Optimal. Leaf size=72 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 \sqrt{c}}+\frac{3}{4} b \sqrt{b x+c x^2}+\frac{\left (b x+c x^2\right )^{3/2}}{2 x} \]
[Out]
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Rubi [A] time = 0.086668, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 \sqrt{c}}+\frac{3}{4} b \sqrt{b x+c x^2}+\frac{\left (b x+c x^2\right )^{3/2}}{2 x} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)^(3/2)/x^2,x]
[Out]
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Rubi in Sympy [A] time = 9.50966, size = 63, normalized size = 0.88 \[ \frac{3 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{4 \sqrt{c}} + \frac{3 b \sqrt{b x + c x^{2}}}{4} + \frac{\left (b x + c x^{2}\right )^{\frac{3}{2}}}{2 x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**(3/2)/x**2,x)
[Out]
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Mathematica [A] time = 0.0775002, size = 71, normalized size = 0.99 \[ \frac{1}{4} \sqrt{x (b+c x)} \left (\frac{3 b^2 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{c} \sqrt{x} \sqrt{b+c x}}+5 b+2 c x\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)^(3/2)/x^2,x]
[Out]
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Maple [A] time = 0.007, size = 99, normalized size = 1.4 \[ 2\,{\frac{ \left ( c{x}^{2}+bx \right ) ^{5/2}}{b{x}^{2}}}-2\,{\frac{c \left ( c{x}^{2}+bx \right ) ^{3/2}}{b}}-{\frac{3\,cx}{2}\sqrt{c{x}^{2}+bx}}-{\frac{3\,b}{4}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){\frac{1}{\sqrt{c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)^(3/2)/x^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((c*x^2 + b*x)^(3/2)/x^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.242955, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{2} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \, \sqrt{c x^{2} + b x}{\left (2 \, c x + 5 \, b\right )} \sqrt{c}}{8 \, \sqrt{c}}, \frac{3 \, b^{2} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) + \sqrt{c x^{2} + b x}{\left (2 \, c x + 5 \, b\right )} \sqrt{-c}}{4 \, \sqrt{-c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(3/2)/x^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)**(3/2)/x**2,x)
[Out]
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GIAC/XCAS [A] time = 0.222239, size = 81, normalized size = 1.12 \[ -\frac{3 \, b^{2}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{8 \, \sqrt{c}} + \frac{1}{4} \, \sqrt{c x^{2} + b x}{\left (2 \, c x + 5 \, b\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(3/2)/x^2,x, algorithm="giac")
[Out]