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

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

[Out]

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

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Rubi [A]  time = 0.0650685, 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 \[ \frac{\sqrt{a+c x^2} (2 A+B x)}{2 c}-\frac{a B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.74127, size = 48, normalized size = 0.86 \[ - \frac{B a \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{2 c^{\frac{3}{2}}} + \frac{\left (2 A + B x\right ) \sqrt{a + c x^{2}}}{2 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0536963, size = 61, normalized size = 1.09 \[ \sqrt{a+c x^2} \left (\frac{A}{c}+\frac{B x}{2 c}\right )-\frac{a B \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{2 c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 55, normalized size = 1. \[{\frac{A}{c}\sqrt{c{x}^{2}+a}}+{\frac{Bx}{2\,c}\sqrt{c{x}^{2}+a}}-{\frac{Ba}{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.344694, size = 1, normalized size = 0.02 \[ \left [\frac{B a \log \left (2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + 2 \, \sqrt{c x^{2} + a}{\left (B x + 2 \, A\right )} \sqrt{c}}{4 \, c^{\frac{3}{2}}}, -\frac{B a \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) - \sqrt{c x^{2} + a}{\left (B x + 2 \, A\right )} \sqrt{-c}}{2 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.12688, size = 70, normalized size = 1.25 \[ A \left (\begin{cases} \frac{x^{2}}{2 \sqrt{a}} & \text{for}\: c = 0 \\\frac{\sqrt{a + c x^{2}}}{c} & \text{otherwise} \end{cases}\right ) + \frac{B \sqrt{a} x \sqrt{1 + \frac{c x^{2}}{a}}}{2 c} - \frac{B a \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.277665, size = 68, normalized size = 1.21 \[ \frac{1}{2} \, \sqrt{c x^{2} + a}{\left (\frac{B x}{c} + \frac{2 \, A}{c}\right )} + \frac{B a{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{2 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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