3.972 \(\int \frac{1}{x \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=46 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{a} \sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.172883, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{a} \sqrt{c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.6223, size = 42, normalized size = 0.91 \[ - \frac{\operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x^{2}}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{\sqrt{a} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0834788, size = 153, normalized size = 3.33 \[ \frac{2 b d x^2 F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}{\sqrt{a+b x^2} \sqrt{c+d x^2} \left (-4 b d x^2 F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )+b c F_1\left (2;\frac{1}{2},\frac{3}{2};3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )+a d F_1\left (2;\frac{3}{2},\frac{1}{2};3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [B]  time = 0.026, size = 103, normalized size = 2.2 \[ -{\frac{1}{2}\ln \left ({\frac{1}{{x}^{2}} \left ( ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac \right ) } \right ) \sqrt{d{x}^{2}+c}\sqrt{b{x}^{2}+a}{\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*ln((a*d*x^2+c*x^2*b+2*(a*c)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+2*a*c
)/x^2)*(d*x^2+c)^(1/2)*(b*x^2+a)^(1/2)/(a*c)^(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(1/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.259627, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} -{\left ({\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt{a c}}{x^{4}}\right )}{4 \, \sqrt{a c}}, -\frac{\arctan \left (\frac{{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt{-a c}}{2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} a c}\right )}{2 \, \sqrt{-a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.229428, size = 120, normalized size = 2.61 \[ -\frac{\sqrt{b d} b \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x),x, algorithm="giac")

[Out]

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