Integrand size = 30, antiderivative size = 360 \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{9/2} \left (a-b x^2\right )} \, dx=-\frac {2 \sqrt {c-d x^2}}{7 a e (e x)^{7/2}}-\frac {2 (7 b c-2 a d) \sqrt {c-d x^2}}{21 a^2 c e^3 (e x)^{3/2}}+\frac {2 d^{3/4} (7 b c-2 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{21 a^2 c^{3/4} e^{9/2} \sqrt {c-d x^2}}+\frac {b \sqrt [4]{c} (b c-a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^3 \sqrt [4]{d} e^{9/2} \sqrt {c-d x^2}}+\frac {b \sqrt [4]{c} (b c-a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^3 \sqrt [4]{d} e^{9/2} \sqrt {c-d x^2}} \] Output:
-2/7*(-d*x^2+c)^(1/2)/a/e/(e*x)^(7/2)-2/21*(-2*a*d+7*b*c)*(-d*x^2+c)^(1/2) /a^2/c/e^3/(e*x)^(3/2)+2/21*d^(3/4)*(-2*a*d+7*b*c)*(1-d*x^2/c)^(1/2)*Ellip ticF(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/a^2/c^(3/4)/e^(9/2)/(-d*x^2+c) ^(1/2)+b*c^(1/4)*(-a*d+b*c)*(1-d*x^2/c)^(1/2)*EllipticPi(d^(1/4)*(e*x)^(1/ 2)/c^(1/4)/e^(1/2),-b^(1/2)*c^(1/2)/a^(1/2)/d^(1/2),I)/a^3/d^(1/4)/e^(9/2) /(-d*x^2+c)^(1/2)+b*c^(1/4)*(-a*d+b*c)*(1-d*x^2/c)^(1/2)*EllipticPi(d^(1/4 )*(e*x)^(1/2)/c^(1/4)/e^(1/2),b^(1/2)*c^(1/2)/a^(1/2)/d^(1/2),I)/a^3/d^(1/ 4)/e^(9/2)/(-d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.22 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.54 \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{9/2} \left (a-b x^2\right )} \, dx=-\frac {2 \sqrt {e x} \left (5 a \left (c-d x^2\right ) \left (3 a c+7 b c x^2-2 a d x^2\right )+5 \left (-21 b^2 c^2+14 a b c d+2 a^2 d^2\right ) x^4 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+b d (7 b c-2 a d) x^6 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{105 a^3 c e^5 x^4 \sqrt {c-d x^2}} \] Input:
Integrate[Sqrt[c - d*x^2]/((e*x)^(9/2)*(a - b*x^2)),x]
Output:
(-2*Sqrt[e*x]*(5*a*(c - d*x^2)*(3*a*c + 7*b*c*x^2 - 2*a*d*x^2) + 5*(-21*b^ 2*c^2 + 14*a*b*c*d + 2*a^2*d^2)*x^4*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + b*d*(7*b*c - 2*a*d)*x^6*Sqrt[1 - (d*x^2)/ c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(105*a^3*c*e^5*x^4*S qrt[c - d*x^2])
Time = 0.83 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.06, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {368, 27, 975, 27, 1053, 25, 1021, 765, 762, 925, 27, 1543, 1542}
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.
| \(\displaystyle \int \frac {\sqrt {c-d x^2}}{(e x)^{9/2} \left (a-b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {\sqrt {c-d x^2}}{e^2 x^4 \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e \int \frac {\sqrt {c-d x^2}}{e^4 x^4 \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}\) |
| \(\Big \downarrow \) 975 |
| \(\displaystyle 2 e \left (\frac {\int \frac {(7 b c-2 a d) e^2-5 b d e^2 x^2}{e^4 x^2 \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{7 a e^2}-\frac {\sqrt {c-d x^2}}{7 a e^2 (e x)^{7/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e \left (\frac {\int \frac {(7 b c-2 a d) e^2-5 b d e^2 x^2}{e^2 x^2 \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{7 a e^4}-\frac {\sqrt {c-d x^2}}{7 a e^2 (e x)^{7/2}}\right )\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle 2 e \left (\frac {-\frac {\int -\frac {\left (21 b^2 c^2-14 a b d c-2 a^2 d^2\right ) e^2-b d (7 b c-2 a d) e^2 x^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (7 b c-2 a d)}{3 a c (e x)^{3/2}}}{7 a e^4}-\frac {\sqrt {c-d x^2}}{7 a e^2 (e x)^{7/2}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 e \left (\frac {\frac {\int \frac {\left (21 b^2 c^2-14 a b d c-2 a^2 d^2\right ) e^2-b d (7 b c-2 a d) e^2 x^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (7 b c-2 a d)}{3 a c (e x)^{3/2}}}{7 a e^4}-\frac {\sqrt {c-d x^2}}{7 a e^2 (e x)^{7/2}}\right )\) |
| \(\Big \downarrow \) 1021 |
