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

Optimal. Leaf size=76 \[ \frac{B \sqrt{a+b x^2+c x^4}}{2 c}-\frac{(b B-2 A c) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 c^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.161301, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{B \sqrt{a+b x^2+c x^4}}{2 c}-\frac{(b B-2 A c) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.4067, size = 66, normalized size = 0.87 \[ \frac{B \sqrt{a + b x^{2} + c x^{4}}}{2 c} + \frac{\left (2 A c - B b\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{4 c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0635454, size = 75, normalized size = 0.99 \[ \frac{(2 A c-b B) \log \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right )}{4 c^{3/2}}+\frac{B \sqrt{a+b x^2+c x^4}}{2 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 93, normalized size = 1.2 \[{\frac{A}{2}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{B}{2\,c}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{bB}{4}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*A*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))/c^(1/2)+1/2*B*(c*x^4+b*x^2
+a)^(1/2)/c-1/4*B*b/c^(3/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(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((B*x^2 + A)*x/sqrt(c*x^4 + b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.313552, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{c x^{4} + b x^{2} + a} B \sqrt{c} -{\left (B b - 2 \, A c\right )} \log \left (-4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c^{2} x^{2} + b c\right )} -{\left (8 \, c^{2} x^{4} + 8 \, b c x^{2} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{8 \, c^{\frac{3}{2}}}, \frac{2 \, \sqrt{c x^{4} + b x^{2} + a} B \sqrt{-c} -{\left (B b - 2 \, A c\right )} \arctan \left (\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{4} + b x^{2} + a} c}\right )}{4 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (A + B x^{2}\right )}{\sqrt{a + b x^{2} + c x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*(A + B*x**2)/sqrt(a + b*x**2 + c*x**4), x)

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GIAC/XCAS [A]  time = 0.296469, size = 93, normalized size = 1.22 \[ \frac{\sqrt{c x^{4} + b x^{2} + a} B}{2 \, c} + \frac{{\left (B b - 2 \, A c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} \sqrt{c} - b \right |}\right )}{4 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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