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

Optimal. Leaf size=73 \[ -\frac{2 A \left (b x+c x^2\right )^{3/2}}{3 b x^3}-\frac{2 B \sqrt{b x+c x^2}}{x}+2 B \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.178339, antiderivative size = 73, 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 \[ -\frac{2 A \left (b x+c x^2\right )^{3/2}}{3 b x^3}-\frac{2 B \sqrt{b x+c x^2}}{x}+2 B \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.0514, size = 66, normalized size = 0.9 \[ - \frac{2 A \left (b x + c x^{2}\right )^{\frac{3}{2}}}{3 b x^{3}} + 2 B \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )} - \frac{2 B \sqrt{b x + c x^{2}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.108684, size = 93, normalized size = 1.27 \[ \frac{2 \sqrt{x (b+c x)} \left (3 b B \sqrt{c} x^{3/2} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )-\sqrt{b+c x} (A (b+c x)+3 b B x)\right )}{3 b x^2 \sqrt{b+c x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 89, normalized size = 1.2 \[ -{\frac{2\,A}{3\,b{x}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-2\,{\frac{B \left ( c{x}^{2}+bx \right ) ^{3/2}}{b{x}^{2}}}+2\,{\frac{Bc\sqrt{c{x}^{2}+bx}}{b}}+B\sqrt{c}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.279367, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, B b \sqrt{c} x^{2} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - 2 \, \sqrt{c x^{2} + b x}{\left (A b +{\left (3 \, B b + A c\right )} x\right )}}{3 \, b x^{2}}, \frac{2 \,{\left (3 \, B b \sqrt{-c} x^{2} \arctan \left (\frac{\sqrt{c x^{2} + b x}}{\sqrt{-c} x}\right ) - \sqrt{c x^{2} + b x}{\left (A b +{\left (3 \, B b + A c\right )} x\right )}\right )}}{3 \, b x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )} \left (A + B x\right )}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x*(b + c*x))*(A + B*x)/x**3, x)

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GIAC/XCAS [A]  time = 0.289839, size = 204, normalized size = 2.79 \[ -B \sqrt{c}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right ) + \frac{2 \,{\left (3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b \sqrt{c} + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A c^{\frac{3}{2}} + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b c + A b^{2} \sqrt{c}\right )}}{3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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