3.3.75 \(\int \frac {\sqrt {-c-d x^2}}{\sqrt {a-b x^2}} \, dx\) [275]

Optimal. Leaf size=90 \[ \frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {-c-d x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {438, 437, 435} \begin {gather*} \frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {-c-d x^2} E\left (\text {ArcSin}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a]*Sqrt[1 - (b*x^2)/a]*Sqrt[-c - d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]
*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c])

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 437

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

Rule 438

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

Rubi steps

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

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Mathematica [A]
time = 0.88, size = 90, normalized size = 1.00 \begin {gather*} \frac {\sqrt {\frac {a-b x^2}{a}} \sqrt {-c-d x^2} E\left (\sin ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} \sqrt {a-b x^2} \sqrt {\frac {c+d x^2}{c}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a - b*x^2)/a]*Sqrt[-c - d*x^2]*EllipticE[ArcSin[Sqrt[b/a]*x], -((a*d)/(b*c))])/(Sqrt[b/a]*Sqrt[a - b*x^
2]*Sqrt[(c + d*x^2)/c])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(167\) vs. \(2(75)=150\).
time = 0.08, size = 168, normalized size = 1.87

method	result	size
default	\(\frac {\sqrt {-d \,x^{2}-c}\, \sqrt {-b \,x^{2}+a}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {-b \,x^{2}+a}{a}}\, \left (a d \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right )+c \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) b -a d \EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right )\right )}{\left (-b d \,x^{4}+a d \,x^{2}-c \,x^{2} b +a c \right ) \sqrt {-\frac {d}{c}}\, b}\)	\(168\)
elliptic	\(\frac {\sqrt {-\left (-b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (-\frac {c \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {b d \,x^{4}-a d \,x^{2}+c \,x^{2} b -a c}}-\frac {d a \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \left (\EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {b d \,x^{4}-a d \,x^{2}+c \,x^{2} b -a c}\, b}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {-d \,x^{2}-c}}\)	\(268\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- c - d x^{2}}}{\sqrt {a - b x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {-d\,x^2-c}}{\sqrt {a-b\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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