Optimal. Leaf size=102 \[ -\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{7/2}}+\frac{5 b^2 \sqrt{b x+c x^2}}{8 c^3}-\frac{5 b x \sqrt{b x+c x^2}}{12 c^2}+\frac{x^2 \sqrt{b x+c x^2}}{3 c} \]
[Out]
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Rubi [A] time = 0.129006, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{7/2}}+\frac{5 b^2 \sqrt{b x+c x^2}}{8 c^3}-\frac{5 b x \sqrt{b x+c x^2}}{12 c^2}+\frac{x^2 \sqrt{b x+c x^2}}{3 c} \]
Antiderivative was successfully verified.
[In] Int[x^3/Sqrt[b*x + c*x^2],x]
[Out]
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Rubi in Sympy [A] time = 14.583, size = 94, normalized size = 0.92 \[ - \frac{5 b^{3} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{8 c^{\frac{7}{2}}} + \frac{5 b^{2} \sqrt{b x + c x^{2}}}{8 c^{3}} - \frac{5 b x \sqrt{b x + c x^{2}}}{12 c^{2}} + \frac{x^{2} \sqrt{b x + c x^{2}}}{3 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(c*x**2+b*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0637374, size = 101, normalized size = 0.99 \[ \frac{\sqrt{c} x \left (15 b^3+5 b^2 c x-2 b c^2 x^2+8 c^3 x^3\right )-15 b^3 \sqrt{x} \sqrt{b+c x} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{24 c^{7/2} \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/Sqrt[b*x + c*x^2],x]
[Out]
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Maple [A] time = 0.007, size = 90, normalized size = 0.9 \[{\frac{{x}^{2}}{3\,c}\sqrt{c{x}^{2}+bx}}-{\frac{5\,bx}{12\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,{b}^{2}}{8\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,{b}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(c*x^2+b*x)^(1/2),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(x^3/sqrt(c*x^2 + b*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226526, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{3} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \,{\left (8 \, c^{2} x^{2} - 10 \, b c x + 15 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{c}}{48 \, c^{\frac{7}{2}}}, -\frac{15 \, b^{3} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (8 \, c^{2} x^{2} - 10 \, b c x + 15 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{-c}}{24 \, \sqrt{-c} c^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/sqrt(c*x^2 + b*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\sqrt{x \left (b + c x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(c*x**2+b*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.228606, size = 104, normalized size = 1.02 \[ \frac{1}{24} \, \sqrt{c x^{2} + b x}{\left (2 \, x{\left (\frac{4 \, x}{c} - \frac{5 \, b}{c^{2}}\right )} + \frac{15 \, b^{2}}{c^{3}}\right )} + \frac{5 \, b^{3}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{16 \, c^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/sqrt(c*x^2 + b*x),x, algorithm="giac")
[Out]