Integrand size = 28, antiderivative size = 257 \[ \int \frac {\sqrt {a-b x^4} \left (A+B x^2+C x^4\right )}{x^{10}} \, dx=-\frac {1}{315} \left (\frac {35 A}{x^9}+\frac {45 B}{x^7}+\frac {63 C}{x^5}\right ) \sqrt {a-b x^4}+\frac {2 A b \sqrt {a-b x^4}}{45 a x^5}+\frac {2 b B \sqrt {a-b x^4}}{21 a x^3}+\frac {2 b (A b+3 a C) \sqrt {a-b x^4}}{15 a^2 x}+\frac {2 b^{5/4} (A b+3 a C) \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{15 a^{5/4} \sqrt {a-b x^4}}-\frac {2 b^{5/4} \left (7 A b+5 \sqrt {a} \sqrt {b} B+21 a C\right ) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{105 a^{5/4} \sqrt {a-b x^4}} \] Output:
-1/315*(35*A/x^9+45*B/x^7+63*C/x^5)*(-b*x^4+a)^(1/2)+2/45*A*b*(-b*x^4+a)^( 1/2)/a/x^5+2/21*b*B*(-b*x^4+a)^(1/2)/a/x^3+2/15*b*(A*b+3*C*a)*(-b*x^4+a)^( 1/2)/a^2/x+2/15*b^(5/4)*(A*b+3*C*a)*(1-b*x^4/a)^(1/2)*EllipticE(b^(1/4)*x/ a^(1/4),I)/a^(5/4)/(-b*x^4+a)^(1/2)-2/105*b^(5/4)*(7*A*b+5*a^(1/2)*b^(1/2) *B+21*C*a)*(1-b*x^4/a)^(1/2)*EllipticF(b^(1/4)*x/a^(1/4),I)/a^(5/4)/(-b*x^ 4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.12 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.40 \[ \int \frac {\sqrt {a-b x^4} \left (A+B x^2+C x^4\right )}{x^{10}} \, dx=-\frac {\sqrt {a-b x^4} \left (35 A \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},-\frac {1}{2},-\frac {5}{4},\frac {b x^4}{a}\right )+45 B x^2 \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},-\frac {1}{2},-\frac {3}{4},\frac {b x^4}{a}\right )+63 C x^4 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {1}{2},-\frac {1}{4},\frac {b x^4}{a}\right )\right )}{315 x^9 \sqrt {1-\frac {b x^4}{a}}} \] Input:
Integrate[(Sqrt[a - b*x^4]*(A + B*x^2 + C*x^4))/x^10,x]
Output:
-1/315*(Sqrt[a - b*x^4]*(35*A*Hypergeometric2F1[-9/4, -1/2, -5/4, (b*x^4)/ a] + 45*B*x^2*Hypergeometric2F1[-7/4, -1/2, -3/4, (b*x^4)/a] + 63*C*x^4*Hy pergeometric2F1[-5/4, -1/2, -1/4, (b*x^4)/a]))/(x^9*Sqrt[1 - (b*x^4)/a])
Time = 1.21 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.607, Rules used = {2364, 27, 2374, 9, 27, 1605, 27, 1605, 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.
