Integrand size = 25, antiderivative size = 167 \[ \int \frac {A+B x^2+C x^4}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {x \left (A+\frac {a C}{c}+B x^2\right )}{2 a \sqrt {a-c x^4}}-\frac {B \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{a} c^{3/4} \sqrt {a-c x^4}}+\frac {\left (\sqrt {a} B \sqrt {c}+A c-a C\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 a^{3/4} c^{5/4} \sqrt {a-c x^4}} \] Output:
1/2*x*(A+a*C/c+B*x^2)/a/(-c*x^4+a)^(1/2)-1/2*B*(1-c*x^4/a)^(1/2)*EllipticE (c^(1/4)*x/a^(1/4),I)/a^(1/4)/c^(3/4)/(-c*x^4+a)^(1/2)+1/2*(a^(1/2)*B*c^(1 /2)+A*c-a*C)*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/a^(3/4)/c^(5 /4)/(-c*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.11 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.70 \[ \int \frac {A+B x^2+C x^4}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {3 (A c+a C) x+3 (A c-a C) x \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {c x^4}{a}\right )+2 B c x^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},\frac {c x^4}{a}\right )}{6 a c \sqrt {a-c x^4}} \] Input:
Integrate[(A + B*x^2 + C*x^4)/(a - c*x^4)^(3/2),x]
Output:
(3*(A*c + a*C)*x + 3*(A*c - a*C)*x*Sqrt[1 - (c*x^4)/a]*Hypergeometric2F1[1 /4, 1/2, 5/4, (c*x^4)/a] + 2*B*c*x^3*Sqrt[1 - (c*x^4)/a]*Hypergeometric2F1 [3/4, 3/2, 7/4, (c*x^4)/a])/(6*a*c*Sqrt[a - c*x^4])
Time = 0.42 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2397, 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.
| \(\displaystyle \int \frac {A+B x^2+C x^4}{\left (a-c x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2397 |
| \(\displaystyle \frac {\int \frac {-B c x^2+A c-a C}{\sqrt {a-c x^4}}dx}{2 a c}+\frac {x \left (a C+A c+B c x^2\right )}{2 a c \sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \frac {\left (\sqrt {a} B \sqrt {c}-a C+A c\right ) \int \frac {1}{\sqrt {a-c x^4}}dx-\sqrt {a} B \sqrt {c} \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-c x^4}}dx}{2 a c}+\frac {x \left (a C+A c+B c x^2\right )}{2 a c \sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (\sqrt {a} B \sqrt {c}-a C+A c\right ) \int \frac {1}{\sqrt {a-c x^4}}dx-B \sqrt {c} \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{2 a c}+\frac {x \left (a C+A c+B c x^2\right )}{2 a c \sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\frac {\sqrt {1-\frac {c x^4}{a}} \left (\sqrt {a} B \sqrt {c}-a C+A c\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {a-c x^4}}-B \sqrt {c} \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{2 a c}+\frac {x \left (a C+A c+B c x^2\right )}{2 a c \sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {a} B \sqrt {c}-a C+A c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-B \sqrt {c} \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{2 a c}+\frac {x \left (a C+A c+B c x^2\right )}{2 a c \sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {a} B \sqrt {c}-a C+A c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-\frac {B \sqrt {c} \sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {a-c x^4}}}{2 a c}+\frac {x \left (a C+A c+B c x^2\right )}{2 a c \sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {a} B \sqrt {c}-a C+A c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-\frac {\sqrt {a} B \sqrt {c} \sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {\frac {\sqrt {c} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}}dx}{\sqrt {a-c x^4}}}{2 a c}+\frac {x \left (a C+A c+B c x^2\right )}{2 a c \sqrt {a-c x^4}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {a} B \sqrt {c}-a C+A c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-\frac {a^{3/4} B \sqrt [4]{c} \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt {a-c x^4}}}{2 a c}+\frac {x \left (a C+A c+B c x^2\right )}{2 a c \sqrt {a-c x^4}}\) |
Input:
Int[(A + B*x^2 + C*x^4)/(a - c*x^4)^(3/2),x]
Output:
(x*(A*c + a*C + B*c*x^2))/(2*a*c*Sqrt[a - c*x^4]) + (-((a^(3/4)*B*c^(1/4)* Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/Sqrt[a - c *x^4]) + (a^(1/4)*(Sqrt[a]*B*Sqrt[c] + A*c - a*C)*Sqrt[1 - (c*x^4)/a]*Elli pticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)*Sqrt[a - c*x^4]))/(2*a*c)
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_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x]}, S imp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) Int[(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]
