Optimal. Leaf size=29 \[ -\frac{(a+b x)^2}{2 a c x \sqrt{c x^2}} \]
[Out]
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Rubi [A] time = 0.0157361, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{(a+b x)^2}{2 a c x \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)/(c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 6.38163, size = 36, normalized size = 1.24 \[ - \frac{a \sqrt{c x^{2}}}{2 c^{2} x^{3}} - \frac{b \sqrt{c x^{2}}}{c^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)/(c*x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.00393579, size = 22, normalized size = 0.76 \[ \frac{x (-a-2 b x)}{2 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)/(c*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.004, size = 17, normalized size = 0.6 \[ -{\frac{x \left ( 2\,bx+a \right ) }{2} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)/(c*x^2)^(3/2),x)
[Out]
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Maxima [A] time = 1.34866, size = 31, normalized size = 1.07 \[ -\frac{b}{\sqrt{c x^{2}} c} - \frac{a}{2 \, c^{\frac{3}{2}} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/(c*x^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.210717, size = 28, normalized size = 0.97 \[ -\frac{\sqrt{c x^{2}}{\left (2 \, b x + a\right )}}{2 \, c^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/(c*x^2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.03309, size = 34, normalized size = 1.17 \[ - \frac{a x}{2 c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} - \frac{b x^{2}}{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)/(c*x**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.538674, size = 4, normalized size = 0.14 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/(c*x^2)^(3/2),x, algorithm="giac")
[Out]