Optimal. Leaf size=89 \[ \frac{3 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{5/2}}-\frac{3 b^2 (b+2 c x) \sqrt{b x+c x^2}}{64 c^2}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c} \]
[Out]
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Rubi [A] time = 0.0701835, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{3 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{5/2}}-\frac{3 b^2 (b+2 c x) \sqrt{b x+c x^2}}{64 c^2}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 7.56327, size = 82, normalized size = 0.92 \[ \frac{3 b^{4} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{64 c^{\frac{5}{2}}} - \frac{3 b^{2} \left (b + 2 c x\right ) \sqrt{b x + c x^{2}}}{64 c^{2}} + \frac{\left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{8 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.113121, size = 100, normalized size = 1.12 \[ \frac{\sqrt{x (b+c x)} \left (\frac{3 b^4 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}+\sqrt{c} \left (-3 b^3+2 b^2 c x+24 b c^2 x^2+16 c^3 x^3\right )\right )}{64 c^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0., size = 95, normalized size = 1.1 \[{\frac{2\,cx+b}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{b}^{2}x}{32\,c}\sqrt{c{x}^{2}+bx}}-{\frac{3\,{b}^{3}}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{4}}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((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((c*x^2 + b*x)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226008, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{4} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \,{\left (16 \, c^{3} x^{3} + 24 \, b c^{2} x^{2} + 2 \, b^{2} c x - 3 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{c}}{128 \, c^{\frac{5}{2}}}, \frac{3 \, b^{4} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (16 \, c^{3} x^{3} + 24 \, b c^{2} x^{2} + 2 \, b^{2} c x - 3 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{-c}}{64 \, \sqrt{-c} c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (b x + c x^{2}\right )^{\frac{3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.221031, size = 112, normalized size = 1.26 \[ -\frac{3 \, b^{4}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{5}{2}}} + \frac{1}{64} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \, c x + 3 \, b\right )} x + \frac{b^{2}}{c}\right )} x - \frac{3 \, b^{3}}{c^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(3/2),x, algorithm="giac")
[Out]