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

Optimal. Leaf size=49 \[ \frac{2 (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{3/2} \sqrt{c}}-\frac{2 A}{b \sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0732803, antiderivative size = 49, 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 \[ \frac{2 (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{3/2} \sqrt{c}}-\frac{2 A}{b \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.68382, size = 46, normalized size = 0.94 \[ - \frac{2 A}{b \sqrt{x}} - \frac{2 \left (A c - B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{b^{\frac{3}{2}} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0643486, size = 49, normalized size = 1. \[ \frac{2 (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{3/2} \sqrt{c}}-\frac{2 A}{b \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 53, normalized size = 1.1 \[ -2\,{\frac{Ac}{b\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) }+2\,{\frac{B}{\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) }-2\,{\frac{A}{b\sqrt{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.30049, size = 1, normalized size = 0.02 \[ \left [-\frac{{\left (B b - A c\right )} \sqrt{x} \log \left (-\frac{2 \, b c \sqrt{x} - \sqrt{-b c}{\left (c x - b\right )}}{c x + b}\right ) + 2 \, \sqrt{-b c} A}{\sqrt{-b c} b \sqrt{x}}, -\frac{2 \,{\left ({\left (B b - A c\right )} \sqrt{x} \arctan \left (\frac{b}{\sqrt{b c} \sqrt{x}}\right ) + \sqrt{b c} A\right )}}{\sqrt{b c} b \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{x^{\frac{3}{2}} \left (b + c x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)/(x**(3/2)*(b + c*x)), x)

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GIAC/XCAS [A]  time = 0.268328, size = 53, normalized size = 1.08 \[ \frac{2 \,{\left (B b - A c\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{\sqrt{b c} b} - \frac{2 \, A}{b \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(B*b - A*c)*arctan(c*sqrt(x)/sqrt(b*c))/(sqrt(b*c)*b) - 2*A/(b*sqrt(x))