Optimal. Leaf size=77 \[ \frac{b^2 x \log (x)}{a^3 \sqrt{c x^2}}-\frac{b^2 x \log (a+b x)}{a^3 \sqrt{c x^2}}+\frac{b}{a^2 \sqrt{c x^2}}-\frac{1}{2 a x \sqrt{c x^2}} \]
[Out]
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Rubi [A] time = 0.0536256, antiderivative size = 77, 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 \[ \frac{b^2 x \log (x)}{a^3 \sqrt{c x^2}}-\frac{b^2 x \log (a+b x)}{a^3 \sqrt{c x^2}}+\frac{b}{a^2 \sqrt{c x^2}}-\frac{1}{2 a x \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*Sqrt[c*x^2]*(a + b*x)),x]
[Out]
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Rubi in Sympy [A] time = 20.203, size = 82, normalized size = 1.06 \[ - \frac{\sqrt{c x^{2}}}{2 a c x^{3}} + \frac{b \sqrt{c x^{2}}}{a^{2} c x^{2}} + \frac{b^{2} \sqrt{c x^{2}} \log{\left (x \right )}}{a^{3} c x} - \frac{b^{2} \sqrt{c x^{2}} \log{\left (a + b x \right )}}{a^{3} c x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(b*x+a)/(c*x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0237383, size = 52, normalized size = 0.68 \[ \frac{c x \left (-2 b^2 x^2 \log (a+b x)-a (a-2 b x)+2 b^2 x^2 \log (x)\right )}{2 a^3 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*Sqrt[c*x^2]*(a + b*x)),x]
[Out]
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Maple [A] time = 0.007, size = 51, normalized size = 0.7 \[{\frac{2\,{b}^{2}\ln \left ( x \right ){x}^{2}-2\,{b}^{2}\ln \left ( bx+a \right ){x}^{2}+2\,abx-{a}^{2}}{2\,x{a}^{3}}{\frac{1}{\sqrt{c{x}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(b*x+a)/(c*x^2)^(1/2),x)
[Out]
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Maxima [A] time = 1.36181, size = 74, normalized size = 0.96 \[ -\frac{b^{2} \log \left (b x + a\right )}{a^{3} \sqrt{c}} + \frac{b^{2} \log \left (x\right )}{a^{3} \sqrt{c}} + \frac{2 \, b \sqrt{c} x - a \sqrt{c}}{2 \, a^{2} c x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^2)*(b*x + a)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.219372, size = 63, normalized size = 0.82 \[ \frac{{\left (2 \, b^{2} x^{2} \log \left (\frac{x}{b x + a}\right ) + 2 \, a b x - a^{2}\right )} \sqrt{c x^{2}}}{2 \, a^{3} c x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^2)*(b*x + a)*x^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \sqrt{c x^{2}} \left (a + b x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(b*x+a)/(c*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.517077, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^2)*(b*x + a)*x^2),x, algorithm="giac")
[Out]