Integrand size = 28, antiderivative size = 157 \[ \int \frac {A+B x^2+C x^4}{x^2 \sqrt {a-b x^4}} \, dx=-\frac {A \sqrt {a-b x^4}}{a x}-\frac {(A b-a C) \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} b^{3/4} \sqrt {a-b x^4}}+\frac {\left (A b+\sqrt {a} \sqrt {b} B-a C\right ) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{a} b^{3/4} \sqrt {a-b x^4}} \] Output:
-A*(-b*x^4+a)^(1/2)/a/x-(A*b-C*a)*(1-b*x^4/a)^(1/2)*EllipticE(b^(1/4)*x/a^ (1/4),I)/a^(1/4)/b^(3/4)/(-b*x^4+a)^(1/2)+(A*b+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)/a^(1/4)/b^(3/4)/(-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 = 103, normalized size of antiderivative = 0.66 \[ \int \frac {A+B x^2+C x^4}{x^2 \sqrt {a-b x^4}} \, dx=\frac {\sqrt {1-\frac {b x^4}{a}} \left (-3 A \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},\frac {b x^4}{a}\right )+3 B x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {b x^4}{a}\right )+C x^4 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {b x^4}{a}\right )\right )}{3 x \sqrt {a-b x^4}} \] Input:
Integrate[(A + B*x^2 + C*x^4)/(x^2*Sqrt[a - b*x^4]),x]
Output:
(Sqrt[1 - (b*x^4)/a]*(-3*A*Hypergeometric2F1[-1/4, 1/2, 3/4, (b*x^4)/a] + 3*B*x^2*Hypergeometric2F1[1/4, 1/2, 5/4, (b*x^4)/a] + C*x^4*Hypergeometric 2F1[1/2, 3/4, 7/4, (b*x^4)/a]))/(3*x*Sqrt[a - b*x^4])
Time = 0.72 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {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.
| \(\displaystyle \int \frac {A+B x^2+C x^4}{x^2 \sqrt {a-b x^4}} \, dx\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle -\frac {\int -\frac {2 \left (a B x-(A b-a C) x^3\right )}{x \sqrt {a-b x^4}}dx}{2 a}-\frac {A \sqrt {a-b x^4}}{a x}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle -\frac {\int -\frac {2 \left (a B-(A b-a C) x^2\right )}{\sqrt {a-b x^4}}dx}{2 a}-\frac {A \sqrt {a-b x^4}}{a x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a B-(A b-a C) x^2}{\sqrt {a-b x^4}}dx}{a}-\frac {A \sqrt {a-b x^4}}{a x}\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \frac {\frac {\sqrt {a} \left (\sqrt {a} \sqrt {b} B-a C+A b\right ) \int \frac {1}{\sqrt {a-b x^4}}dx}{\sqrt {b}}-\frac {\sqrt {a} (A b-a C) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-b x^4}}dx}{\sqrt {b}}}{a}-\frac {A \sqrt {a-b x^4}}{a x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\sqrt {a} \left (\sqrt {a} \sqrt {b} B-a C+A b\right ) \int \frac {1}{\sqrt {a-b x^4}}dx}{\sqrt {b}}-\frac {(A b-a C) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx}{\sqrt {b}}}{a}-\frac {A \sqrt {a-b x^4}}{a x}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^4}{a}} \left (\sqrt {a} \sqrt {b} B-a C+A b\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {b} \sqrt {a-b x^4}}-\frac {(A b-a C) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx}{\sqrt {b}}}{a}-\frac {A \sqrt {a-b x^4}}{a x}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (\sqrt {a} \sqrt {b} B-a C+A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^4}}-\frac {(A b-a C) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx}{\sqrt {b}}}{a}-\frac {A \sqrt {a-b x^4}}{a x}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (\sqrt {a} \sqrt {b} B-a C+A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^4}}-\frac {\sqrt {1-\frac {b x^4}{a}} (A b-a C) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {b} \sqrt {a-b x^4}}}{a}-\frac {A \sqrt {a-b x^4}}{a x}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (\sqrt {a} \sqrt {b} B-a C+A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^4}}-\frac {\sqrt {a} \sqrt {1-\frac {b x^4}{a}} (A b-a C) \int \frac {\sqrt {\frac {\sqrt {b} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}}dx}{\sqrt {b} \sqrt {a-b x^4}}}{a}-\frac {A \sqrt {a-b x^4}}{a x}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (\sqrt {a} \sqrt {b} B-a C+A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^4}}-\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} (A b-a C) E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{b^{3/4} \sqrt {a-b x^4}}}{a}-\frac {A \sqrt {a-b x^4}}{a x}\) |
Input:
