3.990 \(\int \frac{x}{\sqrt{a-b x^2} \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=47 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a-b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{\sqrt{b} \sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.126019, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a-b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{\sqrt{b} \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.8084, size = 41, normalized size = 0.87 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{d} \sqrt{a - b x^{2}}} \right )}}{\sqrt{b} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(-b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)

[Out]

atan(sqrt(b)*sqrt(c + d*x**2)/(sqrt(d)*sqrt(a - b*x**2)))/(sqrt(b)*sqrt(d))

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Mathematica [C]  time = 0.126204, size = 72, normalized size = 1.53 \[ \frac{i \log \left (2 \sqrt{a-b x^2} \sqrt{c+d x^2}-\frac{i \left (-a d+b c+2 b d x^2\right )}{\sqrt{b} \sqrt{d}}\right )}{2 \sqrt{b} \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

((I/2)*Log[2*Sqrt[a - b*x^2]*Sqrt[c + d*x^2] - (I*(b*c - a*d + 2*b*d*x^2))/(Sqrt
[b]*Sqrt[d])])/(Sqrt[b]*Sqrt[d])

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Maple [B]  time = 0.057, size = 108, normalized size = 2.3 \[{\frac{1}{2}\arctan \left ({\frac{2\,bd{x}^{2}-ad+bc}{2\,bd}\sqrt{bd}{\frac{1}{\sqrt{-bd{x}^{4}+ad{x}^{2}-c{x}^{2}b+ac}}}} \right ) \sqrt{-b{x}^{2}+a}\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{-bd{x}^{4}+ad{x}^{2}-c{x}^{2}b+ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arctan(1/2*(b*d)^(1/2)*(2*b*d*x^2-a*d+b*c)/b/d/(-b*d*x^4+a*d*x^2-b*c*x^2+a*c
)^(1/2))*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(b*d)^(1/2)/(-b*d*x^4+a*d*x^2-b*c*x^2+
a*c)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241622, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (4 \,{\left (2 \, b^{2} d^{2} x^{2} + b^{2} c d - a b d^{2}\right )} \sqrt{-b x^{2} + a} \sqrt{d x^{2} + c} +{\left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt{-b d}\right )}{4 \, \sqrt{-b d}}, \frac{\arctan \left (\frac{{\left (2 \, b d x^{2} + b c - a d\right )} \sqrt{b d}}{2 \, \sqrt{-b x^{2} + a} \sqrt{d x^{2} + c} b d}\right )}{2 \, \sqrt{b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)),x, algorithm="fricas")

[Out]

[1/4*log(4*(2*b^2*d^2*x^2 + b^2*c*d - a*b*d^2)*sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)
+ (8*b^2*d^2*x^4 + b^2*c^2 - 6*a*b*c*d + a^2*d^2 + 8*(b^2*c*d - a*b*d^2)*x^2)*sq
rt(-b*d))/sqrt(-b*d), 1/2*arctan(1/2*(2*b*d*x^2 + b*c - a*d)*sqrt(b*d)/(sqrt(-b*
x^2 + a)*sqrt(d*x^2 + c)*b*d))/sqrt(b*d)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.239551, size = 77, normalized size = 1.64 \[ \frac{b{\rm ln}\left ({\left | -\sqrt{-b x^{2} + a} \sqrt{-b d} + \sqrt{b^{2} c +{\left (b x^{2} - a\right )} b d + a b d} \right |}\right )}{\sqrt{-b d}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)),x, algorithm="giac")

[Out]

b*ln(abs(-sqrt(-b*x^2 + a)*sqrt(-b*d) + sqrt(b^2*c + (b*x^2 - a)*b*d + a*b*d)))/
(sqrt(-b*d)*abs(b))