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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]