\(\int \frac {\sqrt {e x} \sqrt {c-d x^2}}{(a-b x^2)^2} \, dx\) [899]

Optimal result

Integrand size = 30, antiderivative size = 417 \[ \int \frac {\sqrt {e x} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\frac {(e x)^{3/2} \sqrt {c-d x^2}}{2 a e \left (a-b x^2\right )}-\frac {c^{3/4} \sqrt [4]{d} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a b \sqrt {c-d x^2}}+\frac {c^{3/4} \sqrt [4]{d} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a b \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c+a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{3/2} b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c+a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{3/2} b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {477, 480, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \[ \int \frac {\sqrt {e x} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=-\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (a d+b c) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{3/2} b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (a d+b c) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{3/2} b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {c^{3/4} \sqrt [4]{d} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a b \sqrt {c-d x^2}}-\frac {c^{3/4} \sqrt [4]{d} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a b \sqrt {c-d x^2}}+\frac {(e x)^{3/2} \sqrt {c-d x^2}}{2 a e \left (a-b x^2\right )} \]

[In]

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

[Out]

((e*x)^(3/2)*Sqrt[c - d*x^2])/(2*a*e*(a - b*x^2)) - (c^(3/4)*d^(1/4)*Sqrt[e]*Sqrt[1 - (d*x^2)/c]*EllipticE[Arc
Sin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a*b*Sqrt[c - d*x^2]) + (c^(3/4)*d^(1/4)*Sqrt[e]*Sqrt[1 - (
d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a*b*Sqrt[c - d*x^2]) - (c^(1/4)*(b*
c + a*d)*Sqrt[e]*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^(3/2)*b^(3/2)*d^(1/4)*Sqrt[c - d*x^2]) + (c^(1/4)*(b*c + a*d)*Sqrt[e]*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^(3/2)*b^(3/2)*d^(1/4)*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 480

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-(e*x)^
(m + 1))*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*e*n*(p + 1))), x] + Dist[1/(a*n*(p + 1)), Int[(e*x)^m*(a + b*x^
n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m + n*(p + 1) + 1) + d*(m + n*(p + 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] && LtQ[0, 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 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 {x^2 \sqrt {c-\frac {d x^4}{e^2}}}{\left (a-\frac {b x^4}{e^2}\right )^2} \, dx,x,\sqrt {e x}\right )}{e} \\ & = \frac {(e x)^{3/2} \sqrt {c-d x^2}}{2 a e \left (a-b x^2\right )}-\frac {\text {Subst}\left (\int \frac {x^2 \left (-c-\frac {d x^4}{e^2}\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 e} \\ & = \frac {(e x)^{3/2} \sqrt {c-d x^2}}{2 a e \left (a-b x^2\right )}-\frac {\text {Subst}\left (\int \left (\frac {d x^2}{b \sqrt {c-\frac {d x^4}{e^2}}}-\frac {(b c+a d) x^2}{b \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 e} \\ & = \frac {(e x)^{3/2} \sqrt {c-d x^2}}{2 a e \left (a-b x^2\right )}-\frac {d \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a b e}+\frac {(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 b e} \\ & = \frac {(e x)^{3/2} \sqrt {c-d x^2}}{2 a e \left (a-b x^2\right )}+\frac {\left (\sqrt {c} \sqrt {d}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a b}-\frac {\left (\sqrt {c} \sqrt {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 b}+\frac {((b c+a d) e) \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 b^{3/2}}-\frac {((b c+a d) e) \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 b^{3/2}} \\ & = \frac {(e x)^{3/2} \sqrt {c-d x^2}}{2 a e \left (a-b x^2\right )}+\frac {\left (\sqrt {c} \sqrt {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 b \sqrt {c-d x^2}}-\frac {\left (\sqrt {c} \sqrt {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 b \sqrt {c-d x^2}}+\frac {\left ((b c+a d) e \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 b^{3/2} \sqrt {c-d x^2}}-\frac {\left ((b c+a d) e \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 b^{3/2} \sqrt {c-d x^2}} \\ & = \frac {(e x)^{3/2} \sqrt {c-d x^2}}{2 a e \left (a-b x^2\right )}+\frac {c^{3/4} \sqrt [4]{d} \sqrt {e} \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 b \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c+a d) \sqrt {e} \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^{3/2} b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c+a d) \sqrt {e} \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^{3/2} b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\left (\sqrt {c} \sqrt {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 b \sqrt {c-d x^2}} \\ & = \frac {(e x)^{3/2} \sqrt {c-d x^2}}{2 a e \left (a-b x^2\right )}-\frac {c^{3/4} \sqrt [4]{d} \sqrt {e} \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 b \sqrt {c-d x^2}}+\frac {c^{3/4} \sqrt [4]{d} \sqrt {e} \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 b \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c+a d) \sqrt {e} \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^{3/2} b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c+a d) \sqrt {e} \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^{3/2} b^{3/2} \sqrt [4]{d} \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 = 11.12 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.39 \[ \int \frac {\sqrt {e x} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\frac {\sqrt {e x} \left (21 a x \left (-c+d x^2\right )+7 c x \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 d x^3 \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{42 a^2 \left (-a+b x^2\right ) \sqrt {c-d x^2}} \]

