Integrand size = 28, antiderivative size = 180 \[ \int \frac {x^2 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\frac {x \left (a B+(A b+a C) x^2\right )}{2 a b \sqrt {a-b x^4}}-\frac {(A b+3 a C) \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{a} b^{7/4} \sqrt {a-b x^4}}+\frac {\left (A b-\sqrt {a} \sqrt {b} B+3 a C\right ) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{a} b^{7/4} \sqrt {a-b x^4}} \] Output:
1/2*x*(a*B+(A*b+C*a)*x^2)/a/b/(-b*x^4+a)^(1/2)-1/2*(A*b+3*C*a)*(1-b*x^4/a) ^(1/2)*EllipticE(b^(1/4)*x/a^(1/4),I)/a^(1/4)/b^(7/4)/(-b*x^4+a)^(1/2)+1/2 *(A*b-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^(1/4)/b^(7/4)/(-b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.08 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.66 \[ \int \frac {x^2 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\frac {3 a x \left (B-2 C x^2\right )-3 a B x \sqrt {1-\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {b x^4}{a}\right )+2 (A b+3 a C) x^3 \sqrt {1-\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},\frac {b x^4}{a}\right )}{6 a b \sqrt {a-b x^4}} \] Input:
Integrate[(x^2*(A + B*x^2 + C*x^4))/(a - b*x^4)^(3/2),x]
Output:
(3*a*x*(B - 2*C*x^2) - 3*a*B*x*Sqrt[1 - (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, (b*x^4)/a] + 2*(A*b + 3*a*C)*x^3*Sqrt[1 - (b*x^4)/a]*Hypergeomet ric2F1[3/4, 3/2, 7/4, (b*x^4)/a])/(6*a*b*Sqrt[a - b*x^4])
Time = 0.76 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {2367, 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.
| \(\displaystyle \int \frac {x^2 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2367 |
| \(\displaystyle \frac {x \left (x^2 (a C+A b)+a B\right )}{2 a b \sqrt {a-b x^4}}-\frac {\int \frac {b \left ((A b+3 a C) x^2+a B\right )}{\sqrt {a-b x^4}}dx}{2 a b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \left (x^2 (a C+A b)+a B\right )}{2 a b \sqrt {a-b x^4}}-\frac {\int \frac {(A b+3 a C) x^2+a B}{\sqrt {a-b x^4}}dx}{2 a b}\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \frac {x \left (x^2 (a C+A b)+a B\right )}{2 a b \sqrt {a-b x^4}}-\frac {\frac {\sqrt {a} (3 a C+A b) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-b x^4}}dx}{\sqrt {b}}-\frac {\sqrt {a} \left (-\sqrt {a} \sqrt {b} B+3 a C+A b\right ) \int \frac {1}{\sqrt {a-b x^4}}dx}{\sqrt {b}}}{2 a b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \left (x^2 (a C+A b)+a B\right )}{2 a b \sqrt {a-b x^4}}-\frac {\frac {(3 a C+A b) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx}{\sqrt {b}}-\frac {\sqrt {a} \left (-\sqrt {a} \sqrt {b} B+3 a C+A b\right ) \int \frac {1}{\sqrt {a-b x^4}}dx}{\sqrt {b}}}{2 a b}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {x \left (x^2 (a C+A b)+a B\right )}{2 a b \sqrt {a-b x^4}}-\frac {\frac {(3 a C+A b) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx}{\sqrt {b}}-\frac {\sqrt {a} \sqrt {1-\frac {b x^4}{a}} \left (-\sqrt {a} \sqrt {b} B+3 a C+A b\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {b} \sqrt {a-b x^4}}}{2 a b}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {x \left (x^2 (a C+A b)+a B\right )}{2 a b \sqrt {a-b x^4}}-\frac {\frac {(3 a C+A b) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx}{\sqrt {b}}-\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (-\sqrt {a} \sqrt {b} B+3 a C+A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^4}}}{2 a b}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {x \left (x^2 (a C+A b)+a B\right )}{2 a b \sqrt {a-b x^4}}-\frac {\frac {\sqrt {1-\frac {b x^4}{a}} (3 a C+A b) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {b} \sqrt {a-b