Int[(A + B*x^2 + C*x^4)/(x^2*Sqrt[a - b*x^4]),x]
Output:
-((A*Sqrt[a - b*x^4])/(a*x)) + (-((a^(3/4)*(A*b - a*C)*Sqrt[1 - (b*x^4)/a] *EllipticE[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(b^(3/4)*Sqrt[a - b*x^4])) + (a^(3/4)*(A*b + Sqrt[a]*Sqrt[b]*B - a*C)*Sqrt[1 - (b*x^4)/a]*EllipticF[Arc Sin[(b^(1/4)*x)/a^(1/4)], -1])/(b^(3/4)*Sqrt[a - b*x^4]))/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[(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 = 1.40 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.15
| method | result | size |
| elliptic | \(-\frac {A \sqrt {-b \,x^{4}+a}}{a x}+\frac {B \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\left (C -\frac {A b}{a}\right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, \sqrt {b}}\) | \(181\) |
| risch | \(-\frac {A \sqrt {-b \,x^{4}+a}}{a x}-\frac {-\frac {\left (A b -C a \right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, \sqrt {b}}-\frac {B a \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}}{a}\) | \(188\) |
| default | \(\frac {B \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}+A \left (-\frac {\sqrt {-b \,x^{4}+a}}{a x}+\frac {\sqrt {b}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {a}\, \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )-\frac {C \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, \sqrt {b}}\) | \(261\) |
Input:
int((C*x^4+B*x^2+A)/x^2/(-b*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-A*(-b*x^4+a)^(1/2)/a/x+B/(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)-(C-A/a*b)*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)*(EllipticF(x*(1/a^(1/2)*b^(1/2))^(1/2),I)-EllipticE(x*(1/a^(1/2)*b^(1/2 ))^(1/2),I))
\[ \int \frac {A+B x^2+C x^4}{x^2 \sqrt {a-b x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {-b x^{4} + a} x^{2}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/x^2/(-b*x^4+a)^(1/2),x, algorithm="fricas")
Output:
integral(-(C*x^4 + B*x^2 + A)*sqrt(-b*x^4 + a)/(b*x^6 - a*x^2), x)
Time = 1.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.80 \[ \int \frac {A+B x^2+C x^4}{x^2 \sqrt {a-b x^4}} \, dx=\frac {A \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} x \Gamma \left (\frac {3}{4}\right )} + \frac {B x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} + \frac {C x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {7}{4}\right )} \] Input:
integrate((C*x**4+B*x**2+A)/x**2/(-b*x**4+a)**(1/2),x)
Output:
A*gamma(-1/4)*hyper((-1/4, 1/2), (3/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*sq rt(a)*x*gamma(3/4)) + B*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), b*x**4*exp_ polar(2*I*pi)/a)/(4*sqrt(a)*gamma(5/4)) + C*x**3*gamma(3/4)*hyper((1/2, 3/ 4), (7/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(a)*gamma(7/4))
\[ \int \frac {A+B x^2+C x^4}{x^2 \sqrt {a-b x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {-b x^{4} + a} x^{2}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/x^2/(-b*x^4+a)^(1/2),x, algorithm="maxima")
Output:
integrate((C*x^4 + B*x^2 + A)/(sqrt(-b*x^4 + a)*x^2), x)
\[ \int \frac {A+B x^2+C x^4}{x^2 \sqrt {a-b x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {-b x^{4} + a} x^{2}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/x^2/(-b*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate((C*x^4 + B*x^2 + A)/(sqrt(-b*x^4 + a)*x^2), x)
Timed out. \[ \int \frac {A+B x^2+C x^4}{x^2 \sqrt {a-b x^4}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{x^2\,\sqrt {a-b\,x^4}} \,d x \] Input:
int((A + B*x^2 + C*x^4)/(x^2*(a - b*x^4)^(1/2)),x)
Output:
int((A + B*x^2 + C*x^4)/(x^2*(a - b*x^4)^(1/2)), x)
\[ \int \frac {A+B x^2+C x^4}{x^2 \sqrt {a-b x^4}} \, dx=\left (\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,x^{6}+a \,x^{2}}d x \right ) a +\left (\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,x^{4}+a}d x \right ) b +\left (\int \frac {\sqrt {-b \,x^{4}+a}\, x^{2}}{-b \,x^{4}+a}d x \right ) c \] Input:
int((C*x^4+B*x^2+A)/x^2/(-b*x^4+a)^(1/2),x)
Output:
int(sqrt(a - b*x**4)/(a*x**2 - b*x**6),x)*a + int(sqrt(a - b*x**4)/(a - b* x**4),x)*b + int((sqrt(a - b*x**4)*x**2)/(a - b*x**4),x)*c