Optimal. Leaf size=105 \[ -\frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{7/2}}+\frac{5 b^2 (b+2 c x) \sqrt{b x+c x^2}}{64 c^3}-\frac{5 b \left (b x+c x^2\right )^{3/2}}{24 c^2}+\frac{x \left (b x+c x^2\right )^{3/2}}{4 c} \]
[Out]
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Rubi [A] time = 0.117302, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ -\frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{7/2}}+\frac{5 b^2 (b+2 c x) \sqrt{b x+c x^2}}{64 c^3}-\frac{5 b \left (b x+c x^2\right )^{3/2}}{24 c^2}+\frac{x \left (b x+c x^2\right )^{3/2}}{4 c} \]
Antiderivative was successfully verified.
[In] Int[x^2*Sqrt[b*x + c*x^2],x]
[Out]
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Rubi in Sympy [A] time = 14.1051, size = 97, normalized size = 0.92 \[ - \frac{5 b^{4} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{64 c^{\frac{7}{2}}} + \frac{5 b^{2} \left (b + 2 c x\right ) \sqrt{b x + c x^{2}}}{64 c^{3}} - \frac{5 b \left (b x + c x^{2}\right )^{\frac{3}{2}}}{24 c^{2}} + \frac{x \left (b x + c x^{2}\right )^{\frac{3}{2}}}{4 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(c*x**2+b*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.106459, size = 100, normalized size = 0.95 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (15 b^3-10 b^2 c x+8 b c^2 x^2+48 c^3 x^3\right )-\frac{15 b^4 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}\right )}{192 c^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2*Sqrt[b*x + c*x^2],x]
[Out]
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Maple [A] time = 0.007, size = 107, normalized size = 1. \[{\frac{x}{4\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,b}{24\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{2}x}{32\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,{b}^{3}}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,{b}^{4}}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(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(sqrt(c*x^2 + b*x)*x^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.234667, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{4} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \,{\left (48 \, c^{3} x^{3} + 8 \, b c^{2} x^{2} - 10 \, b^{2} c x + 15 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{c}}{384 \, c^{\frac{7}{2}}}, -\frac{15 \, b^{4} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (48 \, c^{3} x^{3} + 8 \, b c^{2} x^{2} - 10 \, b^{2} c x + 15 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{-c}}{192 \, \sqrt{-c} c^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)*x^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x^{2} \sqrt{x \left (b + c x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(c*x**2+b*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.2221, size = 115, normalized size = 1.1 \[ \frac{1}{192} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (6 \, x + \frac{b}{c}\right )} x - \frac{5 \, b^{2}}{c^{2}}\right )} x + \frac{15 \, b^{3}}{c^{3}}\right )} + \frac{5 \, b^{4}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)*x^2,x, algorithm="giac")
[Out]