3.321 \(\int \frac{(A+B x) \sqrt{a+c x^2}}{x^3} \, dx\)

Optimal. Leaf size=80 \[ -\frac{\sqrt{a+c x^2} (A+2 B x)}{2 x^2}-\frac{A c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 \sqrt{a}}+B \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.180248, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{\sqrt{a+c x^2} (A+2 B x)}{2 x^2}-\frac{A c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 \sqrt{a}}+B \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.6836, size = 73, normalized size = 0.91 \[ - \frac{A c \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{2 \sqrt{a}} + B \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )} - \frac{\left (A + 2 B x\right ) \sqrt{a + c x^{2}}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**(1/2)/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.18394, size = 96, normalized size = 1.2 \[ \frac{1}{2} \left (-\frac{\sqrt{a+c x^2} (A+2 B x)}{x^2}-\frac{A c \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )}{\sqrt{a}}+\frac{A c \log (x)}{\sqrt{a}}+2 B \sqrt{c} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 121, normalized size = 1.5 \[ -{\frac{A}{2\,a{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{Ac}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{Ac}{2\,a}\sqrt{c{x}^{2}+a}}-{\frac{B}{ax} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Bcx}{a}\sqrt{c{x}^{2}+a}}+B\sqrt{c}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^(1/2)/x^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.366555, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, B \sqrt{a} \sqrt{c} x^{2} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + A c x^{2} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) - 2 \, \sqrt{c x^{2} + a}{\left (2 \, B x + A\right )} \sqrt{a}}{4 \, \sqrt{a} x^{2}}, \frac{4 \, B \sqrt{a} \sqrt{-c} x^{2} \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) + A c x^{2} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) - 2 \, \sqrt{c x^{2} + a}{\left (2 \, B x + A\right )} \sqrt{a}}{4 \, \sqrt{a} x^{2}}, -\frac{A c x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) - B \sqrt{-a} \sqrt{c} x^{2} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + \sqrt{c x^{2} + a}{\left (2 \, B x + A\right )} \sqrt{-a}}{2 \, \sqrt{-a} x^{2}}, \frac{2 \, B \sqrt{-a} \sqrt{-c} x^{2} \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) - A c x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) - \sqrt{c x^{2} + a}{\left (2 \, B x + A\right )} \sqrt{-a}}{2 \, \sqrt{-a} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 11.2989, size = 107, normalized size = 1.34 \[ - \frac{A \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{2 x} - \frac{A c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{2 \sqrt{a}} - \frac{B \sqrt{a}}{x \sqrt{1 + \frac{c x^{2}}{a}}} + B \sqrt{c} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )} - \frac{B c x}{\sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+a)**(1/2)/x**3,x)

[Out]

-A*sqrt(c)*sqrt(a/(c*x**2) + 1)/(2*x) - A*c*asinh(sqrt(a)/(sqrt(c)*x))/(2*sqrt(a
)) - B*sqrt(a)/(x*sqrt(1 + c*x**2/a)) + B*sqrt(c)*asinh(sqrt(c)*x/sqrt(a)) - B*c
*x/(sqrt(a)*sqrt(1 + c*x**2/a))

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GIAC/XCAS [A]  time = 0.280155, size = 220, normalized size = 2.75 \[ \frac{A c \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - B \sqrt{c}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right ) + \frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} A c + 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} B a \sqrt{c} +{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} A a c - 2 \, B a^{2} \sqrt{c}}{{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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