Integrand size = 28, antiderivative size = 223 \[ \int \frac {x^6 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\frac {x \left (a B+(A b+a C) x^2\right )}{2 b^2 \sqrt {a-b x^4}}+\frac {B x \sqrt {a-b x^4}}{3 b^2}+\frac {C x^3 \sqrt {a-b x^4}}{5 b^2}-\frac {3 a^{3/4} (5 A b+7 a C) \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{10 b^{11/4} \sqrt {a-b x^4}}+\frac {a^{3/4} \left (45 A b-25 \sqrt {a} \sqrt {b} B+63 a C\right ) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{30 b^{11/4} \sqrt {a-b x^4}} \] Output:
1/2*x*(a*B+(A*b+C*a)*x^2)/b^2/(-b*x^4+a)^(1/2)+1/3*B*x*(-b*x^4+a)^(1/2)/b^ 2+1/5*C*x^3*(-b*x^4+a)^(1/2)/b^2-3/10*a^(3/4)*(5*A*b+7*C*a)*(1-b*x^4/a)^(1 /2)*EllipticE(b^(1/4)*x/a^(1/4),I)/b^(11/4)/(-b*x^4+a)^(1/2)+1/30*a^(3/4)* (45*A*b-25*a^(1/2)*b^(1/2)*B+63*C*a)*(1-b*x^4/a)^(1/2)*EllipticF(b^(1/4)*x /a^(1/4),I)/b^(11/4)/(-b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.17 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.62 \[ \int \frac {x^6 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\frac {25 a B x-30 A b x^3-42 a C x^3-10 b B x^5-6 b C x^7-25 a B x \sqrt {1-\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {b x^4}{a}\right )+6 (5 A b+7 a C) x^3 \sqrt {1-\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},\frac {b x^4}{a}\right )}{30 b^2 \sqrt {a-b x^4}} \] Input:
Integrate[(x^6*(A + B*x^2 + C*x^4))/(a - b*x^4)^(3/2),x]
Output:
(25*a*B*x - 30*A*b*x^3 - 42*a*C*x^3 - 10*b*B*x^5 - 6*b*C*x^7 - 25*a*B*x*Sq rt[1 - (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, (b*x^4)/a] + 6*(5*A*b + 7*a*C)*x^3*Sqrt[1 - (b*x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, (b*x^4)/a ])/(30*b^2*Sqrt[a - b*x^4])
Time = 1.27 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.12, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2367, 25, 2427, 25, 2427, 25, 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.
| \(\displaystyle \int \frac {x^6 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2367 |
| \(\displaystyle \frac {\int -\frac {2 a b^2 C x^6+2 a b^2 B x^4+3 a b (A b+a C) x^2+a^2 b B}{\sqrt {a-b x^4}}dx}{2 a b^3}+\frac {x \left (x^2 (a C+A b)+a B\right )}{2 b^2 \sqrt {a-b x^4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x \left (x^2 (a C+A b)+a B\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {\int \frac {2 a b^2 C x^6+2 a b^2 B x^4+3 a b (A b+a C) x^2+a^2 b B}{\sqrt {a-b x^4}}dx}{2 a b^3}\) |
| \(\Big \downarrow \) 2427 |
| \(\displaystyle \frac {x \left (x^2 (a C+A b)+a B\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {-\frac {\int -\frac {10 a b^3 B x^4+3 a b^2 (5 A b+7 a C) x^2+5 a^2 b^2 B}{\sqrt {a-b x^4}}dx}{5 b}-\frac {2}{5} a b C x^3 \sqrt {a-b x^4}}{2 a b^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x \left (x^2 (a C+A b)+a B\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {\frac {\int \frac {10 a b^3 B x^4+3 a b^2 (5 A b+7 a C) x^2+5 a^2 b^2 B}{\sqrt {a-b x^4}}dx}{5 b}-\frac {2}{5} a b C x^3 \sqrt {a-b x^4}}{2 a b^3}\) |
| \(\Big \downarrow \) 2427 |
| \(\displaystyle \frac {x \left (x^2 (a C+A b)+a B\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {\frac {-\frac {\int -\frac {a b^3 \left (9 (5 A b+7 a C) x^2+25 a B\right )}{\sqrt {a-b x^4}}dx}{3 b}-\frac {10}{3} a b^2 B x \sqrt {a-b x^4}}{5 b}-\frac {2}{5} a b C x^3 \sqrt {a-b x^4}}{2 a b^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x \left (x^2 (a C+A b)+a B\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {\frac {\frac {\int \frac {a b^3 \left (9 (5 A b+7 a C) x^2+25 a B\right )}{\sqrt {a-b x^4}}dx}{3 b}-\frac {10}{3} a b^2 B x \sqrt {a-b x^4}}{5 b}-\frac {2}{5} a b C x^3 \sqrt {a-b x^4}}{2 a b^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \left (x^2 (a C+A b)+a B\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {\frac {\frac {1}{3} a b^2 \int \frac {9 (5 A b+7 a C) x^2+25 a B}{\sqrt {a-b x^4}}dx-\frac {10}{3} a b^2 B x \sqrt {a-b x^4}}{5 b}-\frac {2}{5} a b C x^3 \sqrt {a-b x^4}}{2 a b^3}\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \frac {x \left (x^2 (a C+A b)+a B\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {\frac {\frac {1}{3} a b^2 \left (\sqrt {a} \left (25 \sqrt {a} B-\frac {9 (7 a C+5 A b)}{\sqrt {b}}\right ) \int \frac {1}{\sqrt {a-b x^4}}dx+\frac {9 \sqrt {a} (7 a C+5 A b) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-b x^4}}dx}{\sqrt {b}}\right )-\frac {10}{3} a b^2 B x \sqrt {a-b x^4}}{5 b}-\frac {2}{5} a b C x^3 \sqrt {a-b x^4}}{2 a b^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \left (x^2 (a C+A b)+a B\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {\frac {\frac {1}{3} a b^2 \left (\sqrt {a} \left (25 \sqrt {a} B-\frac {9 (7 a C+5 A b)}{\sqrt {b}}\right ) \int \frac {1}{\sqrt {a-b x^4}}dx+\frac {9 (7 a C+5 A b) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx}{\sqrt {b}}\right )-\frac {10}{3} a b^2 B x \sqrt {a-b x^4}}{5 b}-\frac {2}{5} a b C x^3 \sqrt {a-b x^4}}{2 a b^3}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {x \left (x^2 (a C+A b)+a B\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {\frac {\frac {1}{3} a b^2 \left (\frac {\sqrt {a} \sqrt {1-\frac {b x^4}{a}} \left (25 \sqrt {a} B-\frac {9 (7 a C+5 A b)}{\sqrt {b}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {a-b x^4}}+\frac {9 (7 a C+5 A b) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx}{\sqrt {b}}\right )-\frac {10}{3} a b^2 B x \sqrt {a-b x^4}}{5 b}-\frac {2}{5} a b C x^3 \sqrt {a-b x^4}}{2 a b^3}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {x \left (x^2 (a C+A b)+a B\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {\frac {\frac {1}{3} a b^2 \left (\frac {9 (7 a C+5 A b) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx}{\sqrt {b}}+\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (25 \sqrt {a} B-\frac {9 (7 a C+5 A b)}{\sqrt {b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )-\frac {10}{3} a b^2 B x \sqrt {a-b x^4}}{5 b}-\frac {2}{5} a b C x^3 \sqrt {a-b x^4}}{2 a b^3}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {x \left (x^2 (a C+A b)+a B\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {\frac {\frac {1}{3} a b^2 \left (\frac {9 \sqrt {1-\frac {b x^4}{a}} (7 a C+5 A b) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {b} \sqrt {a-b x^4}}+\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (25 \sqrt {a} B-\frac {9 (7 a C+5 A b)}{\sqrt {b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )-\frac {10}{3} a b^2 B x \sqrt {a-b x^4}}{5 b}-\frac {2}{5} a b C x^3 \sqrt {a-b x^4}}{2 a b^3}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {x \left (x^2 (a C+A b)+a B\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {\frac {\frac {1}{3} a b^2 \left (\frac {9 \sqrt {a} \sqrt {1-\frac {b x^4}{a}} (7 a C+5 A b) \int \frac {\sqrt {\frac {\sqrt {b} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}}dx}{\sqrt {b} \sqrt {a-b x^4}}+\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (25 \sqrt {a} B-\frac {9 (7 a C+5 A b)}{\sqrt {b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )-\frac {10}{3} a b^2 B x \sqrt {a-b x^4}}{5 b}-\frac {2}{5} a b C x^3 \sqrt {a-b x^4}}{2 a b^3}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {x \left (x^2 (a C+A b)+a B\right )}{2 b^2 \sqrt {a-b x^4}}-\frac {\frac {\frac {1}{3} a b^2 \left (\frac {9 a^{3/4} \sqrt {1-\frac {b x^4}{a}} (7 a C+5 A b) E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{b^{3/4} \sqrt {a-b x^4}}+\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (25 \sqrt {a} B-\frac {9 (7 a C+5 A b)}{\sqrt {b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )-\frac {10}{3} a b^2 B x \sqrt {a-b x^4}}{5 b}-\frac {2}{5} a b C x^3 \sqrt {a-b x^4}}{2 a b^3}\) |
Input:
