Integrand size = 30, antiderivative size = 397 \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=-\frac {d}{c (b c-a d) e (e x)^{3/2} \sqrt {c-d x^2}}-\frac {(2 b c-5 a d) \sqrt {c-d x^2}}{3 a c^2 (b c-a d) e (e x)^{3/2}}+\frac {d^{3/4} (2 b c-5 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{3 a c^{7/4} (b c-a d) e^{5/2} \sqrt {c-d x^2}}+\frac {b^2 \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^2 \sqrt [4]{d} (b c-a d) e^{5/2} \sqrt {c-d x^2}}+\frac {b^2 \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^2 \sqrt [4]{d} (b c-a d) e^{5/2} \sqrt {c-d x^2}} \] Output:
-d/c/(-a*d+b*c)/e/(e*x)^(3/2)/(-d*x^2+c)^(1/2)-1/3*(-5*a*d+2*b*c)*(-d*x^2+ c)^(1/2)/a/c^2/(-a*d+b*c)/e/(e*x)^(3/2)+1/3*d^(3/4)*(-5*a*d+2*b*c)*(1-d*x^ 2/c)^(1/2)*EllipticF(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/a/c^(7/4)/(-a* d+b*c)/e^(5/2)/(-d*x^2+c)^(1/2)+b^2*c^(1/4)*(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^2 /d^(1/4)/(-a*d+b*c)/e^(5/2)/(-d*x^2+c)^(1/2)+b^2*c^(1/4)*(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^2/d^(1/4)/(-a*d+b*c)/e^(5/2)/(-d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.18 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.50 \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\frac {x \left (5 a \left (a d \left (2 c-5 d x^2\right )-2 b c \left (c-d x^2\right )\right )+5 \left (6 b^2 c^2+2 a b c d-5 a^2 d^2\right ) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+b d (-2 b c+5 a d) x^4 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{15 a^2 c^2 (b c-a d) (e x)^{5/2} \sqrt {c-d x^2}} \] Input:
Integrate[1/((e*x)^(5/2)*(a - b*x^2)*(c - d*x^2)^(3/2)),x]
Output:
(x*(5*a*(a*d*(2*c - 5*d*x^2) - 2*b*c*(c - d*x^2)) + 5*(6*b^2*c^2 + 2*a*b*c *d - 5*a^2*d^2)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2) /c, (b*x^2)/a] + b*d*(-2*b*c + 5*a*d)*x^4*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4 , 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(15*a^2*c^2*(b*c - a*d)*(e*x)^(5/2) *Sqrt[c - d*x^2])
Time = 0.85 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.01, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {368, 27, 972, 25, 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.
| \(\displaystyle \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {1}{x^2 \left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e \int \frac {1}{e^2 x^2 \left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}\) |
| \(\Big \downarrow \) 972 |
| \(\displaystyle 2 e \left (-\frac {\int -\frac {5 b d x^2 e^2+(2 b c-5 a d) e^2}{e^4 x^2 \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 c (b c-a d)}-\frac {d}{2 c e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 e \left (\frac {\int \frac {5 b d x^2 e^2+(2 b c-5 a d) e^2}{e^4 x^2 \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 c (b c-a d)}-\frac {d}{2 c e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e \left (\frac {\int \frac {5 b d x^2 e^2+(2 b c-5 a d) e^2}{e^2 x^2 \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 c e^2 (b c-a d)}-\frac {d}{2 c e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle 2 e \left (\frac {-\frac {\int -\frac {\left (6 b^2 c^2+2 a b d c-5 a^2 d^2\right ) e^2-b d (2 b c-5 a d) e^2 x^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (2 b c-5 a d)}{3 a c (e x)^{3/2}}}{2 c e^2 (b c-a d)}-\frac {d}{2 c e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 e \left (\frac {\frac {\int \frac {\left (6 b^2 c^2+2 a b d c-5 a^2 d^2\right ) e^2-b d (2 b c-5 a d) e^2 x^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (2 b c-5 a d)}{3 a c (e x)^{3/2}}}{2 c e^2 (b c-a d)}-\frac {d}{2 c e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1021 |
