\(\int \frac {(c-d x^2)^{3/2}}{(e x)^{3/2} (a-b x^2)^2} \, dx\) [908]

Optimal result

Integrand size = 30, antiderivative size = 519 \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=-\frac {(5 b c-a d) \sqrt {c-d x^2}}{2 a^2 b e \sqrt {e x}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e \sqrt {e x} \left (a-b x^2\right )}-\frac {c^{3/4} \sqrt [4]{d} (5 b c-a d) \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 b e^{3/2} \sqrt {c-d x^2}}+\frac {c^{3/4} \sqrt [4]{d} (5 b c-a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a^2 b e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} \left (5 b^2 c^2-4 a b c d-a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \left (5 b^2 c^2-4 a b c d-a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}} \]

[Out]

-1/2*(-a*d+5*b*c)*(-d*x^2+c)^(1/2)/a^2/b/e/(e*x)^(1/2)+1/2*(-a*d+b*c)*(-d*x^2+c)^(1/2)/a/b/e/(-b*x^2+a)/(e*x)^
(1/2)-1/2*c^(3/4)*d^(1/4)*(-a*d+5*b*c)*EllipticE(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)*(1-d*x^2/c)^(1/2)/a^2/
b/e^(3/2)/(-d*x^2+c)^(1/2)+1/2*c^(3/4)*d^(1/4)*(-a*d+5*b*c)*EllipticF(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)*(
1-d*x^2/c)^(1/2)/a^2/b/e^(3/2)/(-d*x^2+c)^(1/2)-1/4*c^(1/4)*(-a^2*d^2-4*a*b*c*d+5*b^2*c^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)/a^(5/2)/b^(3/2)/d^(1/4)/e^(3
/2)/(-d*x^2+c)^(1/2)+1/4*c^(1/4)*(-a^2*d^2-4*a*b*c*d+5*b^2*c^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)/a^(5/2)/b^(3/2)/d^(1/4)/e^(3/2)/(-d*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {477, 479, 597, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=\frac {c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} (5 b c-a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a^2 b e^{3/2} \sqrt {c-d x^2}}-\frac {c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} (5 b c-a d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 b e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt {c-d x^2} (5 b c-a d)}{2 a^2 b e \sqrt {e x}}-\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \left (-a^2 d^2-4 a b c d+5 b^2 c^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \left (-a^2 d^2-4 a b c d+5 b^2 c^2\right ) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt {c-d x^2} (b c-a d)}{2 a b e \sqrt {e x} \left (a-b x^2\right )} \]

[In]

Int[(c - d*x^2)^(3/2)/((e*x)^(3/2)*(a - b*x^2)^2),x]

[Out]

-1/2*((5*b*c - a*d)*Sqrt[c - d*x^2])/(a^2*b*e*Sqrt[e*x]) + ((b*c - a*d)*Sqrt[c - d*x^2])/(2*a*b*e*Sqrt[e*x]*(a
- b*x^2)) - (c^(3/4)*d^(1/4)*(5*b*c - a*d)*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*
Sqrt[e])], -1])/(2*a^2*b*e^(3/2)*Sqrt[c - d*x^2]) + (c^(3/4)*d^(1/4)*(5*b*c - a*d)*Sqrt[1 - (d*x^2)/c]*Ellipti
cF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a^2*b*e^(3/2)*Sqrt[c - d*x^2]) - (c^(1/4)*(5*b^2*c^2
- 4*a*b*c*d - a^2*d^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])/(4*a^(5/2)*b^(3/2)*d^(1/4)*e^(3/2)*Sqrt[c - d*x^2]) + (c^(1/4)*(5*b^2*c^2
- 4*a*b*c*d - a^2*d^2)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sq
rt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*a^(5/2)*b^(3/2)*d^(1/4)*e^(3/2)*Sqrt[c - d*x^2])

Rule 227

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[Rt[-b, 4]*(x/Rt[a, 4])], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 230

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[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]

Rule 313

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Dist[-q^(-1), Int[1/Sqrt[a + b*x^4]
, x], x] + Dist[1/q, Int[(1 + q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 479

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-(c*b -
a*d))*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*b*e*n*(p + 1))), x] + Dist[1/(a*b*n*(p + 1)),
Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(
p + 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 504

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s
= Denominator[Rt[-a/b, 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), Int[1/
((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 597

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] + Dist[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]

