac {A \left (a-b x^4\right )^{3/2}}{3 a x^3}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {-\frac {2}{3} a \left (\frac {3 a^{3/4} \sqrt [4]{b} B \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (-3 \sqrt {a} \sqrt {b} B-a C+A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )-\frac {\sqrt {a-b x^4} \left (x^2 (A b-a C)+3 a B\right )}{3 x}}{a}-\frac {A \left (a-b x^4\right )^{3/2}}{3 a x^3}\) |
Input:
Int[(Sqrt[a - b*x^4]*(A + B*x^2 + C*x^4))/x^4,x]
Output:
-1/3*(A*(a - b*x^4)^(3/2))/(a*x^3) + (-1/3*((3*a*B + (A*b - a*C)*x^2)*Sqrt [a - b*x^4])/x - (2*a*((3*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)*(A*b - 3*Sqrt [a]*Sqrt[b]*B - 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)/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[(f*x)^(m + 1)*(a + c*x^4)^p*((d*(m + 4*p + 3) + e*(m + 1)* x^2)/(f*(m + 1)*(m + 4*p + 3))), x] + Simp[4*(p/(f^2*(m + 1)*(m + 4*p + 3)) ) Int[(f*x)^(m + 2)*(a + c*x^4)^(p - 1)*(a*e*(m + 1) - c*d*(m + 4*p + 3)* x^2), x], x] /; FreeQ[{a, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1] && m + 4*p + 3 != 0 && 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 = 1.75 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.07
| method | result | size |
| elliptic | \(-\frac {A \sqrt {-b \,x^{4}+a}}{3 x^{3}}-\frac {B \sqrt {-b \,x^{4}+a}}{x}+\frac {C x \sqrt {-b \,x^{4}+a}}{3}+\frac {\left (-\frac {2 A b}{3}+\frac {2 C a}{3}\right ) \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}+\frac {2 B \sqrt {b}\, \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\) | \(208\) |
| risch | \(-\frac {\sqrt {-b \,x^{4}+a}\, \left (-C \,x^{4}+3 B \,x^{2}+A \right )}{3 x^{3}}-\frac {2 A b \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}+\frac {2 C a \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}+\frac {2 B \sqrt {b}\, \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\) | \(251\) |
| default | \(C \left (\frac {x \sqrt {-b \,x^{4}+a}}{3}+\frac {2 a \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )+A \left (-\frac {\sqrt {-b \,x^{4}+a}}{3 x^{3}}-\frac {2 b \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )+B \left (-\frac {\sqrt {-b \,x^{4}+a}}{x}+\frac {2 \sqrt {b}\, \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )\) | \(271\) |
Input:
int((-b*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/x^4,x,method=_RETURNVERBOSE)
Output:
-1/3*A*(-b*x^4+a)^(1/2)/x^3-B*(-b*x^4+a)^(1/2)/x+1/3*C*x*(-b*x^4+a)^(1/2)+ (-2/3*A*b+2/3*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)+2*B*b^(1/2)*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))
\[ \int \frac {\sqrt {a-b x^4} \left (A+B x^2+C x^4\right )}{x^4} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} \sqrt {-b x^{4} + a}}{x^{4}} \,d x } \] Input:
integrate((-b*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/x^4,x, algorithm="fricas")
Output:
integral((C*x^4 + B*x^2 + A)*sqrt(-b*x^4 + a)/x^4, x)
Time = 1.80 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {a-b x^4} \left (A+B x^2+C x^4\right )}{x^4} \, dx=\frac {i A \sqrt {b} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {a}{b x^{4}}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} + \frac {i B \sqrt {b} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {a}{b x^{4}}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {C \sqrt {a} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \] Input:
integrate((-b*x**4+a)**(1/2)*(C*x**4+B*x**2+A)/x**4,x)
Output:
I*A*sqrt(b)*gamma(-1/4)*hyper((-1/2, 1/4), (5/4,), a/(b*x**4))/(4*x*gamma( 3/4)) + I*B*sqrt(b)*x*gamma(1/4)*hyper((-1/2, -1/4), (3/4,), a/(b*x**4))/( 4*gamma(5/4)) + C*sqrt(a)*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), b*x**4*e xp_polar(2*I*pi)/a)/(4*gamma(5/4))
\[ \int \frac {\sqrt {a-b x^4} \left (A+B x^2+C x^4\right )}{x^4} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} \sqrt {-b x^{4} + a}}{x^{4}} \,d x } \] Input:
integrate((-b*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/x^4,x, algorithm="maxima")
Output:
integrate((C*x^4 + B*x^2 + A)*sqrt(-b*x^4 + a)/x^4, x)
\[ \int \frac {\sqrt {a-b x^4} \left (A+B x^2+C x^4\right )}{x^4} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} \sqrt {-b x^{4} + a}}{x^{4}} \,d x } \] Input:
integrate((-b*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/x^4,x, algorithm="giac")
Output:
integrate((C*x^4 + B*x^2 + A)*sqrt(-b*x^4 + a)/x^4, x)
Timed out. \[ \int \frac {\sqrt {a-b x^4} \left (A+B x^2+C x^4\right )}{x^4} \, dx=\int \frac {\sqrt {a-b\,x^4}\,\left (C\,x^4+B\,x^2+A\right )}{x^4} \,d x \] Input:
int(((a - b*x^4)^(1/2)*(A + B*x^2 + C*x^4))/x^4,x)
Output:
int(((a - b*x^4)^(1/2)*(A + B*x^2 + C*x^4))/x^4, x)
\[ \int \frac {\sqrt {a-b x^4} \left (A+B x^2+C x^4\right )}{x^4} \, dx=\frac {-3 \sqrt {-b \,x^{4}+a}\, a b +2 \sqrt {-b \,x^{4}+a}\, a c +3 \sqrt {-b \,x^{4}+a}\, b^{2} x^{2}+\sqrt {-b \,x^{4}+a}\, b c \,x^{4}-6 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,x^{8}+a \,x^{4}}d x \right ) a^{2} b \,x^{3}+6 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,x^{8}+a \,x^{4}}d x \right ) a^{2} c \,x^{3}+6 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,x^{6}+a \,x^{2}}d x \right ) a \,b^{2} x^{3}}{3 b \,x^{3}} \] Input:
int((-b*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/x^4,x)
Output:
( - 3*sqrt(a - b*x**4)*a*b + 2*sqrt(a - b*x**4)*a*c + 3*sqrt(a - b*x**4)*b **2*x**2 + sqrt(a - b*x**4)*b*c*x**4 - 6*int(sqrt(a - b*x**4)/(a*x**4 - b* x**8),x)*a**2*b*x**3 + 6*int(sqrt(a - b*x**4)/(a*x**4 - b*x**8),x)*a**2*c* x**3 + 6*int(sqrt(a - b*x**4)/(a*x**2 - b*x**6),x)*a*b**2*x**3)/(3*b*x**3)