\(\int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} (a-b x^2)} \, dx\) [870]

Optimal result

Integrand size = 30, antiderivative size = 392 \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=-\frac {2 \sqrt {c-d x^2}}{a e \sqrt {e x}}-\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a e^{3/2} \sqrt {c-d x^2}}+\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^{3/2} \sqrt {b} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^{3/2} \sqrt {b} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 392, 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, 486, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=-\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^{3/2} \sqrt {b} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^{3/2} \sqrt {b} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a e^{3/2} \sqrt {c-d x^2}}-\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a e^{3/2} \sqrt {c-d x^2}}-\frac {2 \sqrt {c-d x^2}}{a e \sqrt {e x}} \]

[In]

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

[Out]

(-2*Sqrt[c - d*x^2])/(a*e*Sqrt[e*x]) - (2*c^(3/4)*d^(1/4)*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e
*x])/(c^(1/4)*Sqrt[e])], -1])/(a*e^(3/2)*Sqrt[c - d*x^2]) + (2*c^(3/4)*d^(1/4)*Sqrt[1 - (d*x^2)/c]*EllipticF[A
rcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(a*e^(3/2)*Sqrt[c - d*x^2]) - (c^(1/4)*(b*c - a*d)*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])/(a^(3/2)*Sqrt[b]*d^(1/4)*e^(3/2)*Sqrt[c - d*x^2]) + (c^(1/4)*(b*c - a*d)*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])/(a^(3/2)*Sqrt[b]*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 486

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*(m + 1))), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*
x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*b*(m + 1) + n*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) + b*n*(p + q + 1))*x^n, x
], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[0, q, 1] && LtQ[m, -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 {\sqrt {c-\frac {d x^4}{e^2}}}{x^2 \left (a-\frac {b x^4}{e^2}\right )} \, dx,x,\sqrt {e x}\right )}{e} \\ & = -\frac {2 \sqrt {c-d x^2}}{a e \sqrt {e x}}+\frac {2 \text {Subst}\left (\int \frac {x^2 \left (\frac {b c-2 a d}{e^2}+\frac {b d x^4}{e^4}\right )}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{a e} \\ & = -\frac {2 \sqrt {c-d x^2}}{a e \sqrt {e x}}+\frac {2 \text {Subst}\left (\int \left (-\frac {d x^2}{e^2 \sqrt {c-\frac {d x^4}{e^2}}}+\frac {(b c-a d) x^2}{e^2 \left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}}\right ) \, dx,x,\sqrt {e x}\right )}{a e} \\ & = -\frac {2 \sqrt {c-d x^2}}{a e \sqrt {e x}}-\frac {(2 d) \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{a e^3}+\frac {(2 (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 )}{a e^3} \\ & = -\frac {2 \sqrt {c-d x^2}}{a e \sqrt {e x}}+\frac {\left (2 \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 )}{a e^2}-\frac {\left (2 \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 )}{a e^2}+\frac {(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 )}{a \sqrt {b} e}-\frac {(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 )}{a \sqrt {b} e} \\ & = -\frac {2 \sqrt {c-d x^2}}{a e \sqrt {e x}}+\frac {\left (2 \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 )}{a e^2 \sqrt {c-d x^2}}-\frac {\left (2 \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 )}{a e^2 \sqrt {c-d x^2}}+\frac {\left ((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 )}{a \sqrt {b} e \sqrt {c-d x^2}}-\frac {\left ((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 )}{a \sqrt {b} e \sqrt {c-d x^2}} \\ & = -\frac {2 \sqrt {c-d x^2}}{a e \sqrt {e x}}+\frac {2 c^{3/4} \sqrt [4]{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 )}{a e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (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 )}{a^{3/2} \sqrt {b} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (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 )}{a^{3/2} \sqrt {b} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}-\frac {\left (2 \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 )}{a e^2 \sqrt {c-d x^2}} \\ & = -\frac {2 \sqrt {c-d x^2}}{a e \sqrt {e x}}-\frac {2 c^{3/4} \sqrt [4]{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 )}{a e^{3/2} \sqrt {c-d x^2}}+\frac {2 c^{3/4} \sqrt [4]{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 )}{a e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (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 )}{a^{3/2} \sqrt {b} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (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 )}{a^{3/2} \sqrt {b} \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 = 11.09 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.36 \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=\frac {x \left (-42 a \left (c-d x^2\right )+14 (b c-2 a d) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+6 b d x^4 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{21 a^2 (e x)^{3/2} \sqrt {c-d x^2}} \]

[In]

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

[Out]

(x*(-42*a*(c - d*x^2) + 14*(b*c - 2*a*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] + 6*b*d*x^4*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(21*a^2*(e*x)^(3/2)*Sq
rt[c - d*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(818\) vs. \(2(292)=584\).

Time = 3.10 (sec) , antiderivative size = 819, normalized size of antiderivative = 2.09

method	result	size
elliptic	\(\frac {\sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (-\frac {2 \left (-d e \,x^{2}+c e \right )}{e^{2} a \sqrt {x \left (-d e \,x^{2}+c e \right )}}+\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\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{e a \sqrt {-d e \,x^{3}+c e x}}-\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}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{e a \sqrt {-d e \,x^{3}+c e x}}+\frac {\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 e b \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {\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}{2 a e d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {\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 e b \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}-\frac {\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}{2 a e 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 {-d \,x^{2}+c}}\)	\(819\)
default	\(\text {Expression too large to display}\)	\(1263\)

[In]

int((-d*x^2+c)^(1/2)/(e*x)^(3/2)/(-b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

((-d*x^2+c)*e*x)^(1/2)/(e*x)^(1/2)/(-d*x^2+c)^(1/2)*(-2*(-d*e*x^2+c*e)/e^2/a/(x*(-d*e*x^2+c*e))^(1/2)+2/e/a*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)*Ellipti
cE(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-1/e/a*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))+1/2/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))-1/2/a/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+1/2/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))-1/2/a/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)

Fricas [F(-1)]

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

[In]

integrate((-d*x^2+c)^(1/2)/(e*x)^(3/2)/(-b*x^2+a),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

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

[In]

integrate((-d*x**2+c)**(1/2)/(e*x)**(3/2)/(-b*x**2+a),x)

[Out]

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

Maxima [F]

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

[In]

integrate((-d*x^2+c)^(1/2)/(e*x)^(3/2)/(-b*x^2+a),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((-d*x^2+c)^(1/2)/(e*x)^(3/2)/(-b*x^2+a),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int((c - d*x^2)^(1/2)/((e*x)^(3/2)*(a - b*x^2)),x)

[Out]

int((c - d*x^2)^(1/2)/((e*x)^(3/2)*(a - b*x^2)), x)