Optimal. Leaf size=72 \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 a^{3/2}}-\frac{A \sqrt{a+b x+c x^2}}{a x} \]
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Rubi [A] time = 0.126628, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 a^{3/2}}-\frac{A \sqrt{a+b x+c x^2}}{a x} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^2*Sqrt[a + b*x + c*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 12.735, size = 61, normalized size = 0.85 \[ - \frac{A \sqrt{a + b x + c x^{2}}}{a x} + \frac{\left (A b - 2 B a\right ) \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{2 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x**2/(c*x**2+b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.254457, size = 86, normalized size = 1.19 \[ \frac{x (A b-2 a B) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )+\log (x) (2 a B x-A b x)-2 \sqrt{a} A \sqrt{a+x (b+c x)}}{2 a^{3/2} x} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^2*Sqrt[a + b*x + c*x^2]),x]
[Out]
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Maple [A] time = 0.015, size = 94, normalized size = 1.3 \[ -{\frac{A}{ax}\sqrt{c{x}^{2}+bx+a}}+{\frac{Ab}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{B\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^2/(c*x^2+b*x+a)^(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((B*x + A)/(sqrt(c*x^2 + b*x + a)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.32781, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (2 \, B a - A b\right )} x \log \left (-\frac{4 \,{\left (a b x + 2 \, a^{2}\right )} \sqrt{c x^{2} + b x + a} +{\left (8 \, a b x +{\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{2}}\right ) + 4 \, \sqrt{c x^{2} + b x + a} A \sqrt{a}}{4 \, a^{\frac{3}{2}} x}, -\frac{{\left (2 \, B a - A b\right )} x \arctan \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{2} + b x + a} a}\right ) + 2 \, \sqrt{c x^{2} + b x + a} A \sqrt{-a}}{2 \, \sqrt{-a} a x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(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. \[ \int \frac{A + B x}{x^{2} \sqrt{a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/x**2/(c*x**2+b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.28044, size = 149, normalized size = 2.07 \[ \frac{{\left (2 \, B a - A b\right )} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} A b + 2 \, A a \sqrt{c}}{{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} - a\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(c*x^2 + b*x + a)*x^2),x, algorithm="giac")
[Out]