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

Optimal. Leaf size=57 \[ -\frac{2 \sqrt{b x+c x^2} (3 b B-2 A c)}{3 b^2 x}-\frac{2 A \sqrt{b x+c x^2}}{3 b x^2} \]

[Out]

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

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Rubi [A]  time = 0.133351, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2 \sqrt{b x+c x^2} (3 b B-2 A c)}{3 b^2 x}-\frac{2 A \sqrt{b x+c x^2}}{3 b x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.60237, size = 51, normalized size = 0.89 \[ - \frac{2 A \sqrt{b x + c x^{2}}}{3 b x^{2}} + \frac{4 \left (A c - \frac{3 B b}{2}\right ) \sqrt{b x + c x^{2}}}{3 b^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*sqrt(b*x + c*x**2)/(3*b*x**2) + 4*(A*c - 3*B*b/2)*sqrt(b*x + c*x**2)/(3*b**
2*x)

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Mathematica [A]  time = 0.0625528, size = 35, normalized size = 0.61 \[ -\frac{2 \sqrt{x (b+c x)} (A (b-2 c x)+3 b B x)}{3 b^2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 39, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -2\,Acx+3\,xBb+Ab \right ) }{3\,{b}^{2}x}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.27672, size = 46, normalized size = 0.81 \[ -\frac{2 \, \sqrt{c x^{2} + b x}{\left (A b +{\left (3 \, B b - 2 \, A c\right )} x\right )}}{3 \, b^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.28057, size = 103, normalized size = 1.81 \[ \frac{2 \,{\left (3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A \sqrt{c} + A b\right )}}{3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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