Optimal. Leaf size=66 \[ -\frac{a^2}{5 c x^4 \sqrt{c x^2}}-\frac{a b}{2 c x^3 \sqrt{c x^2}}-\frac{b^2}{3 c x^2 \sqrt{c x^2}} \]
[Out]
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Rubi [A] time = 0.0367929, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{a^2}{5 c x^4 \sqrt{c x^2}}-\frac{a b}{2 c x^3 \sqrt{c x^2}}-\frac{b^2}{3 c x^2 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^2/(x^3*(c*x^2)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 16.5554, size = 61, normalized size = 0.92 \[ - \frac{a^{2} \sqrt{c x^{2}}}{5 c^{2} x^{6}} - \frac{a b \sqrt{c x^{2}}}{2 c^{2} x^{5}} - \frac{b^{2} \sqrt{c x^{2}}}{3 c^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**2/x**3/(c*x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0194348, size = 33, normalized size = 0.5 \[ \frac{c \left (-6 a^2-15 a b x-10 b^2 x^2\right )}{30 \left (c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^2/(x^3*(c*x^2)^(3/2)),x]
[Out]
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Maple [A] time = 0.006, size = 32, normalized size = 0.5 \[ -{\frac{10\,{b}^{2}{x}^{2}+15\,abx+6\,{a}^{2}}{30\,{x}^{2}} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^2/x^3/(c*x^2)^(3/2),x)
[Out]
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Maxima [A] time = 1.3691, size = 45, normalized size = 0.68 \[ -\frac{b^{2}}{3 \, c^{\frac{3}{2}} x^{3}} - \frac{a b}{2 \, c^{\frac{3}{2}} x^{4}} - \frac{a^{2}}{5 \, c^{\frac{3}{2}} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2/((c*x^2)^(3/2)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.209676, size = 46, normalized size = 0.7 \[ -\frac{{\left (10 \, b^{2} x^{2} + 15 \, a b x + 6 \, a^{2}\right )} \sqrt{c x^{2}}}{30 \, c^{2} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2/((c*x^2)^(3/2)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.26511, size = 56, normalized size = 0.85 \[ - \frac{a^{2}}{5 c^{\frac{3}{2}} x^{2} \left (x^{2}\right )^{\frac{3}{2}}} - \frac{a b}{2 c^{\frac{3}{2}} x \left (x^{2}\right )^{\frac{3}{2}}} - \frac{b^{2}}{3 c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**2/x**3/(c*x**2)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{2}}{\left (c x^{2}\right )^{\frac{3}{2}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2/((c*x^2)^(3/2)*x^3),x, algorithm="giac")
[Out]