| \(\displaystyle \int \frac {\sqrt {a-b x^4} \left (A+B x^2+C x^4\right )}{x^{10}} \, dx\) |
| \(\Big \downarrow \) 2364 |
| \(\displaystyle 2 b \int -\frac {63 C x^4+45 B x^2+35 A}{315 x^6 \sqrt {a-b x^4}}dx-\frac {1}{315} \sqrt {a-b x^4} \left (\frac {35 A}{x^9}+\frac {45 B}{x^7}+\frac {63 C}{x^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{315} b \int \frac {63 C x^4+45 B x^2+35 A}{x^6 \sqrt {a-b x^4}}dx-\frac {1}{315} \sqrt {a-b x^4} \left (\frac {35 A}{x^9}+\frac {45 B}{x^7}+\frac {63 C}{x^5}\right )\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle -\frac {2}{315} b \left (-\frac {\int -\frac {30 \left (7 (A b+3 a C) x^3+15 a B x\right )}{x^5 \sqrt {a-b x^4}}dx}{10 a}-\frac {7 A \sqrt {a-b x^4}}{a x^5}\right )-\frac {1}{315} \sqrt {a-b x^4} \left (\frac {35 A}{x^9}+\frac {45 B}{x^7}+\frac {63 C}{x^5}\right )\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle -\frac {2}{315} b \left (-\frac {\int -\frac {30 \left (7 (A b+3 a C) x^2+15 a B\right )}{x^4 \sqrt {a-b x^4}}dx}{10 a}-\frac {7 A \sqrt {a-b x^4}}{a x^5}\right )-\frac {1}{315} \sqrt {a-b x^4} \left (\frac {35 A}{x^9}+\frac {45 B}{x^7}+\frac {63 C}{x^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{315} b \left (\frac {3 \int \frac {7 (A b+3 a C) x^2+15 a B}{x^4 \sqrt {a-b x^4}}dx}{a}-\frac {7 A \sqrt {a-b x^4}}{a x^5}\right )-\frac {1}{315} \sqrt {a-b x^4} \left (\frac {35 A}{x^9}+\frac {45 B}{x^7}+\frac {63 C}{x^5}\right )\) |
| \(\Big \downarrow \) 1605 |
| \(\displaystyle -\frac {2}{315} b \left (\frac {3 \left (-\frac {\int -\frac {3 a \left (5 b B x^2+7 (A b+3 a C)\right )}{x^2 \sqrt {a-b x^4}}dx}{3 a}-\frac {5 B \sqrt {a-b x^4}}{x^3}\right )}{a}-\frac {7 A \sqrt {a-b x^4}}{a x^5}\right )-\frac {1}{315} \sqrt {a-b x^4} \left (\frac {35 A}{x^9}+\frac {45 B}{x^7}+\frac {63 C}{x^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{315} b \left (\frac {3 \left (\int \frac {5 b B x^2+7 (A b+3 a C)}{x^2 \sqrt {a-b x^4}}dx-\frac {5 B \sqrt {a-b x^4}}{x^3}\right )}{a}-\frac {7 A \sqrt {a-b x^4}}{a x^5}\right )-\frac {1}{315} \sqrt {a-b x^4} \left (\frac {35 A}{x^9}+\frac {45 B}{x^7}+\frac {63 C}{x^5}\right )\) |
| \(\Big \downarrow \) 1605 |
| \(\displaystyle -\frac {2}{315} b \left (\frac {3 \left (-\frac {\int -\frac {b \left (5 a B-7 (A b+3 a C) x^2\right )}{\sqrt {a-b x^4}}dx}{a}-\frac {7 \sqrt {a-b x^4} (3 a C+A b)}{a x}-\frac {5 B \sqrt {a-b x^4}}{x^3}\right )}{a}-\frac {7 A \sqrt {a-b x^4}}{a x^5}\right )-\frac {1}{315} \sqrt {a-b x^4} \left (\frac {35 A}{x^9}+\frac {45 B}{x^7}+\frac {63 C}{x^5}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2}{315} b \left (\frac {3 \left (\frac {\int \frac {b \left (5 a B-7 (A b+3 a C) x^2\right )}{\sqrt {a-b x^4}}dx}{a}-\frac {7 \sqrt {a-b x^4} (3 a C+A b)}{a x}-\frac {5 B \sqrt {a-b x^4}}{x^3}\right )}{a}-\frac {7 A \sqrt {a-b x^4}}{a x^5}\right )-\frac {1}{315} \sqrt {a-b x^4} \left (\frac {35 A}{x^9}+\frac {45 B}{x^7}+\frac {63 C}{x^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{315} b \left (\frac {3 \left (\frac {b \int \frac {5 a B-7 (A b+3 a C) x^2}{\sqrt {a-b x^4}}dx}{a}-\frac {7 \sqrt {a-b x^4} (3 a C+A b)}{a x}-\frac {5 B \sqrt {a-b x^4}}{x^3}\right )}{a}-\frac {7 A \sqrt {a-b x^4}}{a x^5}\right )-\frac {1}{315} \sqrt {a-b x^4} \left (\frac {35 A}{x^9}+\frac {45 B}{x^7}+\frac {63 C}{x^5}\right )\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle -\frac {2}{315} b \left (\frac {3 \left (\frac {b \left (\frac {\sqrt {a} \left (5 \sqrt {a} \sqrt {b} B+21 a C+7 A b\right ) \int \frac {1}{\sqrt {a-b x^4}}dx}{\sqrt {b}}-\frac {7 \sqrt {a} (3 a C+A b) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-b x^4}}dx}{\sqrt {b}}\right )}{a}-\frac {7 \sqrt {a-b x^4} (3 a C+A b)}{a x}-\frac {5 B \sqrt {a-b x^4}}{x^3}\right )}{a}-\frac {7 A \sqrt {a-b x^4}}{a x^5}\right )-\frac {1}{315} \sqrt {a-b x^4} \left (\frac {35 A}{x^9}+\frac {45 B}{x^7}+\frac {63 C}{x^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{315} b \left (\frac {3 \left (\frac {b \left (\frac {\sqrt {a} \left (5 \sqrt {a} \sqrt {b} B+21 a C+7 A b\right ) \int \frac {1}{\sqrt {a-b x^4}}dx}{\sqrt {b}}-\frac {7 (3 a C+A b) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx}{\sqrt {b}}\right )}{a}-\frac {7 \sqrt {a-b x^4} (3 a C+A b)}{a x}-\frac {5 B \sqrt {a-b x^4}}{x^3}\right )}{a}-\frac {7 A \sqrt {a-b x^4}}{a x^5}\right )-\frac {1}{315} \sqrt {a-b x^4} \left (\frac {35 A}{x^9}+\frac {45 B}{x^7}+\frac {63 C}{x^5}\right )\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle -\frac {2}{315} b \left (\frac {3 \left (\frac {b \left (\frac {\sqrt {a} \sqrt {1-\frac {b x^4}{a}} \left (5 \sqrt {a} \sqrt {b} B+21 a C+7 A b\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {b} \sqrt {a-b x^4}}-\frac {7 (3 a C+A b) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx}{\sqrt {b}}\right )}{a}-\frac {7 \sqrt {a-b x^4} (3 a C+A b)}{a x}-\frac {5 B \sqrt {a-b x^4}}{x^3}\right )}{a}-\frac {7 A \sqrt {a-b x^4}}{a x^5}\right )-\frac {1}{315} \sqrt {a-b x^4} \left (\frac {35 A}{x^9}+\frac {45 B}{x^7}+\frac {63 C}{x^5}\right )\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle -\frac {2}{315} b \left (\frac {3 \left (\frac {b \left (\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (5 \sqrt {a} \sqrt {b} B+21 a C+7 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^4}}-\frac {7 (3 a C+A b) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx}{\sqrt {b}}\right )}{a}-\frac {7 \sqrt {a-b x^4} (3 a C+A b)}{a x}-\frac {5 B \sqrt {a-b x^4}}{x^3}\right )}{a}-\frac {7 A \sqrt {a-b x^4}}{a x^5}\right )-\frac {1}{315} \sqrt {a-b x^4} \left (\frac {35 A}{x^9}+\frac {45 B}{x^7}+\frac {63 C}{x^5}\right )\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle -\frac {2}{315} b \left (\frac {3 \left (\frac {b \left (\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (5 \sqrt {a} \sqrt {b} B+21 a C+7 