| \(\displaystyle 2 e \left (\frac {\frac {6 b^2 c^2 e^2 \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}+d (2 b c-5 a d) \int \frac {1}{\sqrt {c-d x^2}}d\sqrt {e x}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (2 b c-5 a d)}{3 a c (e x)^{3/2}}}{2 c e^2 (b c-a d)}-\frac {d}{2 c e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle 2 e \left (\frac {\frac {6 b^2 c^2 e^2 \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}+\frac {d \sqrt {1-\frac {d x^2}{c}} (2 b c-5 a d) \int \frac {1}{\sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (2 b c-5 a d)}{3 a c (e x)^{3/2}}}{2 c e^2 (b c-a d)}-\frac {d}{2 c e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle 2 e \left (\frac {\frac {6 b^2 c^2 e^2 \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (2 b c-5 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (2 b c-5 a d)}{3 a c (e x)^{3/2}}}{2 c e^2 (b c-a d)}-\frac {d}{2 c e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle 2 e \left (\frac {\frac {6 b^2 c^2 e^2 \left (\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}+\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}\right )+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (2 b c-5 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (2 b c-5 a d)}{3 a c (e x)^{3/2}}}{2 c e^2 (b c-a d)}-\frac {d}{2 c e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e \left (\frac {\frac {6 b^2 c^2 e^2 \left (\frac {\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}+\frac {\int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}\right )+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (2 b c-5 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (2 b c-5 a d)}{3 a c (e x)^{3/2}}}{2 c e^2 (b c-a d)}-\frac {d}{2 c e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle 2 e \left (\frac {\frac {6 b^2 c^2 e^2 \left (\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}+\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}\right )+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (2 b c-5 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (2 b c-5 a d)}{3 a c (e x)^{3/2}}}{2 c e^2 (b c-a d)}-\frac {d}{2 c e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle 2 e \left (\frac {\frac {6 b^2 c^2 e^2 \left (\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}\right )+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (2 b c-5 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (2 b c-5 a d)}{3 a c (e x)^{3/2}}}{2 c e^2 (b c-a d)}-\frac {d}{2 c e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}\right )\) |
Input:
Int[1/((e*x)^(5/2)*(a - b*x^2)*(c - d*x^2)^(3/2)),x]
Output:
2*e*(-1/2*d/(c*(b*c - a*d)*e^2*(e*x)^(3/2)*Sqrt[c - d*x^2]) + (-1/3*((2*b* c - 5*a*d)*Sqrt[c - d*x^2])/(a*c*(e*x)^(3/2)) + ((c^(1/4)*d^(3/4)*(2*b*c - 5*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] + 6*b^2*c^2*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))/(2*c*(b*c - a*d)*e^2))
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[(-b)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x ^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*( b*c - a*d)*(p + 1) + d*b*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{ a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] & & IntBinomialQ[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]
Time = 2.04 (sec) , antiderivative size = 619, normalized size of antiderivative = 1.56
