Integrand size = 30, antiderivative size = 429 \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=-\frac {(7 b c-4 a d) \sqrt {c-d x^2}}{6 a^2 c (b c-a d) e (e x)^{3/2}}+\frac {b \sqrt {c-d x^2}}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right )}+\frac {d^{3/4} (7 b c-4 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{6 a^2 c^{3/4} (b c-a d) e^{5/2} \sqrt {c-d x^2}}+\frac {b \sqrt [4]{c} (7 b c-9 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^3 \sqrt [4]{d} (b c-a d) e^{5/2} \sqrt {c-d x^2}}+\frac {b \sqrt [4]{c} (7 b c-9 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^3 \sqrt [4]{d} (b c-a d) e^{5/2} \sqrt {c-d x^2}} \] Output:
-1/6*(-4*a*d+7*b*c)*(-d*x^2+c)^(1/2)/a^2/c/(-a*d+b*c)/e/(e*x)^(3/2)+1/2*b* (-d*x^2+c)^(1/2)/a/(-a*d+b*c)/e/(e*x)^(3/2)/(-b*x^2+a)+1/6*d^(3/4)*(-4*a*d +7*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^2/c^(3/4)/(-a*d+b*c)/e^(5/2)/(-d*x^2+c)^(1/2)+1/4*b*c^(1/4)*(-9*a*d+7*b *c)*(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/d^(1/4)/(-a*d+b*c)/e^(5/2)/(-d*x^2+c)^( 1/2)+1/4*b*c^(1/4)*(-9*a*d+7*b*c)*(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/d^(1/4)/(- a*d+b*c)/e^(5/2)/(-d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.22 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.55 \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\frac {x \left (-5 a \left (c-d x^2\right ) \left (4 a^2 d+7 b^2 c x^2-4 a b \left (c+d x^2\right )\right )+5 \left (-21 b^2 c^2+20 a b c d+4 a^2 d^2\right ) x^2 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )-b d (-7 b c+4 a d) x^4 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{30 a^3 c (b c-a d) (e x)^{5/2} \left (-a+b x^2\right ) \sqrt {c-d x^2}} \] Input:
Integrate[1/((e*x)^(5/2)*(a - b*x^2)^2*Sqrt[c - d*x^2]),x]
Output:
(x*(-5*a*(c - d*x^2)*(4*a^2*d + 7*b^2*c*x^2 - 4*a*b*(c + d*x^2)) + 5*(-21* b^2*c^2 + 20*a*b*c*d + 4*a^2*d^2)*x^2*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*Appe llF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] - b*d*(-7*b*c + 4*a*d)*x^4*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2 )/a]))/(30*a^3*c*(b*c - a*d)*(e*x)^(5/2)*(-a + b*x^2)*Sqrt[c - d*x^2])
Time = 0.90 (sec) , antiderivative size = 424, normalized size of antiderivative = 0.99, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {368, 27, 972, 27, 1053, 25, 1021, 765, 762, 925, 27, 1543, 1542}
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.
