3.893 \(\int \frac{1}{\sqrt{e x} (a-b x^2) (c-d x^2)^{3/2}} \, dx\)

Optimal. Leaf size=328 \[ -\frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),-1\right )}{c^{3/4} \sqrt{e} \sqrt{c-d x^2} (b c-a d)}-\frac{d \sqrt{e x}}{c e \sqrt{c-d x^2} (b c-a d)}+\frac{b \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2} (b c-a d)}+\frac{b \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2} (b c-a d)} \]

[Out]

-((d*Sqrt[e*x])/(c*(b*c - a*d)*e*Sqrt[c - d*x^2])) - (d^(3/4)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sq
rt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(c^(3/4)*(b*c - a*d)*Sqrt[e]*Sqrt[c - d*x^2]) + (b*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])/(a*d
^(1/4)*(b*c - a*d)*Sqrt[e]*Sqrt[c - d*x^2]) + (b*c^(1/4)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[c])/(Sqr
t[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(a*d^(1/4)*(b*c - a*d)*Sqrt[e]*Sqrt[c - d*x
^2])

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Rubi [A]  time = 0.48782, antiderivative size = 328, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {466, 414, 523, 224, 221, 409, 1219, 1218} \[ -\frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{c^{3/4} \sqrt{e} \sqrt{c-d x^2} (b c-a d)}-\frac{d \sqrt{e x}}{c e \sqrt{c-d x^2} (b c-a d)}+\frac{b \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2} (b c-a d)}+\frac{b \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[e*x]*(a - b*x^2)*(c - d*x^2)^(3/2)),x]

[Out]

-((d*Sqrt[e*x])/(c*(b*c - a*d)*e*Sqrt[c - d*x^2])) - (d^(3/4)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sq
rt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(c^(3/4)*(b*c - a*d)*Sqrt[e]*Sqrt[c - d*x^2]) + (b*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])/(a*d
^(1/4)*(b*c - a*d)*Sqrt[e]*Sqrt[c - d*x^2]) + (b*c^(1/4)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[c])/(Sqr
t[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(a*d^(1/4)*(b*c - a*d)*Sqrt[e]*Sqrt[c - d*x
^2])

Rule 466

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 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 523

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(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]

Rule 224

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 221

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 409

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1
- Rt[-(d/c), 2]*x^2)), x], x] + Dist[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]

Rule 1219

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]

Rule 1218

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-(c/a), 4]}, Simp[(1*Ellipt
icPi[-(e/(d*q^2)), ArcSin[q*x], -1])/(d*Sqrt[a]*q), x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{e x} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{\left (a-\frac{b x^4}{e^2}\right ) \left (c-\frac{d x^4}{e^2}\right )^{3/2}} \, dx,x,\sqrt{e x}\right )}{e}\\ &=-\frac{d \sqrt{e x}}{c (b c-a d) e \sqrt{c-d x^2}}-\frac{e \operatorname{Subst}\left (\int \frac{-\frac{2 b c-a d}{e^2}-\frac{b d x^4}{e^4}}{\left (a-\frac{b x^4}{e^2}\right ) \sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{c (b c-a d)}\\ &=-\frac{d \sqrt{e x}}{c (b c-a d) e \sqrt{c-d x^2}}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{\left (a-\frac{b x^4}{e^2}\right ) \sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{(b c-a d) e}-\frac{d \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{c (b c-a d) e}\\ &=-\frac{d \sqrt{e x}}{c (b c-a d) e \sqrt{c-d x^2}}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{\sqrt{b} x^2}{\sqrt{a} e}\right ) \sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{a (b c-a d) e}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{\sqrt{b} x^2}{\sqrt{a} e}\right ) \sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{a (b c-a d) e}-\frac{\left (d \sqrt{1-\frac{d x^2}{c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{d x^4}{c e^2}}} \, dx,x,\sqrt{e x}\right )}{c (b c-a d) e \sqrt{c-d x^2}}\\ &=-\frac{d \sqrt{e x}}{c (b c-a d) e \sqrt{c-d x^2}}-\frac{d^{3/4} \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 )}{c^{3/4} (b c-a d) \sqrt{e} \sqrt{c-d x^2}}+\frac{\left (b \sqrt{1-\frac{d x^2}{c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{\sqrt{b} x^2}{\sqrt{a} e}\right ) \sqrt{1-\frac{d x^4}{c e^2}}} \, dx,x,\sqrt{e x}\right )}{a (b c-a d) e \sqrt{c-d x^2}}+\frac{\left (b \sqrt{1-\frac{d x^2}{c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{\sqrt{b} x^2}{\sqrt{a} e}\right ) \sqrt{1-\frac{d x^4}{c e^2}}} \, dx,x,\sqrt{e x}\right )}{a (b c-a d) e \sqrt{c-d x^2}}\\ &=-\frac{d \sqrt{e x}}{c (b c-a d) e \sqrt{c-d x^2}}-\frac{d^{3/4} \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 )}{c^{3/4} (b c-a d) \sqrt{e} \sqrt{c-d x^2}}+\frac{b \sqrt [4]{c} \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 \sqrt [4]{d} (b c-a d) \sqrt{e} \sqrt{c-d x^2}}+\frac{b \sqrt [4]{c} \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 \sqrt [4]{d} (b c-a d) \sqrt{e} \sqrt{c-d x^2}}\\ \end{align*}

