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

Optimal. Leaf size=79 \[ \frac{1}{2} \sqrt{a+c x^2} (2 A+B x)-\sqrt{a} A \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{a B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 \sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.177862, antiderivative size = 79, 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 \[ \frac{1}{2} \sqrt{a+c x^2} (2 A+B x)-\sqrt{a} A \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{a B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 \sqrt{c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.2758, size = 70, normalized size = 0.89 \[ - A \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )} + \frac{B a \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{2 \sqrt{c}} + \frac{\left (2 A + B x\right ) \sqrt{a + c x^{2}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.100723, size = 91, normalized size = 1.15 \[ \sqrt{a+c x^2} \left (A+\frac{B x}{2}\right )-\sqrt{a} A \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )+\sqrt{a} A \log (x)+\frac{a B \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{2 \sqrt{c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 78, normalized size = 1. \[{\frac{Bx}{2}\sqrt{c{x}^{2}+a}}+{\frac{Ba}{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}-A\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ) +A\sqrt{c{x}^{2}+a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.348692, size = 1, normalized size = 0.01 \[ \left [\frac{B a \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + 2 \, A \sqrt{a} \sqrt{c} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \, \sqrt{c x^{2} + a}{\left (B x + 2 \, A\right )} \sqrt{c}}{4 \, \sqrt{c}}, \frac{B a \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) + A \sqrt{a} \sqrt{-c} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + \sqrt{c x^{2} + a}{\left (B x + 2 \, A\right )} \sqrt{-c}}{2 \, \sqrt{-c}}, -\frac{4 \, A \sqrt{-a} \sqrt{c} \arctan \left (\frac{a}{\sqrt{c x^{2} + a} \sqrt{-a}}\right ) - B a \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) - 2 \, \sqrt{c x^{2} + a}{\left (B x + 2 \, A\right )} \sqrt{c}}{4 \, \sqrt{c}}, \frac{B a \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) - 2 \, A \sqrt{-a} \sqrt{-c} \arctan \left (\frac{a}{\sqrt{c x^{2} + a} \sqrt{-a}}\right ) + \sqrt{c x^{2} + a}{\left (B x + 2 \, A\right )} \sqrt{-c}}{2 \, \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [A]  time = 10.8139, size = 107, normalized size = 1.35 \[ - A \sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )} + \frac{A a}{\sqrt{c} x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{A \sqrt{c} x}{\sqrt{\frac{a}{c x^{2}} + 1}} + \frac{B \sqrt{a} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{B a \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.274488, size = 105, normalized size = 1.33 \[ \frac{2 \, A a \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{B a{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{2 \, \sqrt{c}} + \frac{1}{2} \, \sqrt{c x^{2} + a}{\left (B x + 2 \, A\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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