Int[(x^6*(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*b^2*Sqrt[a - b*x^4]) - ((-2*a*b*C*x^3*Sqrt[ a - b*x^4])/5 + ((-10*a*b^2*B*x*Sqrt[a - b*x^4])/3 + (a*b^2*((9*a^(3/4)*(5 *A*b + 7*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)*(25*Sqrt[a]*B - (9*(5*A*b + 7*a*C ))/Sqrt[b])*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)/(5*b))/(2*a*b^3)
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]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x ]}, With[{Pqq = Coeff[Pq, x, q]}, Simp[Pqq*x^(q - n + 1)*((a + b*x^n)^(p + 1)/(b*(q + n*p + 1))), x] + Simp[1/(b*(q + n*p + 1)) Int[ExpandToSum[b*(q + n*p + 1)*(Pq - Pqq*x^q) - a*Pqq*(q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x]] /; NeQ[q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || IntegerQ [p + (q + 1)/(2*n)])] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]
Time = 6.10 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.15
| method | result | size |
| elliptic | \(\frac {2 b \left (\frac {\left (A b +C a \right ) x^{3}}{4 b^{3}}+\frac {B a x}{4 b^{3}}\right )}{\sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {C \,x^{3} \sqrt {-b \,x^{4}+a}}{5 b^{2}}+\frac {B x \sqrt {-b \,x^{4}+a}}{3 b^{2}}-\frac {5 B a \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{6 b^{2} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\left (-\frac {3 \left (A b +C a \right )}{2 b^{2}}-\frac {3 C a}{5 b^{2}}\right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, \sqrt {b}}\) | \(256\) |
| default | \(A \left (\frac {x^{3}}{2 b \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {3 \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 b^{\frac {3}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )+B \left (\frac {a x}{2 b^{2} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {x \sqrt {-b \,x^{4}+a}}{3 b^{2}}-\frac {5 a \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{6 b^{2} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )+C \left (\frac {a \,x^{3}}{2 b^{2} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {x^{3} \sqrt {-b \,x^{4}+a}}{5 b^{2}}+\frac {21 a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{10 b^{\frac {5}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )\) | \(356\) |
| risch | \(\frac {x \left (3 C \,x^{2}+5 B \right ) \sqrt {-b \,x^{4}+a}}{15 b^{2}}-\frac {\frac {20 B a \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {15 A \sqrt {b}\, \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {24 C \,a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, \sqrt {b}}+15 a \left (\frac {2 b \left (-\frac {\left (A b +C a \right ) x^{3}}{4 b a}-\frac {B x}{4 b}\right )}{\sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}-\frac {B \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{2 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\left (A b +C a \right ) \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, \sqrt {b}}\right )}{15 b^{2}}\) | \(483\) |
Input:
int(x^6*(C*x^4+B*x^2+A)/(-b*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
2*b*(1/4/b^3*(A*b+C*a)*x^3+1/4/b^3*B*a*x)/(-(x^4-a/b)*b)^(1/2)+1/5*C*x^3*( -b*x^4+a)^(1/2)/b^2+1/3*B*x*(-b*x^4+a)^(1/2)/b^2-5/6*B*a/b^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)-(-3/2*(A*b+C*a)/b ^2-3/5*C/b^2*a)*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))
Time = 0.10 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.98 \[ \int \frac {x^6 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\frac {9 \, {\left ({\left (7 \, C a b + 5 \, A b^{2}\right )} x^{5} - {\left (7 \, C a^{2} + 5 \, A a b\right )} x\right )} \sqrt {-b} \left (\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left ({\left (63 \, C a b + 5 \, {\left (9 \, A + 5 \, B\right )} b^{2}\right )} x^{5} - {\left (63 \, C a^{2} + 5 \, {\left (9 \, A + 5 \, B\right )} a b\right )} x\right )} \sqrt {-b} \left (\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (6 \, C b^{2} x^{8} + 10 \, B b^{2} x^{6} - 25 \, B a b x^{2} + 6 \, {\left (7 \, C a b + 5 \, A b^{2}\right )} x^{4} - 63 \, C a^{2} - 45 \, A a b\right )} \sqrt {-b x^{4} + a}}{30 \, {\left (b^{4} x^{5} - a b^{3} x\right )}} \] Input:
