Optimal. Leaf size=81 \[ \frac{2 A \sqrt{b x+c x^2}}{\sqrt{x}}-2 A \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )+\frac{2 B \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}} \]
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Rubi [A] time = 0.164302, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2 A \sqrt{b x+c x^2}}{\sqrt{x}}-2 A \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )+\frac{2 B \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*Sqrt[b*x + c*x^2])/x^(3/2),x]
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Rubi in Sympy [A] time = 12.0584, size = 75, normalized size = 0.93 \[ - 2 A \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )} + \frac{2 A \sqrt{b x + c x^{2}}}{\sqrt{x}} + \frac{2 B \left (b x + c x^{2}\right )^{\frac{3}{2}}}{3 c x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x)**(1/2)/x**(3/2),x)
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Mathematica [A] time = 0.115209, size = 80, normalized size = 0.99 \[ \frac{2 \sqrt{x} \sqrt{b+c x} \left (\sqrt{b+c x} (3 A c+b B+B c x)-3 A \sqrt{b} c \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )\right )}{3 c \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*Sqrt[b*x + c*x^2])/x^(3/2),x]
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Maple [A] time = 0.018, size = 79, normalized size = 1. \[ -{\frac{2}{3\,c}\sqrt{x \left ( cx+b \right ) } \left ( 3\,A\sqrt{b}c{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) -Bxc\sqrt{cx+b}-3\,A\sqrt{cx+b}c-Bb\sqrt{cx+b} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{cx+b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x)^(1/2)/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(sqrt(c*x^2 + b*x)*(B*x + A)/x^(3/2),x, algorithm="maxima")
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Fricas [A] time = 0.291958, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, B c^{2} x^{3} + 3 \, \sqrt{c x^{2} + b x} A \sqrt{b} c \sqrt{x} \log \left (-\frac{c x^{2} + 2 \, b x - 2 \, \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{x^{2}}\right ) + 2 \,{\left (2 \, B b c + 3 \, A c^{2}\right )} x^{2} + 2 \,{\left (B b^{2} + 3 \, A b c\right )} x}{3 \, \sqrt{c x^{2} + b x} c \sqrt{x}}, \frac{2 \,{\left (B c^{2} x^{3} - 3 \, \sqrt{c x^{2} + b x} A \sqrt{-b} c \sqrt{x} \arctan \left (\frac{b \sqrt{x}}{\sqrt{c x^{2} + b x} \sqrt{-b}}\right ) +{\left (2 \, B b c + 3 \, A c^{2}\right )} x^{2} +{\left (B b^{2} + 3 \, A b c\right )} x\right )}}{3 \, \sqrt{c x^{2} + b x} c \sqrt{x}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)*(B*x + A)/x^(3/2),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )} \left (A + B x\right )}{x^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x)**(1/2)/x**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.275502, size = 139, normalized size = 1.72 \[ \frac{2 \, A b \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{2 \,{\left (3 \, A b c \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + B \sqrt{-b} b^{\frac{3}{2}} + 3 \, A \sqrt{-b} \sqrt{b} c\right )}}{3 \, \sqrt{-b} c} + \frac{2 \,{\left ({\left (c x + b\right )}^{\frac{3}{2}} B c^{2} + 3 \, \sqrt{c x + b} A c^{3}\right )}}{3 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)*(B*x + A)/x^(3/2),x, algorithm="giac")
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