Optimal. Leaf size=57 \[ \frac{1}{3} a^2 x^2 \sqrt{c x^2}+\frac{1}{2} a b x^3 \sqrt{c x^2}+\frac{1}{5} b^2 x^4 \sqrt{c x^2} \]
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Rubi [A] time = 0.0365248, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{1}{3} a^2 x^2 \sqrt{c x^2}+\frac{1}{2} a b x^3 \sqrt{c x^2}+\frac{1}{5} b^2 x^4 \sqrt{c x^2} \]
Antiderivative was successfully verified.
[In] Int[x*Sqrt[c*x^2]*(a + b*x)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x \sqrt{c x^{2}} \left (a + b x\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b*x+a)**2*(c*x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.00772535, size = 35, normalized size = 0.61 \[ \frac{1}{30} x^2 \sqrt{c x^2} \left (10 a^2+15 a b x+6 b^2 x^2\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x*Sqrt[c*x^2]*(a + b*x)^2,x]
[Out]
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Maple [A] time = 0.005, size = 32, normalized size = 0.6 \[{\frac{{x}^{2} \left ( 6\,{b}^{2}{x}^{2}+15\,abx+10\,{a}^{2} \right ) }{30}\sqrt{c{x}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b*x+a)^2*(c*x^2)^(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 + a)^2*x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.197611, size = 45, normalized size = 0.79 \[ \frac{1}{30} \,{\left (6 \, b^{2} x^{4} + 15 \, a b x^{3} + 10 \, a^{2} x^{2}\right )} \sqrt{c x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2)*(b*x + a)^2*x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.01472, size = 60, normalized size = 1.05 \[ \frac{a^{2} \sqrt{c} x^{2} \sqrt{x^{2}}}{3} + \frac{a b \sqrt{c} x^{3} \sqrt{x^{2}}}{2} + \frac{b^{2} \sqrt{c} x^{4} \sqrt{x^{2}}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(b*x+a)**2*(c*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.207698, size = 47, normalized size = 0.82 \[ \frac{1}{30} \,{\left (6 \, b^{2} x^{5}{\rm sign}\left (x\right ) + 15 \, a b x^{4}{\rm sign}\left (x\right ) + 10 \, a^{2} x^{3}{\rm sign}\left (x\right )\right )} \sqrt{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2)*(b*x + a)^2*x,x, algorithm="giac")
[Out]