Optimal. Leaf size=47 \[ -\frac{a^2}{\sqrt{c x^2}}+\frac{2 a b x \log (x)}{\sqrt{c x^2}}+\frac{b^2 x^2}{\sqrt{c x^2}} \]
[Out]
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Rubi [A] time = 0.0290423, antiderivative size = 47, 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 \[ -\frac{a^2}{\sqrt{c x^2}}+\frac{2 a b x \log (x)}{\sqrt{c x^2}}+\frac{b^2 x^2}{\sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^2/(x*Sqrt[c*x^2]),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2} \sqrt{c x^{2}}}{c x^{2}} + \frac{2 a b \sqrt{c x^{2}} \log{\left (x \right )}}{c x} + \frac{\sqrt{c x^{2}} \int b^{2}\, dx}{c x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**2/x/(c*x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0169981, size = 34, normalized size = 0.72 \[ \frac{c x^2 \left (-a^2+2 a b x \log (x)+b^2 x^2\right )}{\left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^2/(x*Sqrt[c*x^2]),x]
[Out]
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Maple [A] time = 0.008, size = 29, normalized size = 0.6 \[{(2\,ab\ln \left ( x \right ) x+{b}^{2}{x}^{2}-{a}^{2}){\frac{1}{\sqrt{c{x}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^2/x/(c*x^2)^(1/2),x)
[Out]
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Maxima [A] time = 1.34546, size = 47, normalized size = 1. \[ \frac{2 \, a b \log \left (x\right )}{\sqrt{c}} + \frac{\sqrt{c x^{2}} b^{2}}{c} - \frac{a^{2}}{\sqrt{c} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2/(sqrt(c*x^2)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.206527, size = 46, normalized size = 0.98 \[ \frac{{\left (b^{2} x^{2} + 2 \, a b x \log \left (x\right ) - a^{2}\right )} \sqrt{c x^{2}}}{c x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2/(sqrt(c*x^2)*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{2}}{x \sqrt{c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**2/x/(c*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.211898, size = 88, normalized size = 1.87 \[ \frac{\sqrt{c x^{2}} b^{2}}{c} - \frac{2 \,{\left (a b{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right ) - \frac{a^{2} \sqrt{c}}{\sqrt{c} x - \sqrt{c x^{2}}}\right )}}{\sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2/(sqrt(c*x^2)*x),x, algorithm="giac")
[Out]