Time = 0.46 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.33
| method | result | size |
| elliptic | \(\frac {2 c \left (\frac {B \,x^{3}}{4 a c}+\frac {\left (A c +C a \right ) x}{4 a \,c^{2}}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\left (-\frac {C}{c}+\frac {A c +C a}{2 a c}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {B \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(222\) |
| default | \(A \left (\frac {x}{2 a \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+B \left (\frac {x^{3}}{2 a \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\right )+C \left (\frac {x}{2 c \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}-\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )\) | \(297\) |
Input:
int((C*x^4+B*x^2+A)/(-c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
2*c*(1/4/a*B/c*x^3+1/4*(A*c+C*a)/a/c^2*x)/(-(x^4-a/c)*c)^(1/2)+(-C/c+1/2*( A*c+C*a)/a/c)/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^( 1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(c^(1/2)/a^(1/2))^(1/ 2),I)+1/2/a^(1/2)*B/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)* (1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)/c^(1/2)*(EllipticF(x*(c^(1/ 2)/a^(1/2))^(1/2),I)-EllipticE(x*(c^(1/2)/a^(1/2))^(1/2),I))
Time = 0.09 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x^2+C x^4}{\left (a-c x^4\right )^{3/2}} \, dx=-\frac {{\left (B c^{2} x^{4} - B a c\right )} \sqrt {a} \left (\frac {c}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (\frac {c}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left ({\left (C a c - {\left (A + B\right )} c^{2}\right )} x^{4} - C a^{2} + {\left (A + B\right )} a c\right )} \sqrt {a} \left (\frac {c}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (\frac {c}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (B c^{2} x^{3} + {\left (C a c + A c^{2}\right )} x\right )} \sqrt {-c x^{4} + a}}{2 \, {\left (a c^{3} x^{4} - a^{2} c^{2}\right )}} \] Input:
integrate((C*x^4+B*x^2+A)/(-c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
-1/2*((B*c^2*x^4 - B*a*c)*sqrt(a)*(c/a)^(3/4)*elliptic_e(arcsin(x*(c/a)^(1 /4)), -1) + ((C*a*c - (A + B)*c^2)*x^4 - C*a^2 + (A + B)*a*c)*sqrt(a)*(c/a )^(3/4)*elliptic_f(arcsin(x*(c/a)^(1/4)), -1) + (B*c^2*x^3 + (C*a*c + A*c^ 2)*x)*sqrt(-c*x^4 + a))/(a*c^3*x^4 - a^2*c^2)
Time = 4.88 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.74 \[ \int \frac {A+B x^2+C x^4}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {A x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {B x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {C x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {3}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {9}{4}\right )} \] Input:
integrate((C*x**4+B*x**2+A)/(-c*x**4+a)**(3/2),x)
Output:
A*x*gamma(1/4)*hyper((1/4, 3/2), (5/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*a* *(3/2)*gamma(5/4)) + B*x**3*gamma(3/4)*hyper((3/4, 3/2), (7/4,), c*x**4*ex p_polar(2*I*pi)/a)/(4*a**(3/2)*gamma(7/4)) + C*x**5*gamma(5/4)*hyper((5/4, 3/2), (9/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*a**(3/2)*gamma(9/4))
\[ \int \frac {A+B x^2+C x^4}{\left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/(-c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((C*x^4 + B*x^2 + A)/(-c*x^4 + a)^(3/2), x)
\[ \int \frac {A+B x^2+C x^4}{\left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/(-c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((C*x^4 + B*x^2 + A)/(-c*x^4 + a)^(3/2), x)
Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (a-c x^4\right )^{3/2}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{{\left (a-c\,x^4\right )}^{3/2}} \,d x \] Input:
int((A + B*x^2 + C*x^4)/(a - c*x^4)^(3/2),x)
Output:
int((A + B*x^2 + C*x^4)/(a - c*x^4)^(3/2), x)
\[ \int \frac {A+B x^2+C x^4}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {\sqrt {-c \,x^{4}+a}\, x +\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) a b -\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) b c \,x^{4}}{-c \,x^{4}+a} \] Input:
int((C*x^4+B*x^2+A)/(-c*x^4+a)^(3/2),x)
Output:
(sqrt(a - c*x**4)*x + int((sqrt(a - c*x**4)*x**2)/(a**2 - 2*a*c*x**4 + c** 2*x**8),x)*a*b - int((sqrt(a - c*x**4)*x**2)/(a**2 - 2*a*c*x**4 + c**2*x** 8),x)*b*c*x**4)/(a - c*x**4)