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

Optimal. Leaf size=83 \[ \frac{\sqrt{b x+c x^2} (2 A c+b B)}{b}+\frac{(2 A c+b B) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{\sqrt{c}}-\frac{2 A \left (b x+c x^2\right )^{3/2}}{b x^2} \]

[Out]

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

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Rubi [A]  time = 0.205286, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\sqrt{b x+c x^2} (2 A c+b B)}{b}+\frac{(2 A c+b B) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{\sqrt{c}}-\frac{2 A \left (b x+c x^2\right )^{3/2}}{b x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.543, size = 78, normalized size = 0.94 \[ - \frac{2 A \left (b x + c x^{2}\right )^{\frac{3}{2}}}{b x^{2}} + \frac{2 \left (A c + \frac{B b}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{\sqrt{c}} + \frac{2 \left (A c + \frac{B b}{2}\right ) \sqrt{b x + c x^{2}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.111101, size = 74, normalized size = 0.89 \[ \frac{\sqrt{x (b+c x)} \left (\frac{\sqrt{x} (2 A c+b B) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{c} \sqrt{b+c x}}-2 A+B x\right )}{x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 113, normalized size = 1.4 \[ -2\,{\frac{A \left ( c{x}^{2}+bx \right ) ^{3/2}}{b{x}^{2}}}+2\,{\frac{Ac\sqrt{c{x}^{2}+bx}}{b}}+A\sqrt{c}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ) +B\sqrt{c{x}^{2}+bx}+{\frac{Bb}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289545, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (B b + 2 \, A c\right )} x \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \, \sqrt{c x^{2} + b x}{\left (B x - 2 \, A\right )} \sqrt{c}}{2 \, \sqrt{c} x}, \frac{{\left (B b + 2 \, A c\right )} x \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) + \sqrt{c x^{2} + b x}{\left (B x - 2 \, A\right )} \sqrt{-c}}{\sqrt{-c} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )} \left (A + B x\right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x*(b + c*x))*(A + B*x)/x**2, x)

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GIAC/XCAS [A]  time = 0.282924, size = 111, normalized size = 1.34 \[ \sqrt{c x^{2} + b x} B - \frac{{\left (B b + 2 \, A c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{2 \, \sqrt{c}} + \frac{2 \, A b}{\sqrt{c} x - \sqrt{c x^{2} + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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