Optimal. Leaf size=64 \[ -\frac{b c \sqrt{c x^2} \log (x)}{a^2 x}+\frac{b c \sqrt{c x^2} \log (a+b x)}{a^2 x}-\frac{c \sqrt{c x^2}}{a x^2} \]
[Out]
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Rubi [A] time = 0.0439269, antiderivative size = 64, 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 \[ -\frac{b c \sqrt{c x^2} \log (x)}{a^2 x}+\frac{b c \sqrt{c x^2} \log (a+b x)}{a^2 x}-\frac{c \sqrt{c x^2}}{a x^2} \]
Antiderivative was successfully verified.
[In] Int[(c*x^2)^(3/2)/(x^5*(a + b*x)),x]
[Out]
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Rubi in Sympy [A] time = 17.3579, size = 58, normalized size = 0.91 \[ - \frac{c \sqrt{c x^{2}}}{a x^{2}} - \frac{b c \sqrt{c x^{2}} \log{\left (x \right )}}{a^{2} x} + \frac{b c \sqrt{c x^{2}} \log{\left (a + b x \right )}}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2)**(3/2)/x**5/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.0171175, size = 34, normalized size = 0.53 \[ -\frac{c^2 (-b x \log (a+b x)+a+b x \log (x))}{a^2 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(c*x^2)^(3/2)/(x^5*(a + b*x)),x]
[Out]
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Maple [A] time = 0.007, size = 33, normalized size = 0.5 \[ -{\frac{b\ln \left ( x \right ) x-b\ln \left ( bx+a \right ) x+a}{{a}^{2}{x}^{4}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2)^(3/2)/x^5/(b*x+a),x)
[Out]
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Maxima [A] time = 1.35481, size = 50, normalized size = 0.78 \[ \frac{b c^{\frac{3}{2}} \log \left (b x + a\right )}{a^{2}} - \frac{b c^{\frac{3}{2}} \log \left (x\right )}{a^{2}} - \frac{c^{\frac{3}{2}}}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(3/2)/((b*x + a)*x^5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.233999, size = 45, normalized size = 0.7 \[ \frac{{\left (b c x \log \left (\frac{b x + a}{x}\right ) - a c\right )} \sqrt{c x^{2}}}{a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(3/2)/((b*x + a)*x^5),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{2}\right )^{\frac{3}{2}}}{x^{5} \left (a + b x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2)**(3/2)/x**5/(b*x+a),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(3/2)/((b*x + a)*x^5),x, algorithm="giac")
[Out]