Optimal. Leaf size=83 \[ -\frac{a^3 x \log (a+b x)}{b^4 \sqrt{c x^2}}+\frac{a^2 x^2}{b^3 \sqrt{c x^2}}-\frac{a x^3}{2 b^2 \sqrt{c x^2}}+\frac{x^4}{3 b \sqrt{c x^2}} \]
[Out]
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Rubi [A] time = 0.0557734, antiderivative size = 83, 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 \[ -\frac{a^3 x \log (a+b x)}{b^4 \sqrt{c x^2}}+\frac{a^2 x^2}{b^3 \sqrt{c x^2}}-\frac{a x^3}{2 b^2 \sqrt{c x^2}}+\frac{x^4}{3 b \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Int[x^4/(Sqrt[c*x^2]*(a + b*x)),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{3} \sqrt{c x^{2}} \log{\left (a + b x \right )}}{b^{4} c x} - \frac{a \sqrt{c x^{2}} \int x\, dx}{b^{2} c x} + \frac{x^{2} \sqrt{c x^{2}}}{3 b c} + \frac{\sqrt{c x^{2}} \int a^{2}\, dx}{b^{3} c x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(b*x+a)/(c*x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.021187, size = 51, normalized size = 0.61 \[ \frac{x \left (b x \left (6 a^2-3 a b x+2 b^2 x^2\right )-6 a^3 \log (a+b x)\right )}{6 b^4 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^4/(Sqrt[c*x^2]*(a + b*x)),x]
[Out]
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Maple [A] time = 0.008, size = 50, normalized size = 0.6 \[ -{\frac{x \left ( -2\,{b}^{3}{x}^{3}+3\,a{b}^{2}{x}^{2}+6\,{a}^{3}\ln \left ( bx+a \right ) -6\,{a}^{2}bx \right ) }{6\,{b}^{4}}{\frac{1}{\sqrt{c{x}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(b*x+a)/(c*x^2)^(1/2),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^4/(sqrt(c*x^2)*(b*x + a)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.217254, size = 73, normalized size = 0.88 \[ \frac{{\left (2 \, b^{3} x^{3} - 3 \, a b^{2} x^{2} + 6 \, a^{2} b x - 6 \, a^{3} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{6 \, b^{4} c x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(sqrt(c*x^2)*(b*x + a)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{c x^{2}} \left (a + b x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(b*x+a)/(c*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.215686, size = 108, normalized size = 1.3 \[ \frac{1}{6} \, \sqrt{c x^{2}}{\left (x{\left (\frac{2 \, x}{b c} - \frac{3 \, a}{b^{2} c}\right )} + \frac{6 \, a^{2}}{b^{3} c}\right )} + \frac{a^{3}{\rm ln}\left ({\left | -{\left (\sqrt{c} x - \sqrt{c x^{2}}\right )} b - 2 \, a \sqrt{c} \right |}\right )}{b^{4} \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(sqrt(c*x^2)*(b*x + a)),x, algorithm="giac")
[Out]