Optimal. Leaf size=69 \[ -\frac{2 \sqrt{c} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{5/2}}-\frac{2 (b B-A c)}{b^2 \sqrt{x}}-\frac{2 A}{3 b x^{3/2}} \]
[Out]
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Rubi [A] time = 0.104422, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{2 \sqrt{c} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{5/2}}-\frac{2 (b B-A c)}{b^2 \sqrt{x}}-\frac{2 A}{3 b x^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^(3/2)*(b*x + c*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 12.6528, size = 65, normalized size = 0.94 \[ - \frac{2 A}{3 b x^{\frac{3}{2}}} + \frac{2 \left (A c - B b\right )}{b^{2} \sqrt{x}} + \frac{2 \sqrt{c} \left (A c - B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{b^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x**(3/2)/(c*x**2+b*x),x)
[Out]
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Mathematica [A] time = 0.12429, size = 64, normalized size = 0.93 \[ \frac{2 \sqrt{c} (A c-b B) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{5/2}}-\frac{2 (A (b-3 c x)+3 b B x)}{3 b^2 x^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^(3/2)*(b*x + c*x^2)),x]
[Out]
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Maple [A] time = 0.016, size = 78, normalized size = 1.1 \[ -{\frac{2\,A}{3\,b}{x}^{-{\frac{3}{2}}}}+2\,{\frac{Ac}{\sqrt{x}{b}^{2}}}-2\,{\frac{B}{b\sqrt{x}}}+2\,{\frac{A{c}^{2}}{{b}^{2}\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) }-2\,{\frac{Bc}{b\sqrt{bc}}\arctan \left ({\frac{c\sqrt{x}}{\sqrt{bc}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^(3/2)/(c*x^2+b*x),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)*x^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.299279, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (B b - A c\right )} x^{\frac{3}{2}} \sqrt{-\frac{c}{b}} \log \left (\frac{c x + 2 \, b \sqrt{x} \sqrt{-\frac{c}{b}} - b}{c x + b}\right ) + 2 \, A b + 6 \,{\left (B b - A c\right )} x}{3 \, b^{2} x^{\frac{3}{2}}}, \frac{2 \,{\left (3 \,{\left (B b - A c\right )} x^{\frac{3}{2}} \sqrt{\frac{c}{b}} \arctan \left (\frac{b \sqrt{\frac{c}{b}}}{c \sqrt{x}}\right ) - A b - 3 \,{\left (B b - A c\right )} x\right )}}{3 \, b^{2} x^{\frac{3}{2}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + b*x)*x^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{x^{\frac{5}{2}} \left (b + c x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/x**(3/2)/(c*x**2+b*x),x)
[Out]
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GIAC/XCAS [A] time = 0.271203, size = 74, normalized size = 1.07 \[ -\frac{2 \,{\left (B b c - A c^{2}\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{\sqrt{b c} b^{2}} - \frac{2 \,{\left (3 \, B b x - 3 \, A c x + A b\right )}}{3 \, b^{2} x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + b*x)*x^(3/2)),x, algorithm="giac")
[Out]