x^4}}-\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (-\sqrt {a} \sqrt {b} B+3 a C+A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^4}}}{2 a b}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {x \left (x^2 (a C+A b)+a B\right )}{2 a b \sqrt {a-b x^4}}-\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^4}{a}} (3 a C+A b) \int \frac {\sqrt {\frac {\sqrt {b} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}}dx}{\sqrt {b} \sqrt {a-b x^4}}-\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (-\sqrt {a} \sqrt {b} B+3 a C+A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^4}}}{2 a b}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {x \left (x^2 (a C+A b)+a B\right )}{2 a b \sqrt {a-b x^4}}-\frac {\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} (3 a C+A b) E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{b^{3/4} \sqrt {a-b x^4}}-\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (-\sqrt {a} \sqrt {b} B+3 a C+A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^4}}}{2 a b}\) |
Input:
Int[(x^2*(A + B*x^2 + C*x^4))/(a - b*x^4)^(3/2),x]
Output:
(x*(a*B + (A*b + a*C)*x^2))/(2*a*b*Sqrt[a - b*x^4]) - ((a^(3/4)*(A*b + 3*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 + 3*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]))/(2*a*b)
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 = m + Expon[Pq, x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1) *x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*x^ m*Pq, a + b*x^n, x]}, Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floo r[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) I nt[(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] && IGtQ[m, 0]
Time = 1.51 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.24
| method | result | size |
| elliptic | \(\frac {2 b \left (\frac {\left (A b +C a \right ) x^{3}}{4 a \,b^{2}}+\frac {B x}{4 b^{2}}\right )}{\sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}-\frac {B \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{2 b \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\left (-\frac {C}{b}-\frac {A b +C a}{2 a b}\right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, \sqrt {b}}\) | \(223\) |
| default | \(A \left (\frac {x^{3}}{2 a \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {\sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, \sqrt {b}}\right )+B \left (\frac {x}{2 b \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}-\frac {\sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{2 b \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )+C \left (\frac {x^{3}}{2 b \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {3 \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 b^{\frac {3}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )\) | \(319\) |
Input:
int(x^2*(C*x^4+B*x^2+A)/(-b*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
2*b*(1/4/a*(A*b+C*a)/b^2*x^3+1/4*B*x/b^2)/(-(x^4-a/b)*b)^(1/2)-1/2*B/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)-(-1/b*C -1/2/a*(A*b+C*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)*(Elliptic F(x*(1/a^(1/2)*b^(1/2))^(1/2),I)-EllipticE(x*(1/a^(1/2)*b^(1/2))^(1/2),I))
Time = 0.11 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.01 \[ \int \frac {x^2 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\frac {{\left ({\left (3 \, C a b + A b^{2}\right )} x^{5} - {\left (3 \, C a^{2} + A a b\right )} x\right )} \sqrt {-b} \left (\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left ({\left (3 \, C a b + {\left (A + B\right )} b^{2}\right )} x^{5} - {\left (3 \, C a^{2} + {\left (A + B\right )} a b\right )} x\right )} \sqrt {-b} \left (\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (2 \, C a b x^{4} - B a b x^{2} - 3 \, C a^{2} - A a b\right )} \sqrt {-b x^{4} + a}}{2 \, {\left (a b^{3} x^{5} - a^{2} b^{2} x\right )}} \] Input:
