Optimal. Leaf size=28 \[ -\frac{c \sqrt{\frac{c}{(a+b x)^2}}}{2 b (a+b x)} \]
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Rubi [A] time = 0.0249929, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{c \sqrt{\frac{c}{(a+b x)^2}}}{2 b (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[(c/(a + b*x)^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 2.14249, size = 22, normalized size = 0.79 \[ - \frac{c \sqrt{\frac{c}{\left (a + b x\right )^{2}}}}{2 b \left (a + b x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c/(b*x+a)**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0135289, size = 25, normalized size = 0.89 \[ -\frac{(a+b x) \left (\frac{c}{(a+b x)^2}\right )^{3/2}}{2 b} \]
Antiderivative was successfully verified.
[In] Integrate[(c/(a + b*x)^2)^(3/2),x]
[Out]
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Maple [A] time = 0.006, size = 22, normalized size = 0.8 \[ -{\frac{bx+a}{2\,b} \left ({\frac{c}{ \left ( bx+a \right ) ^{2}}} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c/(b*x+a)^2)^(3/2),x)
[Out]
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Maxima [A] time = 1.43811, size = 36, normalized size = 1.29 \[ -\frac{c^{\frac{3}{2}}}{2 \,{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c/(b*x + a)^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.214771, size = 49, normalized size = 1.75 \[ -\frac{c \sqrt{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{2 \,{\left (b^{2} x + a b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c/(b*x + a)^2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (\frac{c}{\left (a + b x\right )^{2}}\right )^{\frac{3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c/(b*x+a)**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.215048, size = 28, normalized size = 1. \[ -\frac{c^{\frac{3}{2}}{\rm sign}\left (b x + a\right )}{2 \,{\left (b x + a\right )}^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c/(b*x + a)^2)^(3/2),x, algorithm="giac")
[Out]