Optimal. Leaf size=35 \[ \frac{a x \log (x)}{c \sqrt{c x^2}}+\frac{b x^2}{c \sqrt{c x^2}} \]
[Out]
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Rubi [A] time = 0.0179818, 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 \[ \frac{a x \log (x)}{c \sqrt{c x^2}}+\frac{b x^2}{c \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(x^2*(a + b*x))/(c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a \sqrt{c x^{2}} \log{\left (x \right )}}{c^{2} x} + \frac{\sqrt{c x^{2}} \int b\, dx}{c^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(b*x+a)/(c*x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.00607968, size = 21, normalized size = 0.6 \[ \frac{x^3 (a \log (x)+b x)}{\left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*(a + b*x))/(c*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.004, size = 20, normalized size = 0.6 \[{{x}^{3} \left ( bx+a\ln \left ( x \right ) \right ) \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(b*x+a)/(c*x^2)^(3/2),x)
[Out]
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Maxima [A] time = 1.3457, size = 31, normalized size = 0.89 \[ \frac{b x^{2}}{\sqrt{c x^{2}} c} + \frac{a \log \left (x\right )}{c^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*x^2/(c*x^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.210445, size = 30, normalized size = 0.86 \[ \frac{\sqrt{c x^{2}}{\left (b x + a \log \left (x\right )\right )}}{c^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*x^2/(c*x^2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} \left (a + b x\right )}{\left (c x^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(b*x+a)/(c*x**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.212585, size = 54, normalized size = 1.54 \[ -\frac{\frac{a{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right )}{\sqrt{c}} - \frac{\sqrt{c x^{2}} b}{c}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*x^2/(c*x^2)^(3/2),x, algorithm="giac")
[Out]