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (-x^{2} d +c \right ) e x}\, \left (\frac {d^{2} x}{e^{2} c^{2} \left (a d -b c \right ) \sqrt {-\left (x^{2}-\frac {c}{d}\right ) d e x}}-\frac {2 \sqrt {-d e \,x^{3}+c e x}}{3 c^{2} e^{3} a \,x^{2}}+\frac {d \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {-d e \,x^{3}+c e x}\, c^{2} \left (a d -b c \right ) e^{2}}+\frac {\sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{3 \sqrt {-d e \,x^{3}+c e x}\, c^{2} e^{2} a}+\frac {b^{2} \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \left (a d -b c \right ) e^{2} a \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {b^{2} \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \left (a d -b c \right ) e^{2} a \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{\sqrt {e x}\, \sqrt {-x^{2} d +c}}\) | \(619\) |
| default | \(\frac {b d \left (3 \sqrt {2}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) b^{3} c^{3} x \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}+3 \sqrt {2}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{2} x \sqrt {c d}\, \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \sqrt {a b}-5 \sqrt {2}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} d^{2} x \sqrt {c d}\, \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \sqrt {a b}+7 \sqrt {2}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) a b c d x \sqrt {c d}\, \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \sqrt {a b}-2 \sqrt {2}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{2} x \sqrt {c d}\, \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \sqrt {a b}-3 \sqrt {2}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {a b}\, d +\sqrt {c d}\, b}, \frac {\sqrt {2}}{2}\right ) b^{3} c^{3} x \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}+3 \sqrt {2}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {a b}\, d +\sqrt {c d}\, b}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{2} x \sqrt {c d}\, \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \sqrt {a b}-10 a^{2} d^{3} x^{2} \sqrt {a b}+14 a b c \,d^{2} x^{2} \sqrt {a b}-4 b^{2} c^{2} d \,x^{2} \sqrt {a b}+4 a^{2} c \,d^{2} \sqrt {a b}-8 a b \,c^{2} d \sqrt {a b}+4 b^{2} c^{3} \sqrt {a b}\right )}{6 x \sqrt {-x^{2} d +c}\, c^{2} a \left (\sqrt {c d}\, b -\sqrt {a b}\, d \right ) \left (\sqrt {a b}\, d +\sqrt {c d}\, b \right ) \sqrt {a b}\, \left (a d -b c \right ) e^{2} \sqrt {e x}}\) | \(885\) |
Input:
int(1/(e*x)^(5/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
((-d*x^2+c)*e*x)^(1/2)/(e*x)^(1/2)/(-d*x^2+c)^(1/2)*(d^2/e^2*x/c^2/(a*d-b* c)/(-(x^2-c/d)*d*e*x)^(1/2)-2/3/c^2/e^3/a*(-d*e*x^3+c*e*x)^(1/2)/x^2+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)*EllipticF(((x+1/d*(c*d)^(1/2))*d/ (c*d)^(1/2))^(1/2),1/2*2^(1/2))/c^2/(a*d-b*c)/e^2+1/3*(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))/c^2/e^2/a+1/2*b^2/(a*d-b*c)/e^2/a/(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))-1/2*b^2/(a*d-b*c)/e^2/a/(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))*Ellip ticPi(((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)))
Timed out. \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x)^(5/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=- \int \frac {1}{- a c \left (e x\right )^{\frac {5}{2}} \sqrt {c - d x^{2}} + a d x^{2} \left (e x\right )^{\frac {5}{2}} \sqrt {c - d x^{2}} + b c x^{2} \left (e x\right )^{\frac {5}{2}} \sqrt {c - d x^{2}} - b d x^{4} \left (e x\right )^{\frac {5}{2}} \sqrt {c - d x^{2}}}\, dx \] Input:
integrate(1/(e*x)**(5/2)/(-b*x**2+a)/(-d*x**2+c)**(3/2),x)
Output:
-Integral(1/(-a*c*(e*x)**(5/2)*sqrt(c - d*x**2) + a*d*x**2*(e*x)**(5/2)*sq rt(c - d*x**2) + b*c*x**2*(e*x)**(5/2)*sqrt(c - d*x**2) - b*d*x**4*(e*x)** (5/2)*sqrt(c - d*x**2)), x)
