Optimal. Leaf size=26 \[ -\frac{(a+b x)^3}{3 a x^2 \sqrt{c x^2}} \]
[Out]
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Rubi [A] time = 0.015093, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{(a+b x)^3}{3 a x^2 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^2/(x^3*Sqrt[c*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 12.3658, size = 24, normalized size = 0.92 \[ - \frac{\sqrt{c x^{2}} \left (a + b x\right )^{3}}{3 a c x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**2/x**3/(c*x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0161121, size = 33, normalized size = 1.27 \[ \frac{c \left (-a^2-3 a b x-3 b^2 x^2\right )}{3 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^2/(x^3*Sqrt[c*x^2]),x]
[Out]
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Maple [A] time = 0.006, size = 30, normalized size = 1.2 \[ -{\frac{3\,{b}^{2}{x}^{2}+3\,abx+{a}^{2}}{3\,{x}^{2}}{\frac{1}{\sqrt{c{x}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^2/x^3/(c*x^2)^(1/2),x)
[Out]
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Maxima [A] time = 1.342, size = 45, normalized size = 1.73 \[ -\frac{b^{2}}{\sqrt{c} x} - \frac{a b}{\sqrt{c} x^{2}} - \frac{a^{2}}{3 \, \sqrt{c} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2/(sqrt(c*x^2)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.2129, size = 43, normalized size = 1.65 \[ -\frac{{\left (3 \, b^{2} x^{2} + 3 \, a b x + a^{2}\right )} \sqrt{c x^{2}}}{3 \, c x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2/(sqrt(c*x^2)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.44276, size = 53, normalized size = 2.04 \[ - \frac{a^{2}}{3 \sqrt{c} x^{2} \sqrt{x^{2}}} - \frac{a b}{\sqrt{c} x \sqrt{x^{2}}} - \frac{b^{2}}{\sqrt{c} \sqrt{x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**2/x**3/(c*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{2}}{\sqrt{c x^{2}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2/(sqrt(c*x^2)*x^3),x, algorithm="giac")
[Out]