Rule 598

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> 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]

Rule 1213

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[d/Sqrt[a], Int[Sqrt[1 + e*(x^2/d)]/Sqrt
[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]

Rule 1214

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4], In
t[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] &&
!GtQ[a, 0]

Rule 1232

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]

Rule 1233

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {\left (c-\frac {d x^4}{e^2}\right )^{3/2}}{x^2 \left (a-\frac {b x^4}{e^2}\right )^2} \, dx,x,\sqrt {e x}\right )}{e} \\ & = \frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e \sqrt {e x} \left (a-b x^2\right )}+\frac {e \text {Subst}\left (\int \frac {\frac {c (5 b c-a d)}{e^2}-\frac {d (3 b c+a d) x^4}{e^4}}{x^2 \left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a b} \\ & = -\frac {(5 b c-a d) \sqrt {c-d x^2}}{2 a^2 b e \sqrt {e x}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e \sqrt {e x} \left (a-b x^2\right )}-\frac {e \text {Subst}\left (\int \frac {x^2 \left (-\frac {b c^2 (5 b c-9 a d)}{e^4}-\frac {b c d (5 b c-a d) x^4}{e^6}\right )}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a^2 b c} \\ & = -\frac {(5 b c-a d) \sqrt {c-d x^2}}{2 a^2 b e \sqrt {e x}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e \sqrt {e x} \left (a-b x^2\right )}-\frac {e \text {Subst}\left (\int \left (\frac {c d (5 b c-a d) x^2}{e^4 \sqrt {c-\frac {d x^4}{e^2}}}-\frac {\left (5 b^2 c^3-4 a b c^2 d-a^2 c d^2\right ) x^2}{e^4 \left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}}\right ) \, dx,x,\sqrt {e x}\right )}{2 a^2 b c} \\ & = -\frac {(5 b c-a d) \sqrt {c-d x^2}}{2 a^2 b e \sqrt {e x}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e \sqrt {e x} \left (a-b x^2\right )}-\frac {(d (5 b c-a d)) \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a^2 b e^3}+\frac {((b c-a d) (5 b c+a d)) \text {Subst}\left (\int \frac {x^2}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a^2 b e^3} \\ & = -\frac {(5 b c-a d) \sqrt {c-d x^2}}{2 a^2 b e \sqrt {e x}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e \sqrt {e x} \left (a-b x^2\right )}+\frac {\left (\sqrt {c} \sqrt {d} (5 b c-a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a^2 b e^2}-\frac {\left (\sqrt {c} \sqrt {d} (5 b c-a d)\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a^2 b e^2}+\frac {((b c-a d) (5 b c+a d)) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} x^2\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a^2 b^{3/2} e}-\frac {((b c-a d) (5 b c+a d)) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e+\sqrt {b} x^2\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a^2 b^{3/2} e} \\ & = -\frac {(5 b c-a d) \sqrt {c-d x^2}}{2 a^2 b e \sqrt {e x}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e \sqrt {e x} \left (a-b x^2\right )}+\frac {\left (\sqrt {c} \sqrt {d} (5 b c-a d) \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a^2 b e^2 \sqrt {c-d x^2}}-\frac {\left (\sqrt {c} \sqrt {d} (5 b c-a d) \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a^2 b e^2 \sqrt {c-d x^2}}+\frac {\left ((b c-a d) (5 b c+a d) \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} x^2\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a^2 b^{3/2} e \sqrt {c-d x^2}}-\frac {\left ((b c-a d) (5 b c+a d) \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e+\sqrt {b} x^2\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a^2 b^{3/2} e \sqrt {c-d x^2}} \\ & = -\frac {(5 b c-a d) \sqrt {c-d x^2}}{2 a^2 b e \sqrt {e x}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e \sqrt {e x} \left (a-b x^2\right )}+\frac {c^{3/4} \sqrt [4]{d} (5 b c-a d) \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 b e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d) (5 b c+a d) \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^{5/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) (5 b c+a d) \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^{5/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}-\frac {\left (\sqrt {c} \sqrt {d} (5 b c-a d) \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {\sqrt {d} x^2}{\sqrt {c} e}}}{\sqrt {1-\frac {\sqrt {d} x^2}{\sqrt {c} e}}} \, dx,x,\sqrt {e x}\right )}{2 a^2 b e^2 \sqrt {c-d x^2}} \\ & = -\frac {(5 b c-a d) \sqrt {c-d x^2}}{2 a^2 b e \sqrt {e x}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e \sqrt {e x} \left (a-b x^2\right )}-\frac {c^{3/4} \sqrt [4]{d} (5 b c-a d) \sqrt {1-\frac {d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 b e^{3/2} \sqrt {c-d x^2}}+\frac {c^{3/4} \sqrt [4]{d} (5 b c-a d) \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 b e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d) (5 b c+a d) \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^{5/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) (5 b c+a d) \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^{5/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 11.19 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.38 \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=\frac {x \left (21 a \left (c-d x^2\right ) \left (4 a c-5 b c x^2+a d x^2\right )+7 c (-5 b c+9 a d) x^2 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 d (-5 b c+a d) x^4 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{42 a^3 (e x)^{3/2} \left (-a+b x^2\right ) \sqrt {c-d x^2}} \]