\[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\int { -\frac {1}{{\left (b x^{2} - a\right )} {\left (-d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(e*x)^(5/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x, algorithm="maxima")
Output:
-integrate(1/((b*x^2 - a)*(-d*x^2 + c)^(3/2)*(e*x)^(5/2)), x)
\[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\int { -\frac {1}{{\left (b x^{2} - a\right )} {\left (-d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(e*x)^(5/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x, algorithm="giac")
Output:
integrate(-1/((b*x^2 - a)*(-d*x^2 + c)^(3/2)*(e*x)^(5/2)), x)
Timed out. \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{5/2}\,\left (a-b\,x^2\right )\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \] Input:
int(1/((e*x)^(5/2)*(a - b*x^2)*(c - d*x^2)^(3/2)),x)
Output:
int(1/((e*x)^(5/2)*(a - b*x^2)*(c - d*x^2)^(3/2)), x)
\[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\frac {\sqrt {e}\, \left (-2 \sqrt {-d \,x^{2}+c}+5 \sqrt {x}\, \left (\int \frac {\sqrt {-d \,x^{2}+c}}{\sqrt {x}\, a \,c^{2}-2 \sqrt {x}\, a c d \,x^{2}+\sqrt {x}\, a \,d^{2} x^{4}-\sqrt {x}\, b \,c^{2} x^{2}+2 \sqrt {x}\, b c d \,x^{4}-\sqrt {x}\, b \,d^{2} x^{6}}d x \right ) a c d x -5 \sqrt {x}\, \left (\int \frac {\sqrt {-d \,x^{2}+c}}{\sqrt {x}\, a \,c^{2}-2 \sqrt {x}\, a c d \,x^{2}+\sqrt {x}\, a \,d^{2} x^{4}-\sqrt {x}\, b \,c^{2} x^{2}+2 \sqrt {x}\, b c d \,x^{4}-\sqrt {x}\, b \,d^{2} x^{6}}d x \right ) a \,d^{2} x^{3}+3 \sqrt {x}\, \left (\int \frac {\sqrt {-d \,x^{2}+c}}{\sqrt {x}\, a \,c^{2}-2 \sqrt {x}\, a c d \,x^{2}+\sqrt {x}\, a \,d^{2} x^{4}-\sqrt {x}\, b \,c^{2} x^{2}+2 \sqrt {x}\, b c d \,x^{4}-\sqrt {x}\, b \,d^{2} x^{6}}d x \right ) b \,c^{2} x -3 \sqrt {x}\, \left (\int \frac {\sqrt {-d \,x^{2}+c}}{\sqrt {x}\, a \,c^{2}-2 \sqrt {x}\, a c d \,x^{2}+\sqrt {x}\, a \,d^{2} x^{4}-\sqrt {x}\, b \,c^{2} x^{2}+2 \sqrt {x}\, b c d \,x^{4}-\sqrt {x}\, b \,d^{2} x^{6}}d x \right ) b c d \,x^{3}-5 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}\, x}{-b \,d^{2} x^{6}+a \,d^{2} x^{4}+2 b c d \,x^{4}-2 a c d \,x^{2}-b \,c^{2} x^{2}+a \,c^{2}}d x \right ) b c d x +5 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}\, x}{-b \,d^{2} x^{6}+a \,d^{2} x^{4}+2 b c d \,x^{4}-2 a c d \,x^{2}-b \,c^{2} x^{2}+a \,c^{2}}d x \right ) b \,d^{2} x^{3}\right )}{3 \sqrt {x}\, a c \,e^{3} x \left (-d \,x^{2}+c \right )} \] Input:
int(1/(e*x)^(5/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x)
Output:
(sqrt(e)*( - 2*sqrt(c - d*x**2) + 5*sqrt(x)*int(sqrt(c - d*x**2)/(sqrt(x)* a*c**2 - 2*sqrt(x)*a*c*d*x**2 + sqrt(x)*a*d**2*x**4 - sqrt(x)*b*c**2*x**2 + 2*sqrt(x)*b*c*d*x**4 - sqrt(x)*b*d**2*x**6),x)*a*c*d*x - 5*sqrt(x)*int(s qrt(c - d*x**2)/(sqrt(x)*a*c**2 - 2*sqrt(x)*a*c*d*x**2 + sqrt(x)*a*d**2*x* *4 - sqrt(x)*b*c**2*x**2 + 2*sqrt(x)*b*c*d*x**4 - sqrt(x)*b*d**2*x**6),x)* a*d**2*x**3 + 3*sqrt(x)*int(sqrt(c - d*x**2)/(sqrt(x)*a*c**2 - 2*sqrt(x)*a *c*d*x**2 + sqrt(x)*a*d**2*x**4 - sqrt(x)*b*c**2*x**2 + 2*sqrt(x)*b*c*d*x* *4 - sqrt(x)*b*d**2*x**6),x)*b*c**2*x - 3*sqrt(x)*int(sqrt(c - d*x**2)/(sq rt(x)*a*c**2 - 2*sqrt(x)*a*c*d*x**2 + sqrt(x)*a*d**2*x**4 - sqrt(x)*b*c**2 *x**2 + 2*sqrt(x)*b*c*d*x**4 - sqrt(x)*b*d**2*x**6),x)*b*c*d*x**3 - 5*sqrt (x)*int((sqrt(x)*sqrt(c - d*x**2)*x)/(a*c**2 - 2*a*c*d*x**2 + a*d**2*x**4 - b*c**2*x**2 + 2*b*c*d*x**4 - b*d**2*x**6),x)*b*c*d*x + 5*sqrt(x)*int((sq rt(x)*sqrt(c - d*x**2)*x)/(a*c**2 - 2*a*c*d*x**2 + a*d**2*x**4 - b*c**2*x* *2 + 2*b*c*d*x**4 - b*d**2*x**6),x)*b*d**2*x**3))/(3*sqrt(x)*a*c*e**3*x*(c - d*x**2))