Optimal. Leaf size=45 \[ \frac{x^2}{b c \sqrt{c x^2}}-\frac{a x \log (a+b x)}{b^2 c \sqrt{c x^2}} \]
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Rubi [A] time = 0.0333006, antiderivative size = 45, 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 \[ \frac{x^2}{b c \sqrt{c x^2}}-\frac{a x \log (a+b x)}{b^2 c \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Int[x^4/((c*x^2)^(3/2)*(a + b*x)),x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a \sqrt{c x^{2}} \log{\left (a + b x \right )}}{b^{2} c^{2} x} + \frac{\sqrt{c x^{2}} \int \frac{1}{b}\, dx}{c^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(c*x**2)**(3/2)/(b*x+a),x)
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Mathematica [A] time = 0.0112439, size = 29, normalized size = 0.64 \[ \frac{x^3 (b x-a \log (a+b x))}{b^2 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^4/((c*x^2)^(3/2)*(a + b*x)),x]
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Maple [A] time = 0.006, size = 29, normalized size = 0.6 \[ -{\frac{{x}^{3} \left ( a\ln \left ( bx+a \right ) -bx \right ) }{{b}^{2}} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(c*x^2)^(3/2)/(b*x+a),x)
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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^4/((c*x^2)^(3/2)*(b*x + a)),x, algorithm="maxima")
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Fricas [A] time = 0.220019, size = 41, normalized size = 0.91 \[ \frac{\sqrt{c x^{2}}{\left (b x - a \log \left (b x + a\right )\right )}}{b^{2} c^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/((c*x^2)^(3/2)*(b*x + a)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\left (c x^{2}\right )^{\frac{3}{2}} \left (a + b x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(c*x**2)**(3/2)/(b*x+a),x)
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GIAC/XCAS [A] time = 0.215779, size = 73, normalized size = 1.62 \[ \frac{\frac{a{\rm ln}\left ({\left | -{\left (\sqrt{c} x - \sqrt{c x^{2}}\right )} b - 2 \, a \sqrt{c} \right |}\right )}{b^{2} \sqrt{c}} + \frac{\sqrt{c x^{2}}}{b c}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/((c*x^2)^(3/2)*(b*x + a)),x, algorithm="giac")
[Out]