Integrand size = 30, antiderivative size = 444 \[ \int \frac {(e x)^{9/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=-\frac {c e^3 (e x)^{3/2}}{d (b c-a d) \sqrt {c-d x^2}}+\frac {c^{3/4} (3 b c-2 a d) e^{9/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b d^{7/4} (b c-a d) \sqrt {c-d x^2}}-\frac {c^{3/4} (3 b c-2 a d) e^{9/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b d^{7/4} (b c-a d) \sqrt {c-d x^2}}-\frac {a^{3/2} \sqrt [4]{c} e^{9/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^{3/2} \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}+\frac {a^{3/2} \sqrt [4]{c} e^{9/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^{3/2} \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}} \]
-c*e^3*(e*x)^(3/2)/d/(-a*d+b*c)/(-d*x^2+c)^(1/2)+c^(3/4)*(-2*a*d+3*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^(7/4)/(-a*d+b*c)/(-d*x^2+c)^(1/2)-c^(3/4)*(-2*a*d+3*b*c)*e^(9/2)*Ellipt icF(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)*(1-d*x^2/c)^(1/2)/b/d^(7/4)/(-a *d+b*c)/(-d*x^2+c)^(1/2)-a^(3/2)*c^(1/4)*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)*(1-d*x^2/c)^(1/2 )/b^(3/2)/d^(1/4)/(-a*d+b*c)/(-d*x^2+c)^(1/2)+a^(3/2)*c^(1/4)*e^(9/2)*Elli pticPi(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)*(1-d*x^2/c)^(1/2)/b^(3/2)/d^(1/4)/(-a*d+b*c)/(-d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.14 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.33 \[ \int \frac {(e x)^{9/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=-\frac {e^3 (e x)^{3/2} \left (-7 a c+7 a c \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+(-3 b c+2 a d) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{7 a d (-b c+a d) \sqrt {c-d x^2}} \]
-1/7*(e^3*(e*x)^(3/2)*(-7*a*c + 7*a*c*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/ 2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + (-3*b*c + 2*a*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*d*(-(b*c) + a*d)* Sqrt[c - d*x^2])
Time = 0.77 (sec) , antiderivative size = 430, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {368, 27, 970, 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.
| \(\displaystyle \int \frac {(e x)^{9/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {e^7 x^5}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e \int \frac {e^5 x^5}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}\) |
| \(\Big \downarrow \) 970 |
| \(\displaystyle 2 e \left (\frac {e^2 \int \frac {e x \left (3 a c e^2-(3 b c-2 a d) e^2 x^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 d (b c-a d)}-\frac {c e^2 (e x)^{3/2}}{2 d \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle 2 e \left (\frac {e^2 \int \left (\frac {2 a^2 d x e^3}{b \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}+\frac {(3 b c-2 a d) x e}{b \sqrt {c-d x^2}}\right )d\sqrt {e x}}{2 d (b c-a d)}-\frac {c e^2 (e x)^{3/2}}{2 d \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 e \left (\frac {e^2 \left (-\frac {a^{3/2} \sqrt [4]{c} d^{3/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^{3/2} \sqrt {c-d x^2}}+\frac {a^{3/2} \sqrt [4]{c} d^{3/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^{3/2} \sqrt {c-d x^2}}-\frac {c^{3/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (3 b c-2 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b d^{3/4} \sqrt {c-d x^2}}+\frac {c^{3/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (3 b c-2 a d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b d^{3/4} \sqrt {c-d x^2}}\right )}{2 d (b c-a d)}-\frac {c e^2 (e x)^{3/2}}{2 d \sqrt {c-d x^2} (b c-a d)}\right )\) |
2*e*(-1/2*(c*e^2*(e*x)^(3/2))/(d*(b*c - a*d)*Sqrt[c - d*x^2]) + (e^2*((c^( 3/4)*(3*b*c - 2*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 )*(3*b*c - 2*a*d)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sq rt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(b*d^(3/4)*Sqrt[c - d*x^2]) - (a^(3/2)*c ^(1/4)*d^(3/4)*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])/(b ^(3/2)*Sqrt[c - d*x^2]) + (a^(3/2)*c^(1/4)*d^(3/4)*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])/(b^(3/2)*Sqrt[c - d*x^2])))/(2*d*(b*c - a*d )))
3.9.88.3.1 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_)*((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. \(762\) vs. \(2(344)=688\).
Time = 3.18 (sec) , antiderivative size = 763, normalized size of antiderivative = 1.72
| method | result | size |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (\frac {e^{5} x^{2} c}{d \left (a d -b c \right ) \sqrt {-\left (x^{2}-\frac {c}{d}\right ) d e x}}-\frac {2 c \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, e^{5} E\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{d^{2} \sqrt {-d e \,x^{3}+c e x}\, b}+\frac {c \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, e^{5} F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{d^{2} \sqrt {-d e \,x^{3}+c e x}\, b}+\frac {c^{2} \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, e^{5} E\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{d^{2} \sqrt {-d e \,x^{3}+c e x}\, \left (a d -b c \right )}-\frac {c^{2} \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, e^{5} F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{2 d^{2} \sqrt {-d e \,x^{3}+c e x}\, \left (a d -b c \right )}+\frac {e^{5} a^{2} \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \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 ) b^{2} d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {e^{5} a^{2} \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \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 ) b^{2} d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{e x \sqrt {-d \,x^{2}+c}}\) | \(763\) |
| default | \(\text {Expression too large to display}\) | \(1028\) |
1/e/x*(e*x)^(1/2)/(-d*x^2+c)^(1/2)*((-d*x^2+c)*e*x)^(1/2)*(1/d*e^5*x^2*c/( a*d-b*c)/(-(x^2-c/d)*d*e*x)^(1/2)-2/d^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 /b*EllipticE(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+1/d^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/b*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d )^(1/2))^(1/2),1/2*2^(1/2))+1/d^2*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)*EllipticE(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-1/ 2/d^2*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)*EllipticF(((x+1/d*(c* d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+1/2*e^5*a^2/(a*d-b*c)/b^2/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)^(1/2)),1/2*2^(1/2))+1/2*e^5*a^2/(a*d-b*c)/b^2/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)^(1/2)),1/2*2^(1/2)))
Timed out. \[ \int \frac {(e x)^{9/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(e x)^{9/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=- \int \frac {\left (e x\right )^{\frac {9}{2}}}{- a c \sqrt {c - d x^{2}} + a d x^{2} \sqrt {c - d x^{2}} + b c x^{2} \sqrt {c - d x^{2}} - b d x^{4} \sqrt {c - d x^{2}}}\, dx \]
-Integral((e*x)**(9/2)/(-a*c*sqrt(c - d*x**2) + a*d*x**2*sqrt(c - d*x**2) + b*c*x**2*sqrt(c - d*x**2) - b*d*x**4*sqrt(c - d*x**2)), x)
\[ \int \frac {(e x)^{9/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\int { -\frac {\left (e x\right )^{\frac {9}{2}}}{{\left (b x^{2} - a\right )} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(e x)^{9/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\int { -\frac {\left (e x\right )^{\frac {9}{2}}}{{\left (b x^{2} - a\right )} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(e x)^{9/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\int \frac {{\left (e\,x\right )}^{9/2}}{\left (a-b\,x^2\right )\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \]