Integrand size = 30, antiderivative size = 531 \[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\frac {d (b c+2 a d) (e x)^{3/2}}{2 a c (b c-a d)^2 e \sqrt {c-d x^2}}+\frac {b (e x)^{3/2}}{2 a (b c-a d) e \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\sqrt [4]{d} (b c+2 a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a \sqrt [4]{c} (b c-a d)^2 \sqrt {c-d x^2}}+\frac {\sqrt [4]{d} (b c+2 a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{c} (b c-a d)^2 \sqrt {c-d x^2}}-\frac {\sqrt {b} \sqrt [4]{c} (b c-7 a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{3/2} \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}}+\frac {\sqrt {b} \sqrt [4]{c} (b c-7 a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{3/2} \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}} \]
1/2*d*(2*a*d+b*c)*(e*x)^(3/2)/a/c/(-a*d+b*c)^2/e/(-d*x^2+c)^(1/2)+1/2*b*(e *x)^(3/2)/a/(-a*d+b*c)/e/(-b*x^2+a)/(-d*x^2+c)^(1/2)-1/2*d^(1/4)*(2*a*d+b* c)*EllipticE(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)*e^(1/2)*(1-d*x^2/c)^(1 /2)/a/c^(1/4)/(-a*d+b*c)^2/(-d*x^2+c)^(1/2)+1/2*d^(1/4)*(2*a*d+b*c)*Ellipt icF(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)*e^(1/2)*(1-d*x^2/c)^(1/2)/a/c^( 1/4)/(-a*d+b*c)^2/(-d*x^2+c)^(1/2)-1/4*c^(1/4)*(-7*a*d+b*c)*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)*b^(1/ 2)*e^(1/2)*(1-d*x^2/c)^(1/2)/a^(3/2)/d^(1/4)/(-a*d+b*c)^2/(-d*x^2+c)^(1/2) +1/4*c^(1/4)*(-7*a*d+b*c)*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)*b^(1/2)*e^(1/2)*(1-d*x^2/c)^(1/2)/a^(3/2 )/d^(1/4)/(-a*d+b*c)^2/(-d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.24 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.43 \[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\frac {\sqrt {e x} \left (21 a x \left (-2 a^2 d^2+2 a b d^2 x^2+b^2 c \left (-c+d x^2\right )\right )+7 \left (-b^2 c^2+8 a b c d+2 a^2 d^2\right ) x \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 b d (b c+2 a d) x^3 \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{42 a^2 c (b c-a d)^2 \left (-a+b x^2\right ) \sqrt {c-d x^2}} \]
(Sqrt[e*x]*(21*a*x*(-2*a^2*d^2 + 2*a*b*d^2*x^2 + b^2*c*(-c + d*x^2)) + 7*( -(b^2*c^2) + 8*a*b*c*d + 2*a^2*d^2)*x*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*Appe llF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 3*b*d*(b*c + 2*a*d)*x^3*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^ 2)/a]))/(42*a^2*c*(b*c - a*d)^2*(-a + b*x^2)*Sqrt[c - d*x^2])
Time = 0.93 (sec) , antiderivative size = 519, normalized size of antiderivative = 0.98, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {368, 27, 972, 27, 1049, 27, 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.