[In]

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

[Out]

(Sqrt[e*x]*(21*a*x*(-c + d*x^2) + 7*c*x*(-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*x^3*(-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]))/(42
*a^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. \(820\) vs. \(2(307)=614\).

Time = 3.06 (sec) , antiderivative size = 821, normalized size of antiderivative = 1.97

method	result	size
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (\frac {x \sqrt {-d e \,x^{3}+c e x}}{2 a \left (-b \,x^{2}+a \right )}+\frac {e 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\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{2 a b \sqrt {-d e \,x^{3}+c e x}}-\frac {e 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}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{4 a b \sqrt {-d e \,x^{3}+c e x}}-\frac {e \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 )}{8 b^{2} \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {e \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 ) c}{8 a b d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {e \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 )}{8 b^{2} \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}-\frac {e \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 ) c}{8 a b 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}}\)	\(821\)
default	\(\text {Expression too large to display}\)	\(2522\)

[In]

int((e*x)^(1/2)*(-d*x^2+c)^(1/2)/(-b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/e/x*(e*x)^(1/2)/(-d*x^2+c)^(1/2)*((-d*x^2+c)*e*x)^(1/2)*(1/2*x/a*(-d*e*x^3+c*e*x)^(1/2)/(-b*x^2+a)+1/2*e/a/b
*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)*Elli
pticE(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-1/4*e/a/b*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)*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*(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))-1/8*e/a/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)^(1/2)/(-1/d*(c*d)^(1/2)-1
/b*(a*b)^(1/2)),1/2*2^(1/2))*c-1/8*e/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))-1/8*e/a/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)^(1/2)/(-1/d*(c*
d)^(1/2)+1/b*(a*b)^(1/2)),1/2*2^(1/2))*c)

Fricas [F(-1)]

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

[In]

integrate((e*x)^(1/2)*(-d*x^2+c)^(1/2)/(-b*x^2+a)^2,x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

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

[In]

integrate((e*x)**(1/2)*(-d*x**2+c)**(1/2)/(-b*x**2+a)**2,x)

[Out]

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

Maxima [F]

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

[In]

integrate((e*x)^(1/2)*(-d*x^2+c)^(1/2)/(-b*x^2+a)^2,x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((e*x)^(1/2)*(-d*x^2+c)^(1/2)/(-b*x^2+a)^2,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int(((e*x)^(1/2)*(c - d*x^2)^(1/2))/(a - b*x^2)^2,x)

[Out]

int(((e*x)^(1/2)*(c - d*x^2)^(1/2))/(a - b*x^2)^2, x)