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^4}}-\frac {7 \sqrt {1-\frac {b x^4}{a}} (3 a C+A b) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {b} \sqrt {a-b x^4}}\right )}{a}-\frac {7 \sqrt {a-b x^4} (3 a C+A b)}{a x}-\frac {5 B \sqrt {a-b x^4}}{x^3}\right )}{a}-\frac {7 A \sqrt {a-b x^4}}{a x^5}\right )-\frac {1}{315} \sqrt {a-b x^4} \left (\frac {35 A}{x^9}+\frac {45 B}{x^7}+\frac {63 C}{x^5}\right )\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle -\frac {2}{315} b \left (\frac {3 \left (\frac {b \left (\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (5 \sqrt {a} \sqrt {b} B+21 a C+7 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^4}}-\frac {7 \sqrt {a} \sqrt {1-\frac {b x^4}{a}} (3 a C+A b) \int \frac {\sqrt {\frac {\sqrt {b} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}}dx}{\sqrt {b} \sqrt {a-b x^4}}\right )}{a}-\frac {7 \sqrt {a-b x^4} (3 a C+A b)}{a x}-\frac {5 B \sqrt {a-b x^4}}{x^3}\right )}{a}-\frac {7 A \sqrt {a-b x^4}}{a x^5}\right )-\frac {1}{315} \sqrt {a-b x^4} \left (\frac {35 A}{x^9}+\frac {45 B}{x^7}+\frac {63 C}{x^5}\right )\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {2}{315} b \left (\frac {3 \left (\frac {b \left (\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (5 \sqrt {a} \sqrt {b} B+21 a C+7 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^4}}-\frac {7 a^{3/4} \sqrt {1-\frac {b x^4}{a}} (3 a C+A b) E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{b^{3/4} \sqrt {a-b x^4}}\right )}{a}-\frac {7 \sqrt {a-b x^4} (3 a C+A b)}{a x}-\frac {5 B \sqrt {a-b x^4}}{x^3}\right )}{a}-\frac {7 A \sqrt {a-b x^4}}{a x^5}\right )-\frac {1}{315} \sqrt {a-b x^4} \left (\frac {35 A}{x^9}+\frac {45 B}{x^7}+\frac {63 C}{x^5}\right )\) |
Input:
Int[(Sqrt[a - b*x^4]*(A + B*x^2 + C*x^4))/x^10,x]
Output:
-1/315*(((35*A)/x^9 + (45*B)/x^7 + (63*C)/x^5)*Sqrt[a - b*x^4]) - (2*b*((- 7*A*Sqrt[a - b*x^4])/(a*x^5) + (3*((-5*B*Sqrt[a - b*x^4])/x^3 - (7*(A*b + 3*a*C)*Sqrt[a - b*x^4])/(a*x) + (b*((-7*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)*(7*A*b + 5*Sqrt[a]*Sqrt[b]*B + 21*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])))/a) )/a))/315
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[d*(f*x)^(m + 1)*((a + c*x^4)^(p + 1)/(a*f*(m + 1))), x] + S imp[1/(a*f^2*(m + 1)) Int[(f*x)^(m + 2)*(a + c*x^4)^p*(a*e*(m + 1) - c*d* (m + 4*p + 5)*x^2), x], x] /; FreeQ[{a, c, d, e, f, p}, x] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Module[{u = IntHide[x^m*Pq, x]}, Simp[u*(a + b*x^n)^p, x] - Simp[b*n*p Int[x^(m + n)*(a + b*x^n)^(p - 1)*ExpandToSum[u/x^(m + 1), x], x], x]] /; FreeQ[{a, b} , x] && PolyQ[Pq, x] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m + Expon[Pq, x] + 1 , 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 = 4.14 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(-\frac {\sqrt {-b \,x^{4}+a}\, \left (-42 A \,b^{2} x^{8}-126 C a b \,x^{8}-30 B b \,x^{6} a -14 A a b \,x^{4}+63 C \,a^{2} x^{4}+45 B \,a^{2} x^{2}+35 a^{2} A \right )}{315 x^{9} a^{2}}+\frac {2 b^{2} \left (-\frac {\left (7 A b +21 C a \right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, \sqrt {b}}-\frac {5 B a \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )}{105 a^{2}}\) | \(249\) |