integrate(x^6*(C*x^4+B*x^2+A)/(-b*x^4+a)^(3/2),x, algorithm="fricas")
Output:
1/30*(9*((7*C*a*b + 5*A*b^2)*x^5 - (7*C*a^2 + 5*A*a*b)*x)*sqrt(-b)*(a/b)^( 3/4)*elliptic_e(arcsin((a/b)^(1/4)/x), -1) - ((63*C*a*b + 5*(9*A + 5*B)*b^ 2)*x^5 - (63*C*a^2 + 5*(9*A + 5*B)*a*b)*x)*sqrt(-b)*(a/b)^(3/4)*elliptic_f (arcsin((a/b)^(1/4)/x), -1) + (6*C*b^2*x^8 + 10*B*b^2*x^6 - 25*B*a*b*x^2 + 6*(7*C*a*b + 5*A*b^2)*x^4 - 63*C*a^2 - 45*A*a*b)*sqrt(-b*x^4 + a))/(b^4*x ^5 - a*b^3*x)
Time = 10.65 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.57 \[ \int \frac {x^6 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\frac {A x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {11}{4}\right )} + \frac {B x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {13}{4}\right )} + \frac {C x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {15}{4}\right )} \] Input:
integrate(x**6*(C*x**4+B*x**2+A)/(-b*x**4+a)**(3/2),x)
Output:
A*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)) + B*x**9*gamma(9/4)*hyper((3/2, 9/4), (13/4,), b*x **4*exp_polar(2*I*pi)/a)/(4*a**(3/2)*gamma(13/4)) + C*x**11*gamma(11/4)*hy per((3/2, 11/4), (15/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*a**(3/2)*gamma(15 /4))
\[ \int \frac {x^6 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} x^{6}}{{\left (-b x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^6*(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^6/(-b*x^4 + a)^(3/2), x)
\[ \int \frac {x^6 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} x^{6}}{{\left (-b x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^6*(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^6/(-b*x^4 + a)^(3/2), x)
Timed out. \[ \int \frac {x^6 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\int \frac {x^6\,\left (C\,x^4+B\,x^2+A\right )}{{\left (a-b\,x^4\right )}^{3/2}} \,d x \] Input:
int((x^6*(A + B*x^2 + C*x^4))/(a - b*x^4)^(3/2),x)
Output:
int((x^6*(A + B*x^2 + C*x^4))/(a - b*x^4)^(3/2), x)
\[ \int \frac {x^6 \left (A+B x^2+C x^4\right )}{\left (a-b x^4\right )^{3/2}} \, dx=\frac {-15 \sqrt {-b \,x^{4}+a}\, a b \,x^{3}+25 \sqrt {-b \,x^{4}+a}\, a b x -21 \sqrt {-b \,x^{4}+a}\, a c \,x^{3}-5 \sqrt {-b \,x^{4}+a}\, b^{2} x^{5}-3 \sqrt {-b \,x^{4}+a}\, b c \,x^{7}-25 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{b^{2} x^{8}-2 a b \,x^{4}+a^{2}}d x \right ) a^{3} b +25 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{b^{2} x^{8}-2 a b \,x^{4}+a^{2}}d x \right ) a^{2} b^{2} x^{4}+45 \left (\int \frac {\sqrt {-b \,x^{4}+a}\, x^{2}}{b^{2} x^{8}-2 a b \,x^{4}+a^{2}}d x \right ) a^{3} b +63 \left (\int \frac {\sqrt {-b \,x^{4}+a}\, x^{2}}{b^{2} x^{8}-2 a b \,x^{4}+a^{2}}d x \right ) a^{3} c -45 \left (\int \frac {\sqrt {-b \,x^{4}+a}\, x^{2}}{b^{2} x^{8}-2 a b \,x^{4}+a^{2}}d x \right ) a^{2} b^{2} x^{4}-63 \left (\int \frac {\sqrt {-b \,x^{4}+a}\, x^{2}}{b^{2} x^{8}-2 a b \,x^{4}+a^{2}}d x \right ) a^{2} b c \,x^{4}}{15 b^{2} \left (-b \,x^{4}+a \right )} \] Input:
int(x^6*(C*x^4+B*x^2+A)/(-b*x^4+a)^(3/2),x)
Output:
( - 15*sqrt(a - b*x**4)*a*b*x**3 + 25*sqrt(a - b*x**4)*a*b*x - 21*sqrt(a - b*x**4)*a*c*x**3 - 5*sqrt(a - b*x**4)*b**2*x**5 - 3*sqrt(a - b*x**4)*b*c* x**7 - 25*int(sqrt(a - b*x**4)/(a**2 - 2*a*b*x**4 + b**2*x**8),x)*a**3*b + 25*int(sqrt(a - b*x**4)/(a**2 - 2*a*b*x**4 + b**2*x**8),x)*a**2*b**2*x**4 + 45*int((sqrt(a - b*x**4)*x**2)/(a**2 - 2*a*b*x**4 + b**2*x**8),x)*a**3* b + 63*int((sqrt(a - b*x**4)*x**2)/(a**2 - 2*a*b*x**4 + b**2*x**8),x)*a**3 *c - 45*int((sqrt(a - b*x**4)*x**2)/(a**2 - 2*a*b*x**4 + b**2*x**8),x)*a** 2*b**2*x**4 - 63*int((sqrt(a - b*x**4)*x**2)/(a**2 - 2*a*b*x**4 + b**2*x** 8),x)*a**2*b*c*x**4)/(15*b**2*(a - b*x**4))