Mathematica [C]  time = 0.126273, size = 147, normalized size = 0.45 \[ \frac{b d x^3 \sqrt{1-\frac{d x^2}{c}} F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+5 x \sqrt{1-\frac{d x^2}{c}} (2 b c-a d) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )-5 a d x}{5 a c \sqrt{e x} \sqrt{c-d x^2} (b c-a d)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(Sqrt[e*x]*(a - b*x^2)*(c - d*x^2)^(3/2)),x]

[Out]

(-5*a*d*x + 5*(2*b*c - a*d)*x*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + b*d*x^3*S
qrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a])/(5*a*c*(b*c - a*d)*Sqrt[e*x]*Sqrt[c - d*x
^2])

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Maple [B]  time = 0.03, size = 706, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b*d*(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/
2))^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(
1/2))*b^2*c^2+((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)
^(1/2))^(1/2)*(a*b)^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^
(1/2)*d),1/2*2^(1/2))*(c*d)^(1/2)*b*c-EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a*d
*(a*b)^(1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c
*d)^(1/2))^(1/2)+EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*b*c*(a*b)^(1/2)*(c*d)^(1
/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)-((d*
x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*Elli
pticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*b^2*c^2+
((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*
(a*b)^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2
^(1/2))*(c*d)^(1/2)*b*c-2*x*a*d^2*(a*b)^(1/2)+2*x*b*c*d*(a*b)^(1/2))*(-d*x^2+c)^(1/2)/c/((c*d)^(1/2)*b-(a*b)^(
1/2)*d)/((a*b)^(1/2)*d+(c*d)^(1/2)*b)/(a*b)^(1/2)/(a*d-b*c)/(d*x^2-c)/(e*x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{{\left (b x^{2} - a\right )}{\left (-d x^{2} + c\right )}^{\frac{3}{2}} \sqrt{e x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{- a c \sqrt{e x} \sqrt{c - d x^{2}} + a d x^{2} \sqrt{e x} \sqrt{c - d x^{2}} + b c x^{2} \sqrt{e x} \sqrt{c - d x^{2}} - b d x^{4} \sqrt{e x} \sqrt{c - d x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(1/(-a*c*sqrt(e*x)*sqrt(c - d*x**2) + a*d*x**2*sqrt(e*x)*sqrt(c - d*x**2) + b*c*x**2*sqrt(e*x)*sqrt(c
- d*x**2) - b*d*x**4*sqrt(e*x)*sqrt(c - d*x**2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{{\left (b x^{2} - a\right )}{\left (-d x^{2} + c\right )}^{\frac{3}{2}} \sqrt{e x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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