| elliptic | \(-\frac {A \sqrt {-b \,x^{4}+a}}{9 x^{9}}-\frac {B \sqrt {-b \,x^{4}+a}}{7 x^{7}}+\frac {\left (2 A b -9 C a \right ) \sqrt {-b \,x^{4}+a}}{45 a \,x^{5}}+\frac {2 b B \sqrt {-b \,x^{4}+a}}{21 a \,x^{3}}+\frac {2 b \left (A b +3 C a \right ) \sqrt {-b \,x^{4}+a}}{15 a^{2} x}-\frac {2 B \,b^{2} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{21 a \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {2 b^{\frac {3}{2}} \left (A b +3 C a \right ) \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 a^{\frac {3}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\) | \(274\) |
| default | \(A \left (-\frac {\sqrt {-b \,x^{4}+a}}{9 x^{9}}+\frac {2 b \sqrt {-b \,x^{4}+a}}{45 a \,x^{5}}+\frac {2 b^{2} \sqrt {-b \,x^{4}+a}}{15 a^{2} x}-\frac {2 b^{\frac {5}{2}} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 a^{\frac {3}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )+B \left (-\frac {\sqrt {-b \,x^{4}+a}}{7 x^{7}}+\frac {2 b \sqrt {-b \,x^{4}+a}}{21 a \,x^{3}}-\frac {2 b^{2} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{21 a \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )+C \left (-\frac {\sqrt {-b \,x^{4}+a}}{5 x^{5}}+\frac {2 b \sqrt {-b \,x^{4}+a}}{5 a x}-\frac {2 b^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {a}\, \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )\) | \(378\) |
Input:
int((-b*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/x^10,x,method=_RETURNVERBOSE)
Output:
-1/315*(-b*x^4+a)^(1/2)*(-42*A*b^2*x^8-126*C*a*b*x^8-30*B*a*b*x^6-14*A*a*b *x^4+63*C*a^2*x^4+45*B*a^2*x^2+35*A*a^2)/x^9/a^2+2/105*b^2/a^2*(-(7*A*b+21 *C*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))-5*B*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))
Time = 0.09 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {a-b x^4} \left (A+B x^2+C x^4\right )}{x^{10}} \, dx=\frac {42 \, {\left (3 \, C a b + A b^{2}\right )} \sqrt {a} x^{9} \left (\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - 6 \, {\left ({\left (5 \, B + 21 \, C\right )} a b + 7 \, A b^{2}\right )} \sqrt {a} x^{9} \left (\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (30 \, B a b x^{6} + 42 \, {\left (3 \, C a b + A b^{2}\right )} x^{8} - 45 \, B a^{2} x^{2} - 7 \, {\left (9 \, C a^{2} - 2 \, A a b\right )} x^{4} - 35 \, A a^{2}\right )} \sqrt {-b x^{4} + a}}{315 \, a^{2} x^{9}} \] Input:
integrate((-b*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/x^10,x, algorithm="fricas")
Output:
1/315*(42*(3*C*a*b + A*b^2)*sqrt(a)*x^9*(b/a)^(3/4)*elliptic_e(arcsin(x*(b /a)^(1/4)), -1) - 6*((5*B + 21*C)*a*b + 7*A*b^2)*sqrt(a)*x^9*(b/a)^(3/4)*e lliptic_f(arcsin(x*(b/a)^(1/4)), -1) + (30*B*a*b*x^6 + 42*(3*C*a*b + A*b^2 )*x^8 - 45*B*a^2*x^2 - 7*(9*C*a^2 - 2*A*a*b)*x^4 - 35*A*a^2)*sqrt(-b*x^4 + a))/(a^2*x^9)
Time = 2.18 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {a-b x^4} \left (A+B x^2+C x^4\right )}{x^{10}} \, dx=\frac {i A \sqrt {b} \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {a}{b x^{4}}} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} + \frac {B \sqrt {a} \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} + \frac {i C \sqrt {b} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {a}{b x^{4}}} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} \] Input:
integrate((-b*x**4+a)**(1/2)*(C*x**4+B*x**2+A)/x**10,x)
Output:
I*A*sqrt(b)*gamma(-7/4)*hyper((-1/2, 7/4), (11/4,), a/(b*x**4))/(4*x**7*ga mma(-3/4)) + B*sqrt(a)*gamma(-7/4)*hyper((-7/4, -1/2), (-3/4,), b*x**4*exp _polar(2*I*pi)/a)/(4*x**7*gamma(-3/4)) + I*C*sqrt(b)*gamma(-3/4)*hyper((-1 /2, 3/4), (7/4,), a/(b*x**4))/(4*x**3*gamma(1/4))
\[ \int \frac {\sqrt {a-b x^4} \left (A+B x^2+C x^4\right )}{x^{10}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} \sqrt {-b x^{4} + a}}{x^{10}} \,d x } \] Input:
integrate((-b*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/x^10,x, algorithm="maxima")
Output:
integrate((C*x^4 + B*x^2 + A)*sqrt(-b*x^4 + a)/x^10, x)
\[ \int \frac {\sqrt {a-b x^4} \left (A+B x^2+C x^4\right )}{x^{10}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} \sqrt {-b x^{4} + a}}{x^{10}} \,d x } \] Input:
integrate((-b*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/x^10,x, algorithm="giac")
Output:
integrate((C*x^4 + B*x^2 + A)*sqrt(-b*x^4 + a)/x^10, x)
Timed out. \[ \int \frac {\sqrt {a-b x^4} \left (A+B x^2+C x^4\right )}{x^{10}} \, dx=\int \frac {\sqrt {a-b\,x^4}\,\left (C\,x^4+B\,x^2+A\right )}{x^{10}} \,d x \] Input:
int(((a - b*x^4)^(1/2)*(A + B*x^2 + C*x^4))/x^10,x)
Output:
int(((a - b*x^4)^(1/2)*(A + B*x^2 + C*x^4))/x^10, x)
\[ \int \frac {\sqrt {a-b x^4} \left (A+B x^2+C x^4\right )}{x^{10}} \, dx=\frac {-15 \sqrt {-b \,x^{4}+a}\, a b -10 \sqrt {-b \,x^{4}+a}\, a c -21 \sqrt {-b \,x^{4}+a}\, b^{2} x^{2}-35 \sqrt {-b \,x^{4}+a}\, b c \,x^{4}-30 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,x^{14}+a \,x^{10}}d x \right ) a^{2} b \,x^{9}-90 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,x^{14}+a \,x^{10}}d x \right ) a^{2} c \,x^{9}-42 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,x^{12}+a \,x^{8}}d x \right ) a \,b^{2} x^{9}}{105 b \,x^{9}} \] Input:
int((-b*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/x^10,x)
Output:
( - 15*sqrt(a - b*x**4)*a*b - 10*sqrt(a - b*x**4)*a*c - 21*sqrt(a - b*x**4 )*b**2*x**2 - 35*sqrt(a - b*x**4)*b*c*x**4 - 30*int(sqrt(a - b*x**4)/(a*x* *10 - b*x**14),x)*a**2*b*x**9 - 90*int(sqrt(a - b*x**4)/(a*x**10 - b*x**14 ),x)*a**2*c*x**9 - 42*int(sqrt(a - b*x**4)/(a*x**8 - b*x**12),x)*a*b**2*x* *9)/(105*b*x**9)