Integrand size = 30, antiderivative size = 529 \[ \int \frac {(e x)^{9/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\frac {(2 b c+a d) e^3 (e x)^{3/2}}{2 b (b c-a d)^2 \sqrt {c-d x^2}}+\frac {a e^3 (e x)^{3/2}}{2 b (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {c^{3/4} (2 b c+a d) e^{9/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 b d^{3/4} (b c-a d)^2 \sqrt {c-d x^2}}+\frac {c^{3/4} (2 b c+a d) e^{9/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 b d^{3/4} (b c-a d)^2 \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} (7 b c-a d) e^{9/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b^{3/2} \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} (7 b c-a d) e^{9/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b^{3/2} \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}} \]
1/2*(a*d+2*b*c)*e^3*(e*x)^(3/2)/b/(-a*d+b*c)^2/(-d*x^2+c)^(1/2)+1/2*a*e^3* (e*x)^(3/2)/b/(-a*d+b*c)/(-b*x^2+a)/(-d*x^2+c)^(1/2)-1/2*c^(3/4)*(a*d+2*b* c)*e^(9/2)*EllipticE(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)*(1-d*x^2/c)^(1 /2)/b/d^(3/4)/(-a*d+b*c)^2/(-d*x^2+c)^(1/2)+1/2*c^(3/4)*(a*d+2*b*c)*e^(9/2 )*EllipticF(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)*(1-d*x^2/c)^(1/2)/b/d^( 3/4)/(-a*d+b*c)^2/(-d*x^2+c)^(1/2)+1/4*c^(1/4)*(-a*d+7*b*c)*e^(9/2)*Ellipt icPi(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^(1/2)*(1-d*x^2/c)^(1/2)/b^(3/2)/d^(1/4)/(-a*d+b*c)^2/(-d*x^2+c)^(1/2) -1/4*c^(1/4)*(-a*d+7*b*c)*e^(9/2)*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^(1/2)*(1-d*x^2/c)^(1/2)/b^(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 = 11.19 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.36 \[ \int \frac {(e x)^{9/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\frac {e^3 (e x)^{3/2} \left (7 a \left (-3 a c+2 b c x^2+a d x^2\right )+21 a c \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+(2 b c+a d) x^2 \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{14 a (b c-a d)^2 \left (-a+b x^2\right ) \sqrt {c-d x^2}} \]
(e^3*(e*x)^(3/2)*(7*a*(-3*a*c + 2*b*c*x^2 + a*d*x^2) + 21*a*c*(a - b*x^2)* Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + (2* b*c + a*d)*x^2*(-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]))/(14*a*(b*c - a*d)^2*(-a + b*x^2)*Sqrt[c - d*x^2] )
Time = 0.91 (sec) , antiderivative size = 519, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {368, 27, 970, 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.
| \(\displaystyle \int \frac {(e x)^{9/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {e^9 x^5}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \int \frac {e^5 x^5}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}\) |
| \(\Big \downarrow \) 970 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{3/2}}{4 b \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\int \frac {e x \left ((4 b c-a d) x^2 e^2+3 a c e^2\right )}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1049 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{3/2}}{4 b \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {-\frac {\int -\frac {2 b c e x \left (9 a c e^2-(2 b c+a d) e^2 x^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 c (b c-a d)}-\frac {(e x)^{3/2} (a d+2 b c)}{\sqrt {c-d x^2} (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{3/2}}{4 b \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {b \int \frac {e x \left (9 a c e^2-(2 b c+a d) e^2 x^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b c-a d}-\frac {(e x)^{3/2} (a d+2 b c)}{\sqrt {c-d x^2} (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{3/2}}{4 b \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {b \int \left (\frac {(2 b c+a d) e x}{b \sqrt {c-d x^2}}-\frac {e \left (a^2 d e^2-7 a b c e^2\right ) x}{b \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}\right )d\sqrt {e x}}{b c-a d}-\frac {(e x)^{3/2} (a d+2 b c)}{\sqrt {c-d x^2} (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{3/2}}{4 b \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {b \left (-\frac {\sqrt {a} \sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (7 b c-a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (7 b c-a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {c^{3/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (a d+2 b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b d^{3/4} \sqrt {c-d x^2}}+\frac {c^{3/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (a d+2 b c) E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b d^{3/4} \sqrt {c-d x^2}}\right )}{b c-a d}-\frac {(e x)^{3/2} (a d+2 b c)}{\sqrt {c-d x^2} (b c-a d)}}{4 b (b c-a d)}\right )\) |
2*e^3*((a*e^2*(e*x)^(3/2))/(4*b*(b*c - a*d)*Sqrt[c - d*x^2]*(a*e^2 - b*e^2 *x^2)) - (-(((2*b*c + a*d)*(e*x)^(3/2))/((b*c - a*d)*Sqrt[c - d*x^2])) + ( b*((c^(3/4)*(2*b*c + 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])/(b*d^(3/4)*Sqrt[c - d*x^2]) - (c ^(3/4)*(2*b*c + 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])/(b*d^(3/4)*Sqrt[c - d*x^2]) - (Sqrt[a ]*c^(1/4)*(7*b*c - 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*b^(3/2)*d^(1/4)*Sqrt[c - d*x^2]) + (Sqrt[a]*c^(1/4)*(7*b*c - 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*b^(3/2)*d^(1/4) *Sqrt[c - d*x^2])))/(b*c - a*d))/(4*b*(b*c - a*d)))
3.10.18.3.1 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[(-a)*e^(2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n) ^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Simp[e^(2*n) /(b*n*(b*c - a*d)*(p + 1)) Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d *x^n)^q*Simp[a*c*(m - 2*n + 1) + (a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^ n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[ n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && 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. \(1129\) vs. \(2(413)=826\).
Time = 3.19 (sec) , antiderivative size = 1130, normalized size of antiderivative = 2.14
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1130\) |
| default | \(\text {Expression too large to display}\) | \(2952\) |
1/e/x*(e*x)^(1/2)/(-d*x^2+c)^(1/2)*((-d*x^2+c)*e*x)^(1/2)*(1/2/(a*d-b*c)^2 *a*e^4*x*(-d*e*x^3+c*e*x)^(1/2)/(-b*x^2+a)+e^5*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)*e^5*a/(a*d-b*c)^2/b*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)*e^5*a/(a*d-b*c)^2/b*EllipticF(((x+1/d*(c*d)^(1/2))* d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+1/d*c^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)*e^5 /(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*c^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)*e^5/(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))-1/8*a^2*e^5/(a*d-b*c)^2/ b^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))+7/8*a*e^5/(a*d-b*c)^2/b/ 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/...
Timed out. \[ \int \frac {(e x)^{9/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e x)^{9/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(e x)^{9/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {9}{2}}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(e x)^{9/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {9}{2}}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(e x)^{9/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int \frac {{\left (e\,x\right )}^{9/2}}{{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \]