Optimal. Leaf size=68 \[ -\frac{a^2 c \sqrt{c x^2}}{b^3 x (a+b x)}-\frac{2 a c \sqrt{c x^2} \log (a+b x)}{b^3 x}+\frac{c \sqrt{c x^2}}{b^2} \]
[Out]
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Rubi [A] time = 0.0527329, antiderivative size = 68, 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 \[ -\frac{a^2 c \sqrt{c x^2}}{b^3 x (a+b x)}-\frac{2 a c \sqrt{c x^2} \log (a+b x)}{b^3 x}+\frac{c \sqrt{c x^2}}{b^2} \]
Antiderivative was successfully verified.
[In] Int[(c*x^2)^(3/2)/(x*(a + b*x)^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2} c \sqrt{c x^{2}}}{b^{3} x \left (a + b x\right )} - \frac{2 a c \sqrt{c x^{2}} \log{\left (a + b x \right )}}{b^{3} x} + \frac{c \sqrt{c x^{2}} \int \frac{1}{b^{2}}\, dx}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2)**(3/2)/x/(b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.0105255, size = 55, normalized size = 0.81 \[ \frac{c^2 x \left (-a^2+a b x-2 a (a+b x) \log (a+b x)+b^2 x^2\right )}{b^3 \sqrt{c x^2} (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(c*x^2)^(3/2)/(x*(a + b*x)^2),x]
[Out]
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Maple [A] time = 0.007, size = 62, normalized size = 0.9 \[ -{\frac{2\,\ln \left ( bx+a \right ) xab-{b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( bx+a \right ) -abx+{a}^{2}}{{x}^{3} \left ( bx+a \right ){b}^{3}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2)^(3/2)/x/(b*x+a)^2,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)^2*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.209069, size = 85, normalized size = 1.25 \[ \frac{{\left (b^{2} c x^{2} + a b c x - a^{2} c - 2 \,{\left (a b c x + a^{2} c\right )} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{b^{4} x^{2} + a b^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(3/2)/((b*x + a)^2*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{2}\right )^{\frac{3}{2}}}{x \left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2)**(3/2)/x/(b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.206268, size = 78, normalized size = 1.15 \[ c^{\frac{3}{2}}{\left (\frac{x{\rm sign}\left (x\right )}{b^{2}} - \frac{2 \, a{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (x\right )}{b^{3}} + \frac{{\left (2 \, a{\rm ln}\left ({\left | a \right |}\right ) + a\right )}{\rm sign}\left (x\right )}{b^{3}} - \frac{a^{2}{\rm sign}\left (x\right )}{{\left (b x + a\right )} b^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(3/2)/((b*x + a)^2*x),x, algorithm="giac")
[Out]