3.133 \(\int \frac{x \left (A+B x^2\right )}{\sqrt{b x^2+c x^4}} \, dx\)

Optimal. Leaf size=66 \[ \frac{B \sqrt{b x^2+c x^4}}{2 c}-\frac{(b B-2 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{2 c^{3/2}} \]

[Out]

(B*Sqrt[b*x^2 + c*x^4])/(2*c) - ((b*B - 2*A*c)*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2
+ c*x^4]])/(2*c^(3/2))

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Rubi [A]  time = 0.224472, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{B \sqrt{b x^2+c x^4}}{2 c}-\frac{(b B-2 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{2 c^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(x*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]

[Out]

(B*Sqrt[b*x^2 + c*x^4])/(2*c) - ((b*B - 2*A*c)*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2
+ c*x^4]])/(2*c^(3/2))

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Rubi in Sympy [A]  time = 17.6371, size = 56, normalized size = 0.85 \[ \frac{B \sqrt{b x^{2} + c x^{4}}}{2 c} + \frac{\left (2 A c - B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{b x^{2} + c x^{4}}} \right )}}{2 c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(B*x**2+A)/(c*x**4+b*x**2)**(1/2),x)

[Out]

B*sqrt(b*x**2 + c*x**4)/(2*c) + (2*A*c - B*b)*atanh(sqrt(c)*x**2/sqrt(b*x**2 + c
*x**4))/(2*c**(3/2))

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Mathematica [A]  time = 0.084591, size = 84, normalized size = 1.27 \[ \frac{x \left (\sqrt{b+c x^2} (2 A c-b B) \log \left (\sqrt{c} \sqrt{b+c x^2}+c x\right )+B \sqrt{c} x \left (b+c x^2\right )\right )}{2 c^{3/2} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]

[Out]

(x*(B*Sqrt[c]*x*(b + c*x^2) + (-(b*B) + 2*A*c)*Sqrt[b + c*x^2]*Log[c*x + Sqrt[c]
*Sqrt[b + c*x^2]]))/(2*c^(3/2)*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.009, size = 88, normalized size = 1.3 \[{\frac{x}{2}\sqrt{c{x}^{2}+b} \left ( Bx\sqrt{c{x}^{2}+b}{c}^{{\frac{3}{2}}}+2\,A\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+b} \right ){c}^{2}-Bb\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+b} \right ) c \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(B*x^2+A)/(c*x^4+b*x^2)^(1/2),x)

[Out]

1/2*x*(c*x^2+b)^(1/2)*(B*x*(c*x^2+b)^(1/2)*c^(3/2)+2*A*ln(c^(1/2)*x+(c*x^2+b)^(1
/2))*c^2-B*b*ln(c^(1/2)*x+(c*x^2+b)^(1/2))*c)/(c*x^4+b*x^2)^(1/2)/c^(5/2)

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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((B*x^2 + A)*x/sqrt(c*x^4 + b*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233946, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{c x^{4} + b x^{2}} B c -{\left (B b - 2 \, A c\right )} \sqrt{c} \log \left (-{\left (2 \, c x^{2} + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{4} + b x^{2}} c\right )}{4 \, c^{2}}, \frac{\sqrt{c x^{4} + b x^{2}} B c +{\left (B b - 2 \, A c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + b x^{2}}}\right )}{2 \, c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x/sqrt(c*x^4 + b*x^2),x, algorithm="fricas")

[Out]

[1/4*(2*sqrt(c*x^4 + b*x^2)*B*c - (B*b - 2*A*c)*sqrt(c)*log(-(2*c*x^2 + b)*sqrt(
c) - 2*sqrt(c*x^4 + b*x^2)*c))/c^2, 1/2*(sqrt(c*x^4 + b*x^2)*B*c + (B*b - 2*A*c)
*sqrt(-c)*arctan(sqrt(-c)*x^2/sqrt(c*x^4 + b*x^2)))/c^2]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(B*x**2+A)/(c*x**4+b*x**2)**(1/2),x)

[Out]

Integral(x*(A + B*x**2)/sqrt(x**2*(b + c*x**2)), x)

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GIAC/XCAS [A]  time = 0.242774, size = 90, normalized size = 1.36 \[ \frac{\sqrt{c x^{4} + b x^{2}} B}{2 \, c} + \frac{{\left (B b - 2 \, A c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2}}\right )} \sqrt{c} - b \right |}\right )}{4 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x/sqrt(c*x^4 + b*x^2),x, algorithm="giac")

[Out]

1/2*sqrt(c*x^4 + b*x^2)*B/c + 1/4*(B*b - 2*A*c)*ln(abs(-2*(sqrt(c)*x^2 - sqrt(c*
x^4 + b*x^2))*sqrt(c) - b))/c^(3/2)