| \(\displaystyle \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {e^2}{x^2 \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \int \frac {1}{e^2 x^2 \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}\) |
| \(\Big \downarrow \) 972 |
| \(\displaystyle 2 e^3 \left (\frac {\int \frac {(7 b c-4 a d) e^2-5 b d e^2 x^2}{e^4 x^2 \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 a e^2 (b c-a d)}+\frac {b \sqrt {c-d x^2}}{4 a e^2 (e x)^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\int \frac {(7 b c-4 a d) e^2-5 b d e^2 x^2}{e^2 x^2 \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {c-d x^2}}{4 a e^2 (e x)^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle 2 e^3 \left (\frac {-\frac {\int -\frac {\left (21 b^2 c^2-20 a b d c-4 a^2 d^2\right ) e^2-b d (7 b c-4 a d) e^2 x^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (7 b c-4 a d)}{3 a c (e x)^{3/2}}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {c-d x^2}}{4 a e^2 (e x)^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\int \frac {\left (21 b^2 c^2-20 a b d c-4 a^2 d^2\right ) e^2-b d (7 b c-4 a d) e^2 x^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (7 b c-4 a d)}{3 a c (e x)^{3/2}}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {c-d x^2}}{4 a e^2 (e x)^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1021 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {3 b c e^2 (7 b c-9 a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}+d (7 b c-4 a d) \int \frac {1}{\sqrt {c-d x^2}}d\sqrt {e x}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (7 b c-4 a d)}{3 a c (e x)^{3/2}}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {c-d x^2}}{4 a e^2 (e x)^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {3 b c e^2 (7 b c-9 a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}+\frac {d \sqrt {1-\frac {d x^2}{c}} (7 b c-4 a d) \int \frac {1}{\sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (7 b c-4 a d)}{3 a c (e x)^{3/2}}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {c-d x^2}}{4 a e^2 (e x)^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {3 b c e^2 (7 b c-9 a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (7 b c-4 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (7 b c-4 a d)}{3 a c (e x)^{3/2}}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {c-d x^2}}{4 a e^2 (e x)^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {3 b c e^2 (7 b c-9 a d) \left (\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}+\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}\right )+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (7 b c-4 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (7 b c-4 a d)}{3 a c (e x)^{3/2}}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {c-d x^2}}{4 a e^2 (e x)^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {3 b c e^2 (7 b c-9 a d) \left (\frac {\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}+\frac {\int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}\right )+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (7 b c-4 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (7 b c-4 a d)}{3 a c (e x)^{3/2}}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {c-d x^2}}{4 a e^2 (e x)^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {3 b c e^2 (7 b c-9 a d) \left (\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}+\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}\right )+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (7 b c-4 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} (7 b c-4 a d)}{3 a c (e x)^{3/2}}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {c-d x^2}}{4 a e^2 (e x)^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (7 b c-4 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}+3 b c e^2 (7 b c-9 a d) \left (\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}\right )}{3 a c e^2}-\frac {\sqrt {c-d x^2} (7 b c-4 a d)}{3 a c (e x)^{3/2}}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {c-d x^2}}{4 a e^2 (e x)^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
Input:
Int[1/((e*x)^(5/2)*(a - b*x^2)^2*Sqrt[c - d*x^2]),x]
Output:
2*e^3*((b*Sqrt[c - d*x^2])/(4*a*(b*c - a*d)*e^2*(e*x)^(3/2)*(a*e^2 - b*e^2 *x^2)) + (-1/3*((7*b*c - 4*a*d)*Sqrt[c - d*x^2])/(a*c*(e*x)^(3/2)) + ((c^( 1/4)*d^(3/4)*(7*b*c - 4*a*d)*Sqrt[e]*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] + 3*b*c*(7*b* c - 9*a*d)*e^2*((c^(1/4)*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*a*d^(1/4)*e^(3/2)*Sqrt[c - d*x^2]) + (c^(1/4)*Sqrt[1 - (d*x^2)/c]*Ellip ticPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^( 1/4)*Sqrt[e])], -1])/(2*a*d^(1/4)*e^(3/2)*Sqrt[c - d*x^2])))/(3*a*c*e^2))/ (4*a*(b*c - a*d)*e^4))
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[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Simp[ 1/(2*c) Int[1/(Sqrt[a + b*x^4]*(1 - Rt[-d/c, 2]*x^2)), x], x] + Simp[1/(2 *c) Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
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[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x _)^(n_)]), x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^n], x], x] + Simp[(b* e - a*f)/b Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, n}, 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[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x ], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a) ]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && !GtQ[a, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(947\) vs. \(2(341)=682\).
