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

Optimal. Leaf size=188 \[ \frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2}} \]

[Out]

c^(1/4)*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
/d^(1/4)/e^(1/2)/(-d*x^2+c)^(1/2)+c^(1/4)*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/d^(1/4)/e^(1/2)/(-d*x^2+c)^(1/2)

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Rubi [A]  time = 0.25, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {466, 409, 1219, 1218} \[ \frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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)*Sqrt[e]*Sqrt[c - d*x^2]) + (c^(1/4)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*S
qrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(a*d^(1/4)*Sqrt[e]*Sqrt[c - d*x
^2])

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 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 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]

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]

Rubi steps

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

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Mathematica [C]  time = 0.04, size = 68, normalized size = 0.36 \[ \frac {2 x \sqrt {\frac {c-d x^2}{c}} F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )}{a \sqrt {e x} \sqrt {c-d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{{\left (b x^{2} - a\right )} \sqrt {-d x^{2} + c} \sqrt {e x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.03, size = 346, normalized size = 1.84 \[ -\frac {\left (-\sqrt {c d}\, b \EllipticPi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right )+\sqrt {c d}\, b \EllipticPi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b +\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right )-\sqrt {a b}\, d \EllipticPi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right )-\sqrt {a b}\, d \EllipticPi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b +\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right )\right ) \sqrt {c d}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-d \,x^{2}+c}\, b}{2 \left (\sqrt {c d}\, b -\sqrt {a b}\, d \right ) \left (\sqrt {c d}\, b +\sqrt {a b}\, d \right ) \sqrt {a b}\, \left (d \,x^{2}-c \right ) \sqrt {e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {1}{{\left (b x^{2} - a\right )} \sqrt {-d x^{2} + c} \sqrt {e x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {e\,x}\,\left (a-b\,x^2\right )\,\sqrt {c-d\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {1}{- a \sqrt {e x} \sqrt {c - d x^{2}} + b x^{2} \sqrt {e x} \sqrt {c - d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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