integrate(x^2*(C*x^4+B*x^2+A)/(-b*x^4+a)^(3/2),x, algorithm="fricas")
Output:
1/2*(((3*C*a*b + A*b^2)*x^5 - (3*C*a^2 + A*a*b)*x)*sqrt(-b)*(a/b)^(3/4)*el liptic_e(arcsin((a/b)^(1/4)/x), -1) - ((3*C*a*b + (A + B)*b^2)*x^5 - (3*C* a^2 + (A + B)*a*b)*x)*sqrt(-b)*(a/b)^(3/4)*elliptic_f(arcsin((a/b)^(1/4)/x ), -1) + (2*C*a*b*x^4 - B*a*b*x^2 - 3*C*a^2 - A*a*b)*sqrt(-b*x^4 + a))/(a* b^3*x^5 - a^2*b^2*x)
Time = 4.87 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.70 \[ \int \frac {x^2 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\frac {A x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {B x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {3}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {9}{4}\right )} + \frac {C x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {11}{4}\right )} \] Input:
integrate(x**2*(C*x**4+B*x**2+A)/(-b*x**4+a)**(3/2),x)
Output:
A*x**3*gamma(3/4)*hyper((3/4, 3/2), (7/4,), b*x**4*exp_polar(2*I*pi)/a)/(4 *a**(3/2)*gamma(7/4)) + B*x**5*gamma(5/4)*hyper((5/4, 3/2), (9/4,), b*x**4 *exp_polar(2*I*pi)/a)/(4*a**(3/2)*gamma(9/4)) + C*x**7*gamma(7/4)*hyper((3 /2, 7/4), (11/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*a**(3/2)*gamma(11/4))
\[ \int \frac {x^2 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} x^{2}}{{\left (-b x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^2*(C*x^4+B*x^2+A)/(-b*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((C*x^4 + B*x^2 + A)*x^2/(-b*x^4 + a)^(3/2), x)
\[ \int \frac {x^2 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} x^{2}}{{\left (-b x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^2*(C*x^4+B*x^2+A)/(-b*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((C*x^4 + B*x^2 + A)*x^2/(-b*x^4 + a)^(3/2), x)
Timed out. \[ \int \frac {x^2 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\int \frac {x^2\,\left (C\,x^4+B\,x^2+A\right )}{{\left (a-b\,x^4\right )}^{3/2}} \,d x \] Input:
int((x^2*(A + B*x^2 + C*x^4))/(a - b*x^4)^(3/2),x)
Output:
int((x^2*(A + B*x^2 + C*x^4))/(a - b*x^4)^(3/2), x)
\[ \int \frac {x^2 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\frac {\sqrt {-b \,x^{4}+a}\, b x -\sqrt {-b \,x^{4}+a}\, c \,x^{3}-\left (\int \frac {\sqrt {-b \,x^{4}+a}}{b^{2} x^{8}-2 a b \,x^{4}+a^{2}}d x \right ) a^{2} b +\left (\int \frac {\sqrt {-b \,x^{4}+a}}{b^{2} x^{8}-2 a b \,x^{4}+a^{2}}d x \right ) a \,b^{2} x^{4}+\left (\int \frac {\sqrt {-b \,x^{4}+a}\, x^{2}}{b^{2} x^{8}-2 a b \,x^{4}+a^{2}}d x \right ) a^{2} b +3 \left (\int \frac {\sqrt {-b \,x^{4}+a}\, x^{2}}{b^{2} x^{8}-2 a b \,x^{4}+a^{2}}d x \right ) a^{2} c -\left (\int \frac {\sqrt {-b \,x^{4}+a}\, x^{2}}{b^{2} x^{8}-2 a b \,x^{4}+a^{2}}d x \right ) a \,b^{2} x^{4}-3 \left (\int \frac {\sqrt {-b \,x^{4}+a}\, x^{2}}{b^{2} x^{8}-2 a b \,x^{4}+a^{2}}d x \right ) a b c \,x^{4}}{b \left (-b \,x^{4}+a \right )} \] Input:
int(x^2*(C*x^4+B*x^2+A)/(-b*x^4+a)^(3/2),x)
Output:
(sqrt(a - b*x**4)*b*x - sqrt(a - b*x**4)*c*x**3 - int(sqrt(a - b*x**4)/(a* *2 - 2*a*b*x**4 + b**2*x**8),x)*a**2*b + int(sqrt(a - b*x**4)/(a**2 - 2*a* b*x**4 + b**2*x**8),x)*a*b**2*x**4 + int((sqrt(a - b*x**4)*x**2)/(a**2 - 2 *a*b*x**4 + b**2*x**8),x)*a**2*b + 3*int((sqrt(a - b*x**4)*x**2)/(a**2 - 2 *a*b*x**4 + b**2*x**8),x)*a**2*c - int((sqrt(a - b*x**4)*x**2)/(a**2 - 2*a *b*x**4 + b**2*x**8),x)*a*b**2*x**4 - 3*int((sqrt(a - b*x**4)*x**2)/(a**2 - 2*a*b*x**4 + b**2*x**8),x)*a*b*c*x**4)/(b*(a - b*x**4))