Time = 2.26 (sec) , antiderivative size = 948, normalized size of antiderivative = 2.21
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (-x^{2} d +c \right ) e x}\, \left (-\frac {b^{2} \sqrt {-d e \,x^{3}+c e x}}{2 a^{2} \left (a d -b c \right ) e^{3} \left (-b \,x^{2}+a \right )}-\frac {2 \sqrt {-d e \,x^{3}+c e x}}{3 e^{3} c \,a^{2} x^{2}}-\frac {\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 {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) b}{4 \sqrt {-d e \,x^{3}+c e x}\, \left (a d -b c \right ) a^{2} e^{2}}+\frac {\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 {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{3 \sqrt {-d e \,x^{3}+c e x}\, c \,e^{2} a^{2}}-\frac {9 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 )}{8 \left (a d -b c \right ) a \,e^{2} \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {7 b^{2} \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 ) c}{8 \left (a d -b c \right ) a^{2} e^{2} \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {9 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 )}{8 \left (a d -b c \right ) a \,e^{2} \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}-\frac {7 b^{2} \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 ) c}{8 \left (a d -b c \right ) a^{2} e^{2} \sqrt {a b}\, 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}}\) | \(948\) |
| default | \(\text {Expression too large to display}\) | \(2610\) |
Input:
int(1/(e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
((-d*x^2+c)*e*x)^(1/2)/(e*x)^(1/2)/(-d*x^2+c)^(1/2)*(-1/2*b^2/a^2/(a*d-b*c )/e^3*(-d*e*x^3+c*e*x)^(1/2)/(-b*x^2+a)-2/3/e^3/c/a^2*(-d*e*x^3+c*e*x)^(1/ 2)/x^2-1/4*(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)*EllipticF(((x+1/d*(c*d) ^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*b/(a*d-b*c)/a^2/e^2+1/3*(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)*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/ 2))^(1/2),1/2*2^(1/2))/c/e^2/a^2-9/8/(a*d-b*c)/a/e^2*b/(a*b)^(1/2)*(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))*Ellipt icPi(((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*d-b*c)/a^2/e^2*b^2/(a*b)^(1/2 )/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))*c+9/8/(a*d-b*c)/a/e^2*b/( a*b)^(1/2)*(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))-7/8/(a*d-b*c)/a...
Timed out. \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x, algorithm="fricas ")
Output:
Timed out
\[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int \frac {1}{\left (e x\right )^{\frac {5}{2}} \left (- a + b x^{2}\right )^{2} \sqrt {c - d x^{2}}}\, dx \] Input:
integrate(1/(e*x)**(5/2)/(-b*x**2+a)**2/(-d*x**2+c)**(1/2),x)
Output:
Integral(1/((e*x)**(5/2)*(-a + b*x**2)**2*sqrt(c - d*x**2)), x)
\[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} \sqrt {-d x^{2} + c} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x, algorithm="maxima ")
Output:
integrate(1/((b*x^2 - a)^2*sqrt(-d*x^2 + c)*(e*x)^(5/2)), x)
\[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} \sqrt {-d x^{2} + c} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x, algorithm="giac")
