Optimal. Leaf size=35 \[ \frac{a x^3}{2 \sqrt{c x^2}}+\frac{b x^4}{3 \sqrt{c x^2}} \]
[Out]
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Rubi [A] time = 0.0206731, antiderivative size = 35, 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 \[ \frac{a x^3}{2 \sqrt{c x^2}}+\frac{b x^4}{3 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(x^2*(a + b*x))/Sqrt[c*x^2],x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a \sqrt{c x^{2}} \int x\, dx}{c x} + \frac{b x^{2} \sqrt{c x^{2}}}{3 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(b*x+a)/(c*x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0063251, size = 24, normalized size = 0.69 \[ \frac{x^3 (3 a+2 b x)}{6 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*(a + b*x))/Sqrt[c*x^2],x]
[Out]
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Maple [A] time = 0.004, size = 21, normalized size = 0.6 \[{\frac{{x}^{3} \left ( 2\,bx+3\,a \right ) }{6}{\frac{1}{\sqrt{c{x}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(b*x+a)/(c*x^2)^(1/2),x)
[Out]
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Maxima [A] time = 1.33397, size = 35, normalized size = 1. \[ \frac{\sqrt{c x^{2}} b x^{2}}{3 \, c} + \frac{a x^{2}}{2 \, \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*x^2/sqrt(c*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.195584, size = 31, normalized size = 0.89 \[ \frac{{\left (2 \, b x^{2} + 3 \, a x\right )} \sqrt{c x^{2}}}{6 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*x^2/sqrt(c*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.85788, size = 36, normalized size = 1.03 \[ \frac{a x^{3}}{2 \sqrt{c} \sqrt{x^{2}}} + \frac{b x^{4}}{3 \sqrt{c} \sqrt{x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(b*x+a)/(c*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.210782, size = 32, normalized size = 0.91 \[ \frac{1}{6} \, \sqrt{c x^{2}}{\left (\frac{2 \, b x}{c} + \frac{3 \, a}{c}\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*x^2/sqrt(c*x^2),x, algorithm="giac")
[Out]