Integrand size = 30, antiderivative size = 568 \[ \int \frac {(e x)^{9/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {(2 b c+3 a d) e^3 (e x)^{3/2}}{6 b (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {a e^3 (e x)^{3/2}}{2 b (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {(b c+4 a d) e^3 (e x)^{3/2}}{2 (b c-a d)^3 \sqrt {c-d x^2}}-\frac {c^{3/4} (b c+4 a d) e^{9/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 d^{3/4} (b c-a d)^3 \sqrt {c-d x^2}}+\frac {c^{3/4} (b c+4 a d) e^{9/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 d^{3/4} (b c-a d)^3 \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} (7 b c+3 a d) e^{9/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt {b} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} (7 b c+3 a d) e^{9/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt {b} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}} \] Output:
1/6*(3*a*d+2*b*c)*e^3*(e*x)^(3/2)/b/(-a*d+b*c)^2/(-d*x^2+c)^(3/2)+1/2*a*e^ 3*(e*x)^(3/2)/b/(-a*d+b*c)/(-b*x^2+a)/(-d*x^2+c)^(3/2)+1/2*(4*a*d+b*c)*e^3 *(e*x)^(3/2)/(-a*d+b*c)^3/(-d*x^2+c)^(1/2)-1/2*c^(3/4)*(4*a*d+b*c)*e^(9/2) *(1-d*x^2/c)^(1/2)*EllipticE(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/d^(3/4 )/(-a*d+b*c)^3/(-d*x^2+c)^(1/2)+1/2*c^(3/4)*(4*a*d+b*c)*e^(9/2)*(1-d*x^2/c )^(1/2)*EllipticF(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/d^(3/4)/(-a*d+b*c )^3/(-d*x^2+c)^(1/2)+1/4*a^(1/2)*c^(1/4)*(3*a*d+7*b*c)*e^(9/2)*(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)/b^(1/2)/d^(1/4)/(-a*d+b*c)^3/(-d*x^2+c)^(1/2)-1/4*a^(1/2)* c^(1/4)*(3*a*d+7*b*c)*e^(9/2)*(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)/b^(1/2)/d^(1/4)/(- a*d+b*c)^3/(-d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.30 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.45 \[ \int \frac {(e x)^{9/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=-\frac {e^3 (e x)^{3/2} \left (7 a \left (a^2 d \left (7 c-9 d x^2\right )+b^2 c x^2 \left (-5 c+3 d x^2\right )+4 a b \left (2 c^2-4 c d x^2+3 d^2 x^4\right )\right )+7 a (8 b c+7 a d) \left (-a+b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 b (b c+4 a d) x^2 \left (a-b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{42 a (b c-a d)^3 \left (-a+b x^2\right ) \left (c-d x^2\right )^{3/2}} \] Input:
Integrate[(e*x)^(9/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]
Output:
-1/42*(e^3*(e*x)^(3/2)*(7*a*(a^2*d*(7*c - 9*d*x^2) + b^2*c*x^2*(-5*c + 3*d *x^2) + 4*a*b*(2*c^2 - 4*c*d*x^2 + 3*d^2*x^4)) + 7*a*(8*b*c + 7*a*d)*(-a + b*x^2)*(c - d*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*(b*c + 4*a*d)*x^2*(a - b*x^2)*(c - d*x^2)*Sqrt[1 - (d *x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(a*(b*c - a*d )^3*(-a + b*x^2)*(c - d*x^2)^(3/2))
Time = 1.06 (sec) , antiderivative size = 567, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {368, 27, 970, 1049, 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.
| \(\displaystyle \int \frac {(e x)^{9/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {e^9 x^5}{\left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \int \frac {e^5 x^5}{\left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}\) |
| \(\Big \downarrow \) 970 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{3/2}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\int \frac {e x \left ((4 b c+3 a d) x^2 e^2+3 a c e^2\right )}{\left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1049 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{3/2}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {-\frac {\int -\frac {6 b c e x \left ((2 b c+3 a d) x^2 e^2+5 a c e^2\right )}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{6 c (b c-a d)}-\frac {(e x)^{3/2} (3 a d+2 b c)}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{3/2}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {b \int \frac {e x \left ((2 b c+3 a d) x^2 e^2+5 a c e^2\right )}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b c-a d}-\frac {(e x)^{3/2} (3 a d+2 b c)}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1049 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{3/2}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {b \left (-\frac {\int -\frac {2 c e x \left (a (8 b c+7 a d) e^2-b (b c+4 a d) e^2 x^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 c (b c-a d)}-\frac {(e x)^{3/2} (4 a d+b c)}{\sqrt {c-d x^2} (b c-a d)}\right )}{b c-a d}-\frac {(e x)^{3/2} (3 a d+2 b c)}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{3/2}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {b \left (\frac {\int \frac {e x \left (a (8 b c+7 a d) e^2-b (b c+4 a d) e^2 x^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b c-a d}-\frac {(e x)^{3/2} (4 a d+b c)}{\sqrt {c-d x^2} (b c-a d)}\right )}{b c-a d}-\frac {(e x)^{3/2} (3 a d+2 b c)}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{3/2}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {b \left (\frac {\int \left (\frac {(b c+4 a d) e x}{\sqrt {c-d x^2}}+\frac {e \left (7 a b c e^2+3 a^2 d e^2\right ) x}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}\right )d\sqrt {e x}}{b c-a d}-\frac {(e x)^{3/2} (4 a d+b c)}{\sqrt {c-d x^2} (b c-a d)}\right )}{b c-a d}-\frac {(e x)^{3/2} (3 a d+2 b