Output:
integrate(1/((b*x^2 - a)^2*sqrt(-d*x^2 + c)*(e*x)^(5/2)), x)
Timed out. \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{5/2}\,{\left (a-b\,x^2\right )}^2\,\sqrt {c-d\,x^2}} \,d x \] Input:
int(1/((e*x)^(5/2)*(a - b*x^2)^2*(c - d*x^2)^(1/2)),x)
Output:
int(1/((e*x)^(5/2)*(a - b*x^2)^2*(c - d*x^2)^(1/2)), x)
\[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\frac {\sqrt {e}\, \left (-2 \sqrt {-d \,x^{2}+c}+\sqrt {x}\, \left (\int \frac {\sqrt {-d \,x^{2}+c}}{\sqrt {x}\, a^{2} c -\sqrt {x}\, a^{2} d \,x^{2}-2 \sqrt {x}\, a b c \,x^{2}+2 \sqrt {x}\, a b d \,x^{4}+\sqrt {x}\, b^{2} c \,x^{4}-\sqrt {x}\, b^{2} d \,x^{6}}d x \right ) a^{2} d x +7 \sqrt {x}\, \left (\int \frac {\sqrt {-d \,x^{2}+c}}{\sqrt {x}\, a^{2} c -\sqrt {x}\, a^{2} d \,x^{2}-2 \sqrt {x}\, a b c \,x^{2}+2 \sqrt {x}\, a b d \,x^{4}+\sqrt {x}\, b^{2} c \,x^{4}-\sqrt {x}\, b^{2} d \,x^{6}}d x \right ) a b c x -\sqrt {x}\, \left (\int \frac {\sqrt {-d \,x^{2}+c}}{\sqrt {x}\, a^{2} c -\sqrt {x}\, a^{2} d \,x^{2}-2 \sqrt {x}\, a b c \,x^{2}+2 \sqrt {x}\, a b d \,x^{4}+\sqrt {x}\, b^{2} c \,x^{4}-\sqrt {x}\, b^{2} d \,x^{6}}d x \right ) a b d \,x^{3}-7 \sqrt {x}\, \left (\int \frac {\sqrt {-d \,x^{2}+c}}{\sqrt {x}\, a^{2} c -\sqrt {x}\, a^{2} d \,x^{2}-2 \sqrt {x}\, a b c \,x^{2}+2 \sqrt {x}\, a b d \,x^{4}+\sqrt {x}\, b^{2} c \,x^{4}-\sqrt {x}\, b^{2} d \,x^{6}}d x \right ) b^{2} c \,x^{3}-5 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}\, x}{-b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}-a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) a b d x +5 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}\, x}{-b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}-a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) b^{2} d \,x^{3}\right )}{3 \sqrt {x}\, a c \,e^{3} x \left (-b \,x^{2}+a \right )} \] Input:
int(1/(e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x)
Output:
(sqrt(e)*( - 2*sqrt(c - d*x**2) + sqrt(x)*int(sqrt(c - d*x**2)/(sqrt(x)*a* *2*c - sqrt(x)*a**2*d*x**2 - 2*sqrt(x)*a*b*c*x**2 + 2*sqrt(x)*a*b*d*x**4 + sqrt(x)*b**2*c*x**4 - sqrt(x)*b**2*d*x**6),x)*a**2*d*x + 7*sqrt(x)*int(sq rt(c - d*x**2)/(sqrt(x)*a**2*c - sqrt(x)*a**2*d*x**2 - 2*sqrt(x)*a*b*c*x** 2 + 2*sqrt(x)*a*b*d*x**4 + sqrt(x)*b**2*c*x**4 - sqrt(x)*b**2*d*x**6),x)*a *b*c*x - sqrt(x)*int(sqrt(c - d*x**2)/(sqrt(x)*a**2*c - sqrt(x)*a**2*d*x** 2 - 2*sqrt(x)*a*b*c*x**2 + 2*sqrt(x)*a*b*d*x**4 + sqrt(x)*b**2*c*x**4 - sq rt(x)*b**2*d*x**6),x)*a*b*d*x**3 - 7*sqrt(x)*int(sqrt(c - d*x**2)/(sqrt(x) *a**2*c - sqrt(x)*a**2*d*x**2 - 2*sqrt(x)*a*b*c*x**2 + 2*sqrt(x)*a*b*d*x** 4 + sqrt(x)*b**2*c*x**4 - sqrt(x)*b**2*d*x**6),x)*b**2*c*x**3 - 5*sqrt(x)* int((sqrt(x)*sqrt(c - d*x**2)*x)/(a**2*c - a**2*d*x**2 - 2*a*b*c*x**2 + 2* a*b*d*x**4 + b**2*c*x**4 - b**2*d*x**6),x)*a*b*d*x + 5*sqrt(x)*int((sqrt(x )*sqrt(c - d*x**2)*x)/(a**2*c - a**2*d*x**2 - 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 - b**2*d*x**6),x)*b**2*d*x**3))/(3*sqrt(x)*a*c*e**3*x*(a - b *x**2))