| \(\displaystyle 2 e \left (\frac {\frac {21 b c e^2 (b c-a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}+d (7 b c-2 a d) \int \frac {1}{\sqrt {c-d x^2}}d\sqrt {e x}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (7 b c-2 a d)}{3 a c (e x)^{3/2}}}{7 a e^4}-\frac {\sqrt {c-d x^2}}{7 a e^2 (e x)^{7/2}}\right )\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle 2 e \left (\frac {\frac {21 b c e^2 (b c-a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}+\frac {d \sqrt {1-\frac {d x^2}{c}} (7 b c-2 a d) \int \frac {1}{\sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (7 b c-2 a d)}{3 a c (e x)^{3/2}}}{7 a e^4}-\frac {\sqrt {c-d x^2}}{7 a e^2 (e x)^{7/2}}\right )\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle 2 e \left (\frac {\frac {21 b c e^2 (b c-a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (7 b c-2 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (7 b c-2 a d)}{3 a c (e x)^{3/2}}}{7 a e^4}-\frac {\sqrt {c-d x^2}}{7 a e^2 (e x)^{7/2}}\right )\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle 2 e \left (\frac {\frac {21 b c e^2 (b c-a d) \left (\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}+\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}\right )+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (7 b c-2 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (7 b c-2 a d)}{3 a c (e x)^{3/2}}}{7 a e^4}-\frac {\sqrt {c-d x^2}}{7 a e^2 (e x)^{7/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e \left (\frac {\frac {21 b c e^2 (b c-a d) \left (\frac {\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}+\frac {\int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}\right )+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (7 b c-2 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (7 b c-2 a d)}{3 a c (e x)^{3/2}}}{7 a e^4}-\frac {\sqrt {c-d x^2}}{7 a e^2 (e x)^{7/2}}\right )\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle 2 e \left (\frac {\frac {21 b c e^2 (b c-a d) \left (\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}+\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}\right )+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (7 b c-2 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (7 b c-2 a d)}{3 a c (e x)^{3/2}}}{7 a e^4}-\frac {\sqrt {c-d x^2}}{7 a e^2 (e x)^{7/2}}\right )\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle 2 e \left (\frac {\frac {\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (7 b c-2 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}+21 b c e^2 (b c-a d) \left (\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}\right )}{3 a c e^2}-\frac {\sqrt {c-d x^2} (7 b c-2 a d)}{3 a c (e x)^{3/2}}}{7 a e^4}-\frac {\sqrt {c-d x^2}}{7 a e^2 (e x)^{7/2}}\right )\) |
Input:
Int[Sqrt[c - d*x^2]/((e*x)^(9/2)*(a - b*x^2)),x]
Output:
2*e*(-1/7*Sqrt[c - d*x^2]/(a*e^2*(e*x)^(7/2)) + (-1/3*((7*b*c - 2*a*d)*Sqr t[c - d*x^2])/(a*c*(e*x)^(3/2)) + ((c^(1/4)*d^(3/4)*(7*b*c - 2*a*d)*Sqrt[e ]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e ])], -1])/Sqrt[c - d*x^2] + 21*b*c*(b*c - a*d)*e^2*((c^(1/4)*Sqrt[1 - (d*x ^2)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)* Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a*d^(1/4)*e^(3/2)*Sqrt[c - d*x^2]) + (c^(1/4)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[ d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a*d^(1/4)*e^(3 /2)*Sqrt[c - d*x^2])))/(3*a*c*e^2))/(7*a*e^4))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m ] && IntegerQ[p]
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[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Simp[ 1/(2*c) Int[1/(Sqrt[a + b*x^4]*(1 - Rt[-d/c, 2]*x^2)), x], x] + Simp[1/(2 *c) Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) )^(q_), x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/ (a*e*(m + 1))), x] - Simp[1/(a*e^n*(m + 1)) Int[(e*x)^(m + n)*(a + b*x^n) ^p*(c + d*x^n)^(q - 1)*Simp[c*b*(m + 1) + n*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) + b*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomi alQ[a, b, c, d, e, m, n, p, q, x]
Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x _)^(n_)]), x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^n], x], x] + Simp[(b* e - a*f)/b Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ ))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b *x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 ) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && LtQ[m, -1]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x ], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a) ]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && !GtQ[a, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(887\) vs. \(2(276)=552\).
Time = 1.46 (sec) , antiderivative size = 888, normalized size of antiderivative = 2.47
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (-x^{2} d +c \right ) e x}\, \left (-\frac {2 \sqrt {-d e \,x^{3}+c e x}}{7 e^{5} a \,x^{4}}+\frac {2 \left (2 a d -7 b c \right ) \sqrt {-d e \,x^{3}+c e x}}{21 e^{5} c \,a^{2} x^{2}}-\frac {2 \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) d}{21 c \,e^{4} a \sqrt {-d e \,x^{3}+c e x}}+\frac {\sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) b}{3 e^{4} a^{2} \sqrt {-d e \,x^{3}+c e x}}+\frac {b \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 e^{4} a \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {b^{2} \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) c}{2 e^{4} a^{2} \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {b \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 e^{4} a \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}+\frac {b^{2} \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) c}{2 e^{4} a^{2} \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{\sqrt {e x}\, \sqrt {-x^{2} d +c}}\) | \(888\) |
| default | \(\text {Expression too large to display}\) | \(1374\) |
Input:
int((-d*x^2+c)^(1/2)/(e*x)^(9/2)/(-b*x^2+a),x,method=_RETURNVERBOSE)
Output:
((-d*x^2+c)*e*x)^(1/2)/(e*x)^(1/2)/(-d*x^2+c)^(1/2)*(-2/7/e^5/a*(-d*e*x^3+ c*e*x)^(1/2)/x^4+2/21/e^5/c*(2*a*d-7*b*c)/a^2*(-d*e*x^3+c*e*x)^(1/2)/x^2-2 /21/c/e^4/a*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1 /2)*(-x*d/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+1/d*(c*d )^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*d+1/3/e^4/a^2*(c*d)^(1/2)*(1+x* d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)/ (-d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2) ,1/2*2^(1/2))*b+1/2/e^4/a*b/(a*b)^(1/2)*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^(1 /2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^ (1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2))*EllipticPi(((x+1/d*(c*d)^(1/2))*d /(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2)),1/ 2*2^(1/2))-1/2/e^4/a^2*b^2/(a*b)^(1/2)/d*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^( 1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x) ^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2))*EllipticPi(((x+1/d*(c*d)^(1/2))* d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2)),1 /2*2^(1/2))*c-1/2/e^4/a*b/(a*b)^(1/2)*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^(1/2 )*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1 /2)/(-1/d*(c*d)^(1/2)+1/b*(a*b)^(1/2))*EllipticPi(((x+1/d*(c*d)^(1/2))*d/( c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1/d*(c*d)^(1/2)+1/b*(a*b)^(1/2)),1/2* 2^(1/2))+1/2/e^4/a^2*b^2/(a*b)^(1/2)/d*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^...
Timed out. \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{9/2} \left (a-b x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((-d*x^2+c)^(1/2)/(e*x)^(9/2)/(-b*x^2+a),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\sqrt {c-d x^2}}{(e x)^{9/2} \left (a-b x^2\right )} \, dx=- \int \frac {\sqrt {c - d x^{2}}}{- a \left (e x\right )^{\frac {9}{2}} + b x^{2} \left (e x\right )^{\frac {9}{2}}}\, dx \] Input:
integrate((-d*x**2+c)**(1/2)/(e*x)**(9/2)/(-b*x**2+a),x)
Output:
-Integral(sqrt(c - d*x**2)/(-a*(e*x)**(9/2) + b*x**2*(e*x)**(9/2)), x)
\[ \int \frac {\sqrt {c-d x^2}}{(e x)^{9/2} \left (a-b x^2\right )} \, dx=\int { -\frac {\sqrt {-d x^{2} + c}}{{\left (b x^{2} - a\right )} \left (e x\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((-d*x^2+c)^(1/2)/(e*x)^(9/2)/(-b*x^2+a),x, algorithm="maxima")
Output:
-integrate(sqrt(-d*x^2 + c)/((b*x^2 - a)*(e*x)^(9/2)), x)
\[ \int \frac {\sqrt {c-d x^2}}{(e x)^{9/2} \left (a-b x^2\right )} \, dx=\int { -\frac {\sqrt {-d x^{2} + c}}{{\left (b x^{2} - a\right )} \left (e x\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((-d*x^2+c)^(1/2)/(e*x)^(9/2)/(-b*x^2+a),x, algorithm="giac")
Output:
integrate(-sqrt(-d*x^2 + c)/((b*x^2 - a)*(e*x)^(9/2)), x)
Timed out. \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{9/2} \left (a-b x^2\right )} \, dx=\int \frac {\sqrt {c-d\,x^2}}{{\left (e\,x\right )}^{9/2}\,\left (a-b\,x^2\right )} \,d x \] Input:
int((c - d*x^2)^(1/2)/((e*x)^(9/2)*(a - b*x^2)),x)
Output:
int((c - d*x^2)^(1/2)/((e*x)^(9/2)*(a - b*x^2)), x)
\[ \int \frac {\sqrt {c-d x^2}}{(e x)^{9/2} \left (a-b x^2\right )} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {-d \,x^{2}+c}}{\sqrt {x}\, a \,x^{4}-\sqrt {x}\, b \,x^{6}}d x \right )}{e^{5}} \] Input:
int((-d*x^2+c)^(1/2)/(e*x)^(9/2)/(-b*x^2+a),x)
Output:
(sqrt(e)*int(sqrt(c - d*x**2)/(sqrt(x)*a*x**4 - sqrt(x)*b*x**6),x))/e**5