Optimal. Leaf size=40 \[ \frac{c \sqrt{c x^2}}{b}-\frac{a c \sqrt{c x^2} \log (a+b x)}{b^2 x} \]
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Rubi [A] time = 0.0298071, antiderivative size = 40, 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 \[ \frac{c \sqrt{c x^2}}{b}-\frac{a c \sqrt{c x^2} \log (a+b x)}{b^2 x} \]
Antiderivative was successfully verified.
[In] Int[(c*x^2)^(3/2)/(x^2*(a + b*x)),x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a c \sqrt{c x^{2}} \log{\left (a + b x \right )}}{b^{2} x} + \frac{c \sqrt{c x^{2}} \int \frac{1}{b}\, dx}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2)**(3/2)/x**2/(b*x+a),x)
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Mathematica [A] time = 0.0103214, size = 30, normalized size = 0.75 \[ \frac{c^2 x (b x-a \log (a+b x))}{b^2 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(c*x^2)^(3/2)/(x^2*(a + b*x)),x]
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Maple [A] time = 0.005, size = 29, normalized size = 0.7 \[ -{\frac{a\ln \left ( bx+a \right ) -bx}{{b}^{2}{x}^{3}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2)^(3/2)/x^2/(b*x+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(3/2)/((b*x + a)*x^2),x, algorithm="maxima")
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Fricas [A] time = 0.234928, size = 39, normalized size = 0.98 \[ \frac{{\left (b c x - a c \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{b^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(3/2)/((b*x + a)*x^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{2}\right )^{\frac{3}{2}}}{x^{2} \left (a + b x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2)**(3/2)/x**2/(b*x+a),x)
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GIAC/XCAS [A] time = 0.204419, size = 50, normalized size = 1.25 \[ c^{\frac{3}{2}}{\left (\frac{x{\rm sign}\left (x\right )}{b} - \frac{a{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (x\right )}{b^{2}} + \frac{a{\rm ln}\left ({\left | a \right |}\right ){\rm sign}\left (x\right )}{b^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(3/2)/((b*x + a)*x^2),x, algorithm="giac")
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