Optimal. Leaf size=68 \[ -\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{3/2}}-\frac{2 \sqrt{x} (b B-A c)}{b c \sqrt{b x+c x^2}} \]
[Out]
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Rubi [A] time = 0.137286, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{3/2}}-\frac{2 \sqrt{x} (b B-A c)}{b c \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[x]*(A + B*x))/(b*x + c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 10.2637, size = 60, normalized size = 0.88 \[ - \frac{2 A \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{b^{\frac{3}{2}}} + \frac{2 \sqrt{x} \left (A c - B b\right )}{b c \sqrt{b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*x**(1/2)/(c*x**2+b*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0846726, size = 69, normalized size = 1.01 \[ -\frac{2 \sqrt{x} \left (\sqrt{b} (b B-A c)+A c \sqrt{b+c x} \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )\right )}{b^{3/2} c \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[x]*(A + B*x))/(b*x + c*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.02, size = 63, normalized size = 0.9 \[ -2\,{\frac{\sqrt{x \left ( cx+b \right ) }}{{b}^{3/2}\sqrt{x} \left ( cx+b \right ) c} \left ( A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) c\sqrt{cx+b}-Ac\sqrt{b}+B{b}^{3/2} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*x^(1/2)/(c*x^2+b*x)^(3/2),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)*sqrt(x)/(c*x^2 + b*x)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.301568, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{c x^{2} + b x}{\left (B b - A c\right )} \sqrt{b} \sqrt{x} -{\left (A c^{2} x^{2} + A b c x\right )} \log \left (\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} -{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right )}{{\left (b c^{2} x^{2} + b^{2} c x\right )} \sqrt{b}}, -\frac{2 \,{\left (\sqrt{c x^{2} + b x}{\left (B b - A c\right )} \sqrt{-b} \sqrt{x} +{\left (A c^{2} x^{2} + A b c x\right )} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right )\right )}}{{\left (b c^{2} x^{2} + b^{2} c x\right )} \sqrt{-b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(x)/(c*x^2 + b*x)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x} \left (A + B x\right )}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*x**(1/2)/(c*x**2+b*x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.27696, size = 130, normalized size = 1.91 \[ \frac{2 \, A \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} - \frac{2 \,{\left (B b - A c\right )}}{\sqrt{c x + b} b c} - \frac{2 \,{\left (A \sqrt{b} c \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) - B \sqrt{-b} b + A \sqrt{-b} c\right )}}{\sqrt{-b} b^{\frac{3}{2}} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(x)/(c*x^2 + b*x)^(3/2),x, algorithm="giac")
[Out]