[In]

Integrate[(c - d*x^2)^(3/2)/((e*x)^(3/2)*(a - b*x^2)^2),x]

[Out]

(x*(21*a*(c - d*x^2)*(4*a*c - 5*b*c*x^2 + a*d*x^2) + 7*c*(-5*b*c + 9*a*d)*x^2*(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] + 3*d*(-5*b*c + a*d)*x^4*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*Appe
llF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(42*a^3*(e*x)^(3/2)*(-a + b*x^2)*Sqrt[c - d*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1344\) vs. \(2(403)=806\).

Time = 3.01 (sec) , antiderivative size = 1345, normalized size of antiderivative = 2.59

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1345\)
default	\(\text {Expression too large to display}\)	\(3867\)

[In]

int((-d*x^2+c)^(3/2)/(e*x)^(3/2)/(-b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

((-d*x^2+c)*e*x)^(1/2)/(e*x)^(1/2)/(-d*x^2+c)^(1/2)*(-1/2*(a*d-b*c)/a^2/e^2*x*(-d*e*x^3+c*e*x)^(1/2)/(-b*x^2+a
)-2*(-d*e*x^2+c*e)/e^2*c/a^2/(x*(-d*e*x^2+c*e))^(1/2)-1/2*c*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)/a/e/b*EllipticE(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/
2),1/2*2^(1/2))+1/4*c*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)/a/e/b*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+5/2*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/a^2*EllipticE(((x+1
/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-5/4*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/a^2*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1
/2*2^(1/2))+1/8/e/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/a/e/b*(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))*c-5/8/a^2/e/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))*c^2+1/8/e/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/a/e/b*(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))*c-5/8/a^2/e/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))*c^2)

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=\text {Timed out} \]

[In]

integrate((-d*x^2+c)^(3/2)/(e*x)^(3/2)/(-b*x^2+a)^2,x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=\int \frac {\left (c - d x^{2}\right )^{\frac {3}{2}}}{\left (e x\right )^{\frac {3}{2}} \left (- a + b x^{2}\right )^{2}}\, dx \]

[In]

integrate((-d*x**2+c)**(3/2)/(e*x)**(3/2)/(-b*x**2+a)**2,x)

[Out]

Integral((c - d*x**2)**(3/2)/((e*x)**(3/2)*(-a + b*x**2)**2), x)

Maxima [F]

\[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=\int { \frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} - a\right )}^{2} \left (e x\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate((-d*x^2+c)^(3/2)/(e*x)^(3/2)/(-b*x^2+a)^2,x, algorithm="maxima")

[Out]

integrate((-d*x^2 + c)^(3/2)/((b*x^2 - a)^2*(e*x)^(3/2)), x)

Giac [F]

\[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=\int { \frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} - a\right )}^{2} \left (e x\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate((-d*x^2+c)^(3/2)/(e*x)^(3/2)/(-b*x^2+a)^2,x, algorithm="giac")

[Out]

integrate((-d*x^2 + c)^(3/2)/((b*x^2 - a)^2*(e*x)^(3/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=\int \frac {{\left (c-d\,x^2\right )}^{3/2}}{{\left (e\,x\right )}^{3/2}\,{\left (a-b\,x^2\right )}^2} \,d x \]

[In]

int((c - d*x^2)^(3/2)/((e*x)^(3/2)*(a - b*x^2)^2),x)

[Out]

int((c - d*x^2)^(3/2)/((e*x)^(3/2)*(a - b*x^2)^2), x)