Integrand size = 30, antiderivative size = 493 \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=-\frac {d}{c (b c-a d) e \sqrt {e x} \sqrt {c-d x^2}}-\frac {(2 b c-3 a d) \sqrt {c-d x^2}}{a c^2 (b c-a d) e \sqrt {e x}}-\frac {\sqrt [4]{d} (2 b c-3 a d) \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a c^{5/4} (b c-a d) e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{d} (2 b c-3 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a c^{5/4} (b c-a d) e^{3/2} \sqrt {c-d x^2}}-\frac {b^{3/2} \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^{3/2} \sqrt [4]{d} (b c-a d) e^{3/2} \sqrt {c-d x^2}}+\frac {b^{3/2} \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^{3/2} \sqrt [4]{d} (b c-a d) e^{3/2} \sqrt {c-d x^2}} \] Output:
-d/c/(-a*d+b*c)/e/(e*x)^(1/2)/(-d*x^2+c)^(1/2)-(-3*a*d+2*b*c)*(-d*x^2+c)^( 1/2)/a/c^2/(-a*d+b*c)/e/(e*x)^(1/2)-d^(1/4)*(-3*a*d+2*b*c)*(1-d*x^2/c)^(1/ 2)*EllipticE(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/a/c^(5/4)/(-a*d+b*c)/e ^(3/2)/(-d*x^2+c)^(1/2)+d^(1/4)*(-3*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^(5/4)/(-a*d+b*c)/e^(3/2)/(-d*x ^2+c)^(1/2)-b^(3/2)*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^(3/2)/d^(1/4)/(-a *d+b*c)/e^(3/2)/(-d*x^2+c)^(1/2)+b^(3/2)*c^(1/4)*(1-d*x^2/c)^(1/2)*Ellipti cPi(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^(3/2)/d^(1/4)/(-a*d+b*c)/e^(3/2)/(-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 = 198, normalized size of antiderivative = 0.40 \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\frac {x \left (21 a \left (a d \left (2 c-3 d x^2\right )-2 b c \left (c-d x^2\right )\right )+7 \left (2 b^2 c^2-2 a b c d+3 a^2 d^2\right ) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 b d (2 b c-3 a d) x^4 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{21 a^2 c^2 (b c-a d) (e x)^{3/2} \sqrt {c-d x^2}} \] Input:
Integrate[1/((e*x)^(3/2)*(a - b*x^2)*(c - d*x^2)^(3/2)),x]
Output:
(x*(21*a*(a*d*(2*c - 3*d*x^2) - 2*b*c*(c - d*x^2)) + 7*(2*b^2*c^2 - 2*a*b* c*d + 3*a^2*d^2)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2 )/c, (b*x^2)/a] + 3*b*d*(2*b*c - 3*a*d)*x^4*Sqrt[1 - (d*x^2)/c]*AppellF1[7 /4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(21*a^2*c^2*(b*c - a*d)*(e*x)^(3 /2)*Sqrt[c - d*x^2])
Time = 0.92 (sec) , antiderivative size = 471, normalized size of antiderivative = 0.96, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {368, 27, 972, 25, 27, 1053, 25, 1054, 2009}
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.
