Optimal. Leaf size=68 \[ -\frac{x \log (a+b x)}{a^2 c \sqrt{c x^2}}+\frac{x \log (x)}{a^2 c \sqrt{c x^2}}+\frac{x}{a c \sqrt{c x^2} (a+b x)} \]
[Out]
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Rubi [A] time = 0.0481917, 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 \[ -\frac{x \log (a+b x)}{a^2 c \sqrt{c x^2}}+\frac{x \log (x)}{a^2 c \sqrt{c x^2}}+\frac{x}{a c \sqrt{c x^2} (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[x^2/((c*x^2)^(3/2)*(a + b*x)^2),x]
[Out]
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Rubi in Sympy [A] time = 16.4641, size = 63, normalized size = 0.93 \[ \frac{\sqrt{c x^{2}}}{a c^{2} x \left (a + b x\right )} + \frac{\sqrt{c x^{2}} \log{\left (x \right )}}{a^{2} c^{2} x} - \frac{\sqrt{c x^{2}} \log{\left (a + b x \right )}}{a^{2} c^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(c*x**2)**(3/2)/(b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.0232432, size = 46, normalized size = 0.68 \[ \frac{x^3 (\log (x) (a+b x)-(a+b x) \log (a+b x)+a)}{a^2 \left (c x^2\right )^{3/2} (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/((c*x^2)^(3/2)*(a + b*x)^2),x]
[Out]
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Maple [A] time = 0.007, size = 52, normalized size = 0.8 \[{\frac{{x}^{3} \left ( b\ln \left ( x \right ) x-b\ln \left ( bx+a \right ) x+a\ln \left ( x \right ) -a\ln \left ( bx+a \right ) +a \right ) }{{a}^{2} \left ( bx+a \right ) } \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(c*x^2)^(3/2)/(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(x^2/((c*x^2)^(3/2)*(b*x + a)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.224644, size = 65, normalized size = 0.96 \[ \frac{\sqrt{c x^{2}}{\left ({\left (b x + a\right )} \log \left (\frac{x}{b x + a}\right ) + a\right )}}{a^{2} b c^{2} x^{2} + a^{3} c^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((c*x^2)^(3/2)*(b*x + a)^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (c x^{2}\right )^{\frac{3}{2}} \left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(c*x**2)**(3/2)/(b*x+a)**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (c x^{2}\right )^{\frac{3}{2}}{\left (b x + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((c*x^2)^(3/2)*(b*x + a)^2),x, algorithm="giac")
[Out]