3.24 \(\int \frac{x (A+B x)}{\sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=56 \[ \frac{\sqrt{a+b x^2} (2 A+B x)}{2 b}-\frac{a B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0803692, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\sqrt{a+b x^2} (2 A+B x)}{2 b}-\frac{a B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.39284, size = 48, normalized size = 0.86 \[ - \frac{B a \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{2 b^{\frac{3}{2}}} + \frac{\left (2 A + B x\right ) \sqrt{a + b x^{2}}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0536122, size = 61, normalized size = 1.09 \[ \sqrt{a+b x^2} \left (\frac{A}{b}+\frac{B x}{2 b}\right )-\frac{a B \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{2 b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 55, normalized size = 1. \[{\frac{A}{b}\sqrt{b{x}^{2}+a}}+{\frac{Bx}{2\,b}\sqrt{b{x}^{2}+a}}-{\frac{Ba}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.261594, size = 1, normalized size = 0.02 \[ \left [\frac{B a \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \, \sqrt{b x^{2} + a}{\left (B x + 2 \, A\right )} \sqrt{b}}{4 \, b^{\frac{3}{2}}}, -\frac{B a \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) - \sqrt{b x^{2} + a}{\left (B x + 2 \, A\right )} \sqrt{-b}}{2 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.57541, size = 70, normalized size = 1.25 \[ A \left (\begin{cases} \frac{x^{2}}{2 \sqrt{a}} & \text{for}\: b = 0 \\\frac{\sqrt{a + b x^{2}}}{b} & \text{otherwise} \end{cases}\right ) + \frac{B \sqrt{a} x \sqrt{1 + \frac{b x^{2}}{a}}}{2 b} - \frac{B a \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*Piecewise((x**2/(2*sqrt(a)), Eq(b, 0)), (sqrt(a + b*x**2)/b, True)) + B*sqrt(a
)*x*sqrt(1 + b*x**2/a)/(2*b) - B*a*asinh(sqrt(b)*x/sqrt(a))/(2*b**(3/2))

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GIAC/XCAS [A]  time = 0.219447, size = 68, normalized size = 1.21 \[ \frac{1}{2} \, \sqrt{b x^{2} + a}{\left (\frac{B x}{b} + \frac{2 \, A}{b}\right )} + \frac{B a{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{2 \, b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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