| \(\displaystyle \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {e}{x \left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e \int \frac {1}{e x \left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}\) |
| \(\Big \downarrow \) 972 |
| \(\displaystyle 2 e \left (-\frac {\int -\frac {3 b d x^2 e^2+(2 b c-3 a d) e^2}{e^3 x \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 c (b c-a d)}-\frac {d}{2 c e^2 \sqrt {e x} \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 e \left (\frac {\int \frac {3 b d x^2 e^2+(2 b c-3 a d) e^2}{e^3 x \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 c (b c-a d)}-\frac {d}{2 c e^2 \sqrt {e x} \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e \left (\frac {\int \frac {3 b d x^2 e^2+(2 b c-3 a d) e^2}{e x \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 c e^2 (b c-a d)}-\frac {d}{2 c e^2 \sqrt {e x} \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle 2 e \left (\frac {-\frac {\int -\frac {e x \left (b d (2 b c-3 a d) x^2 e^2+\left (2 b^2 c^2-2 a b d c+3 a^2 d^2\right ) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{a c e^2}-\frac {\sqrt {c-d x^2} (2 b c-3 a d)}{a c \sqrt {e x}}}{2 c e^2 (b c-a d)}-\frac {d}{2 c e^2 \sqrt {e x} \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 e \left (\frac {\frac {\int \frac {e x \left (b d (2 b c-3 a d) x^2 e^2+\left (2 b^2 c^2-2 a b d c+3 a^2 d^2\right ) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{a c e^2}-\frac {\sqrt {c-d x^2} (2 b c-3 a d)}{a c \sqrt {e x}}}{2 c e^2 (b c-a d)}-\frac {d}{2 c e^2 \sqrt {e x} \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle 2 e \left (\frac {\frac {\int \left (\frac {2 b^2 c^2 e^3 x}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}-\frac {d (2 b c-3 a d) e x}{\sqrt {c-d x^2}}\right )d\sqrt {e x}}{a c e^2}-\frac {\sqrt {c-d x^2} (2 b c-3 a d)}{a c \sqrt {e x}}}{2 c e^2 (b c-a d)}-\frac {d}{2 c e^2 \sqrt {e x} \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 e \left (\frac {\frac {-\frac {b^{3/2} c^{9/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {a} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {b^{3/2} c^{9/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {a} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (2 b c-3 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}-\frac {c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (2 b c-3 a d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {c-d x^2}}}{a c e^2}-\frac {\sqrt {c-d x^2} (2 b c-3 a d)}{a c \sqrt {e x}}}{2 c e^2 (b c-a d)}-\frac {d}{2 c e^2 \sqrt {e x} \sqrt {c-d x^2} (b c-a d)}\right )\) |
Input:
Int[1/((e*x)^(3/2)*(a - b*x^2)*(c - d*x^2)^(3/2)),x]
Output:
2*e*(-1/2*d/(c*(b*c - a*d)*e^2*Sqrt[e*x]*Sqrt[c - d*x^2]) + (-(((2*b*c - 3 *a*d)*Sqrt[c - d*x^2])/(a*c*Sqrt[e*x])) + (-((c^(3/4)*d^(1/4)*(2*b*c - 3*a *d)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1 /4)*Sqrt[e])], -1])/Sqrt[c - d*x^2]) + (c^(3/4)*d^(1/4)*(2*b*c - 3*a*d)*e^ (3/2)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sq rt[e])], -1])/Sqrt[c - d*x^2] - (b^(3/2)*c^(9/4)*e^(3/2)*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])/(Sqrt[a]*d^(1/4)*Sqrt[c - d*x^2]) + (b^(3/ 2)*c^(9/4)*e^(3/2)*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])/(Sqrt[a]* d^(1/4)*Sqrt[c - d*x^2]))/(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[((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[((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[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n _)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(786\) vs. \(2(389)=778\).
Time = 1.71 (sec) , antiderivative size = 787, normalized size of antiderivative = 1.60
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (-x^{2} d +c \right ) e x}\, \left (\frac {d^{2} x^{2}}{e \,c^{2} \left (a d -b c \right ) \sqrt {-\left (x^{2}-\frac {c}{d}\right ) d e x}}-\frac {2 \left (-d e \,x^{2}+c e \right )}{c^{2} e^{2} a \sqrt {x \left (-d e \,x^{2}+c e \right )}}+\frac {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 {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {-d e \,x^{3}+c e x}\, \left (a d -b c \right ) e}-\frac {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 c \sqrt {-d e \,x^{3}+c e x}\, \left (a d -b c \right ) e}+\frac {2 \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 {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {-d e \,x^{3}+c e x}\, e a}-\frac {\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 )}{c \sqrt {-d e \,x^{3}+c e x}\, e a}+\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 a \left (a d -b c \right ) e 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 a \left (a d -b c \right ) e 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}}\) | \(787\) |
| default | \(\text {Expression too large to display}\) | \(1047\) |
Input:
int(1/(e*x)^(3/2)/(-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*x^2/c^2/(a*d-b* c)/(-(x^2-c/d)*d*e*x)^(1/2)-2*(-d*e*x^2+c*e)/c^2/e^2/a/(x*(-d*e*x^2+c*e))^ (1/2)+1/c*d*(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)/(a*d-b*c)/e*EllipticE(((x+1/d*(c*d )^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-1/2/c*d*(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)/(a*d-b*c)/e*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2 ^(1/2))+2/c*(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)/e/a*EllipticE(((x+1/d*(c*d)^(1/2)) *d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-1/c*(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)/e/a*Ell ipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+1/2*b/a/(a*d -b*c)/e/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/a/(a*d-b*c)/e /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)))
Timed out. \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x)^(3/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=- \int \frac {1}{- a c \left (e x\right )^{\frac {3}{2}} \sqrt {c - d x^{2}} + a d x^{2} \left (e x\right )^{\frac {3}{2}} \sqrt {c - d x^{2}} + b c x^{2} \left (e x\right )^{\frac {3}{2}} \sqrt {c - d x^{2}} - b d x^{4} \left (e x\right )^{\frac {3}{2}} \sqrt {c - d x^{2}}}\, dx \] Input:
integrate(1/(e*x)**(3/2)/(-b*x**2+a)/(-d*x**2+c)**(3/2),x)
Output:
-Integral(1/(-a*c*(e*x)**(3/2)*sqrt(c - d*x**2) + a*d*x**2*(e*x)**(3/2)*sq rt(c - d*x**2) + b*c*x**2*(e*x)**(3/2)*sqrt(c - d*x**2) - b*d*x**4*(e*x)** (3/2)*sqrt(c - d*x**2)), x)
\[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\int { -\frac {1}{{\left (b x^{2} - a\right )} {\left (-d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(e*x)^(3/2)/(-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)^(3/2)), x)
\[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\int { -\frac {1}{{\left (b x^{2} - a\right )} {\left (-d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(e*x)^(3/2)/(-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)^(3/2)), x)
Timed out. \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{3/2}\,\left (a-b\,x^2\right )\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \] Input:
int(1/((e*x)^(3/2)*(a - b*x^2)*(c - d*x^2)^(3/2)),x)
Output:
int(1/((e*x)^(3/2)*(a - b*x^2)*(c - d*x^2)^(3/2)), x)
\[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\frac {\sqrt {e}\, \left (-2 \sqrt {-d \,x^{2}+c}-3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}\, x^{2}}{-b \,d^{2} x^{6}+a \,d^{2} x^{4}+2 b c d \,x^{4}-2 a c d \,x^{2}-b \,c^{2} x^{2}+a \,c^{2}}d x \right ) b c d +3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}\, x^{2}}{-b \,d^{2} x^{6}+a \,d^{2} x^{4}+2 b c d \,x^{4}-2 a c d \,x^{2}-b \,c^{2} x^{2}+a \,c^{2}}d x \right ) b \,d^{2} x^{2}+3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}}{-b \,d^{2} x^{6}+a \,d^{2} x^{4}+2 b c d \,x^{4}-2 a c d \,x^{2}-b \,c^{2} x^{2}+a \,c^{2}}d x \right ) a c d -3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}}{-b \,d^{2} x^{6}+a \,d^{2} x^{4}+2 b c d \,x^{4}-2 a c d \,x^{2}-b \,c^{2} x^{2}+a \,c^{2}}d x \right ) a \,d^{2} x^{2}+\sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}}{-b \,d^{2} x^{6}+a \,d^{2} x^{4}+2 b c d \,x^{4}-2 a c d \,x^{2}-b \,c^{2} x^{2}+a \,c^{2}}d x \right ) b \,c^{2}-\sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}}{-b \,d^{2} x^{6}+a \,d^{2} x^{4}+2 b c d \,x^{4}-2 a c d \,x^{2}-b \,c^{2} x^{2}+a \,c^{2}}d x \right ) b c d \,x^{2}\right )}{\sqrt {x}\, a c \,e^{2} \left (-d \,x^{2}+c \right )} \] Input:
int(1/(e*x)^(3/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x)
Output:
(sqrt(e)*( - 2*sqrt(c - d*x**2) - 3*sqrt(x)*int((sqrt(x)*sqrt(c - d*x**2)* x**2)/(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 + 3*sqrt(x)*int((sqrt(x)*sqrt(c - d*x**2)*x**2)/(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**2 + 3*sqrt(x)*int((sqrt(x)*sqrt(c - d*x**2))/(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)*a*c *d - 3*sqrt(x)*int((sqrt(x)*sqrt(c - d*x**2))/(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)*a*d**2*x**2 + sqrt (x)*int((sqrt(x)*sqrt(c - d*x**2))/(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**2 - sqrt(x)*int((sqrt(x) *sqrt(c - d*x**2))/(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**2))/(sqrt(x)*a*c*e**2*(c - d*x**2))