3.368 \(\int \frac{x (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=48 \[ \frac{B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}-\frac{A+B x}{c \sqrt{a+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0611734, antiderivative size = 48, 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 \[ \frac{B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}-\frac{A+B x}{c \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(x*(A + B*x))/(a + c*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.74831, size = 41, normalized size = 0.85 \[ \frac{B \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{c^{\frac{3}{2}}} - \frac{A + B x}{c \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(B*x+A)/(c*x**2+a)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.064122, size = 53, normalized size = 1.1 \[ \frac{-A-B x}{c \sqrt{a+c x^2}}+\frac{B \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{c^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x*(A + B*x))/(a + c*x^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.007, size = 54, normalized size = 1.1 \[ -{\frac{A}{c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{Bx}{c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{B\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(B*x+A)/(c*x^2+a)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.294367, size = 1, normalized size = 0.02 \[ \left [-\frac{2 \, \sqrt{c x^{2} + a}{\left (B x + A\right )} \sqrt{c} -{\left (B c x^{2} + B a\right )} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{2 \,{\left (c^{2} x^{2} + a c\right )} \sqrt{c}}, -\frac{\sqrt{c x^{2} + a}{\left (B x + A\right )} \sqrt{-c} -{\left (B c x^{2} + B a\right )} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{{\left (c^{2} x^{2} + a c\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x/(c*x^2 + a)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 12.0381, size = 66, normalized size = 1.38 \[ A \left (\begin{cases} - \frac{1}{c \sqrt{a + c x^{2}}} & \text{for}\: c \neq 0 \\\frac{x^{2}}{2 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + B \left (\frac{\operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{c^{\frac{3}{2}}} - \frac{x}{\sqrt{a} c \sqrt{1 + \frac{c x^{2}}{a}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(B*x+A)/(c*x**2+a)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.277032, size = 65, normalized size = 1.35 \[ -\frac{\frac{B x}{c} + \frac{A}{c}}{\sqrt{c x^{2} + a}} - \frac{B{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x/(c*x^2 + a)^(3/2),x, algorithm="giac")

[Out]

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