3.705 \(\int \frac{x}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{\sqrt{b} \sqrt{b c-a d}} \]

[Out]

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

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Rubi [A]  time = 0.113835, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{\sqrt{b} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.8001, size = 41, normalized size = 0.84 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{\sqrt{b} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0352749, size = 49, normalized size = 1. \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{\sqrt{b} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.015, size = 300, normalized size = 6.1 \[ -{\frac{1}{2\,b}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}-{\frac{1}{2\,b}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244582, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{{\left (b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2}\right )} \sqrt{b^{2} c - a b d} - 4 \,{\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2} +{\left (b^{3} c d - a b^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{4 \, \sqrt{b^{2} c - a b d}}, \frac{\arctan \left (-\frac{{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt{-b^{2} c + a b d}}{2 \,{\left (b^{2} c - a b d\right )} \sqrt{d x^{2} + c}}\right )}{2 \, \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*log(((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*(4*b^2*c*d - 3*a*b*
d^2)*x^2)*sqrt(b^2*c - a*b*d) - 4*(2*b^3*c^2 - 3*a*b^2*c*d + a^2*b*d^2 + (b^3*c*
d - a*b^2*d^2)*x^2)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2))/sqrt(b^2*c - a
*b*d), 1/2*arctan(-1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(-b^2*c + a*b*d)/((b^2*c - a*
b*d)*sqrt(d*x^2 + c)))/sqrt(-b^2*c + a*b*d)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.235547, size = 53, normalized size = 1.08 \[ \frac{\arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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