c)}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{3/2}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {b \left (\frac {-\frac {c^{3/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (4 a d+b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{d^{3/4} \sqrt {c-d x^2}}+\frac {c^{3/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (4 a d+b c) E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{d^{3/4} \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (3 a d+7 b c) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {b} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (3 a d+7 b c) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {b} \sqrt [4]{d} \sqrt {c-d x^2}}}{b c-a d}-\frac {(e x)^{3/2} (4 a d+b c)}{\sqrt {c-d x^2} (b c-a d)}\right )}{b c-a d}-\frac {(e x)^{3/2} (3 a d+2 b c)}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\) |
Input:
Int[(e*x)^(9/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]
Output:
2*e^3*((a*e^2*(e*x)^(3/2))/(4*b*(b*c - a*d)*(c - d*x^2)^(3/2)*(a*e^2 - b*e ^2*x^2)) - (-1/3*((2*b*c + 3*a*d)*(e*x)^(3/2))/((b*c - a*d)*(c - d*x^2)^(3 /2)) + (b*(-(((b*c + 4*a*d)*(e*x)^(3/2))/((b*c - a*d)*Sqrt[c - d*x^2])) + ((c^(3/4)*(b*c + 4*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])/(d^(3/4)*Sqrt[c - d*x^2]) - (c^(3/ 4)*(b*c + 4*a*d)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqr t[e*x])/(c^(1/4)*Sqrt[e])], -1])/(d^(3/4)*Sqrt[c - d*x^2]) - (Sqrt[a]*c^(1 /4)*(7*b*c + 3*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[b]*d^(1/4)*Sqrt[c - d*x^2]) + (Sqrt[a]*c^(1/4)*(7*b*c + 3*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[b]*d^(1/4)*S qrt[c - d*x^2]))/(b*c - a*d)))/(b*c - a*d))/(4*b*(b*c - a*d)))
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. \(1408\) vs. \(2(446)=892\).
Time = 3.36 (sec) , antiderivative size = 1409, normalized size of antiderivative = 2.48
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1409\) |
| default | \(\text {Expression too large to display}\) | \(5114\) |
Input:
int((e*x)^(9/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/e/x*(e*x)^(1/2)/(-d*x^2+c)^(1/2)*((-d*x^2+c)*e*x)^(1/2)*(1/2*d*a*e^4*b/( a*d-b*c)/(a^2*d^2-2*a*b*c*d+b^2*c^2)*x*(-d*e*x^3+c*e*x)^(1/2)/(b*d*x^2-a*d )+1/3*c*e^4/d^2/(a*d-b*c)^2*x*(-d*e*x^3+c*e*x)^(1/2)/(x^2-c/d)^2-1/2*e^5*x ^2*(3*a*d+b*c)/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)/(-(x^2-c/d)*d*e*x)^(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)*a*e^5/(a*d-b*c)/(a^2*d^2-2*a*b*c*d+b^2 *c^2)*EllipticE(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+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)*a*e^5/(a*d-b*c)/(a^2*d^2-2*a*b*c*d+b^2*c^2)*Ell ipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-1/2/d*c^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)*e^5/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)*b*Elli pticE(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+1/4/d*c^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)*e^5/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)*b*Ellip ticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-3/8*e^5*a^2/(a ^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)/b*(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...
Timed out. \[ \int \frac {(e x)^{9/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x)^(9/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {(e x)^{9/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x)**(9/2)/(-b*x**2+a)**2/(-d*x**2+c)**(5/2),x)
Output:
Timed out
\[ \int \frac {(e x)^{9/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {9}{2}}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((e*x)^(9/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x, algorithm="maxima")
Output:
integrate((e*x)^(9/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)), x)
\[ \int \frac {(e x)^{9/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {9}{2}}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((e*x)^(9/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x, algorithm="giac")
Output:
integrate((e*x)^(9/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)), x)
Timed out. \[ \int \frac {(e x)^{9/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int \frac {{\left (e\,x\right )}^{9/2}}{{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{5/2}} \,d x \] Input:
int((e*x)^(9/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x)
Output:
int((e*x)^(9/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)), x)
\[ \int \frac {(e x)^{9/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\sqrt {e}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}\, x^{4}}{-b^{2} d^{3} x^{10}+2 a b \,d^{3} x^{8}+3 b^{2} c \,d^{2} x^{8}-a^{2} d^{3} x^{6}-6 a b c \,d^{2} x^{6}-3 b^{2} c^{2} d \,x^{6}+3 a^{2} c \,d^{2} x^{4}+6 a b \,c^{2} d \,x^{4}+b^{2} c^{3} x^{4}-3 a^{2} c^{2} d \,x^{2}-2 a b \,c^{3} x^{2}+a^{2} c^{3}}d x \right ) e^{4} \] Input:
int((e*x)^(9/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x)
Output:
sqrt(e)*int((sqrt(x)*sqrt(c - d*x**2)*x**4)/(a**2*c**3 - 3*a**2*c**2*d*x** 2 + 3*a**2*c*d**2*x**4 - a**2*d**3*x**6 - 2*a*b*c**3*x**2 + 6*a*b*c**2*d*x **4 - 6*a*b*c*d**2*x**6 + 2*a*b*d**3*x**8 + b**2*c**3*x**4 - 3*b**2*c**2*d *x**6 + 3*b**2*c*d**2*x**8 - b**2*d**3*x**10),x)*e**4