| \(\displaystyle \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {e^5 x}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \int \frac {e x}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}\) |
| \(\Big \downarrow \) 972 |
| \(\displaystyle 2 e^3 \left (\frac {\int \frac {x \left ((b c-4 a d) e^2-3 b d e^2 x^2\right )}{e \left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 a e^2 (b c-a d)}+\frac {b (e x)^{3/2}}{4 a e^2 \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\int \frac {e x \left ((b c-4 a d) e^2-3 b d e^2 x^2\right )}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 a e^4 (b c-a d)}+\frac {b (e x)^{3/2}}{4 a e^2 \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1049 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {d (e x)^{3/2} (2 a d+b c)}{c \sqrt {c-d x^2} (b c-a d)}-\frac {\int -\frac {2 e x \left (b d (b c+2 a d) x^2 e^2+\left (b^2 c^2-8 a b d c-2 a^2 d^2\right ) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 c (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b (e x)^{3/2}}{4 a e^2 \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\int \frac {e x \left (b d (b c+2 a d) x^2 e^2+\left (b^2 c^2-8 a b d c-2 a^2 d^2\right ) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{c (b c-a d)}+\frac {d (e x)^{3/2} (2 a d+b c)}{c \sqrt {c-d x^2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b (e x)^{3/2}}{4 a e^2 \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\int \left (\frac {e \left (b^2 c^2 e^2-7 a b c d e^2\right ) x}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}-\frac {d (b c+2 a d) e x}{\sqrt {c-d x^2}}\right )d\sqrt {e x}}{c (b c-a d)}+\frac {d (e x)^{3/2} (2 a d+b c)}{c \sqrt {c-d x^2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b (e x)^{3/2}}{4 a e^2 \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (2 a d+b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}-\frac {c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (2 a d+b c) E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {c-d x^2}}-\frac {\sqrt {b} c^{5/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (b c-7 a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt {b} c^{5/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (b c-7 a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} \sqrt [4]{d} \sqrt {c-d x^2}}}{c (b c-a d)}+\frac {d (e x)^{3/2} (2 a d+b c)}{c \sqrt {c-d x^2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b (e x)^{3/2}}{4 a e^2 \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
2*e^3*((b*(e*x)^(3/2))/(4*a*(b*c - a*d)*e^2*Sqrt[c - d*x^2]*(a*e^2 - b*e^2 *x^2)) + ((d*(b*c + 2*a*d)*(e*x)^(3/2))/(c*(b*c - a*d)*Sqrt[c - d*x^2]) + (-((c^(3/4)*d^(1/4)*(b*c + 2*a*d)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticE[Ar cSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/Sqrt[c - d*x^2]) + (c^(3 /4)*d^(1/4)*(b*c + 2*a*d)*e^(3/2)*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] - (Sqrt[b]*c^(5/ 4)*(b*c - 7*a*d)*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])/ (2*Sqrt[a]*d^(1/4)*Sqrt[c - d*x^2]) + (Sqrt[b]*c^(5/4)*(b*c - 7*a*d)*e^(3/ 2)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), Arc Sin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*Sqrt[a]*d^(1/4)*Sqrt[c - d*x^2]))/(c*(b*c - a*d)))/(4*a*(b*c - a*d)*e^4))
3.10.22.3.1 Defintions of rubi rules used
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[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p + 1))) , x] + Simp[1/(a*n*(b*c - a*d)*(p + 1)) Int[(g*x)^m*(a + b*x^n)^(p + 1)*( c + d*x^n)^q*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -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. \(1096\) vs. \(2(415)=830\).
Time = 3.20 (sec) , antiderivative size = 1097, normalized size of antiderivative = 2.07
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1097\) |
| default | \(\text {Expression too large to display}\) | \(2938\) |
1/e/x*(e*x)^(1/2)/(-d*x^2+c)^(1/2)*((-d*x^2+c)*e*x)^(1/2)*(1/2*b^2/(a*d-b* c)^2/a*x*(-d*e*x^3+c*e*x)^(1/2)/(-b*x^2+a)+d^2*e*x^2/c/(a*d-b*c)^2/(-(x^2- c/d)*d*e*x)^(1/2)+1/2*c*(d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^( 1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*b*e/(a*d-b*c)^2/a*Ell ipticE(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-1/4*c*(d*x/( c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/ (-d*e*x^3+c*e*x)^(1/2)*b*e/(a*d-b*c)^2/a*EllipticF(((x+1/d*(c*d)^(1/2))*d/ (c*d)^(1/2))^(1/2),1/2*2^(1/2))+d*(d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^ (1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*e/(a*d-b*c) ^2*EllipticE(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-1/2*d* (d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^ (1/2)/(-d*e*x^3+c*e*x)^(1/2)*e/(a*d-b*c)^2*EllipticF(((x+1/d*(c*d)^(1/2))* d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+7/8*e/(a*d-b*c)^2*(c*d)^(1/2)*(d*x/(c*d) ^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(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))-1/8*e/(a*d-b*c)^2/a/d*(c*d)^(1/2)*(d*x/(c*d)^(1/2) +1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(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)...
Timed out. \[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {e x}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {e x}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {e\,x}}{{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \]