3.3.81 \(\int \frac {\sqrt {-c+d x^2}}{\sqrt {-a+b x^2}} \, dx\) [281]

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 = 0.04, 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 {d x^2-c} E\left (\text {ArcSin}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {b x^2-a} \sqrt {1-\frac {d x^2}{c}}} \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]*Sq
rt[-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 = 0.71, 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. \(162\) vs. \(2(75)=150\).
time = 0.08, size = 163, normalized size = 1.81

method	result	size
default	\(\frac {\sqrt {d \,x^{2}-c}\, \sqrt {b \,x^{2}-a}\, \sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {-b \,x^{2}+a}{a}}\, \left (a d \EllipticF \left (x \sqrt {\frac {d}{c}}, \sqrt {\frac {b c}{a d}}\right )-c \EllipticF \left (x \sqrt {\frac {d}{c}}, \sqrt {\frac {b c}{a d}}\right ) b -a d \EllipticE \left (x \sqrt {\frac {d}{c}}, \sqrt {\frac {b c}{a d}}\right )\right )}{\left (b d \,x^{4}-a d \,x^{2}-c \,x^{2} b +a c \right ) \sqrt {\frac {d}{c}}\, b}\)	\(163\)
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (-d \,x^{2}+c \right )}\, \left (-\frac {c \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \EllipticF \left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {b d \,x^{4}-a d \,x^{2}-c \,x^{2} b +a c}}+\frac {d a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \left (\EllipticF \left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )-\EllipticE \left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )\right )}{\sqrt {\frac {d}{c}}\, \sqrt {b d \,x^{4}-a d \,x^{2}-c \,x^{2} b +a c}\, b}\right )}{\sqrt {b \,x^{2}-a}\, \sqrt {d \,x^{2}-c}}\)	\(267\)

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 {b\,x^2-a}} \,d x \end {gather*}

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)

[Out]

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

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