Optimal. Leaf size=34 \[ \frac{x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \sqrt{d x^2}} \]
[Out]
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Rubi [A] time = 0.0232445, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \sqrt{d x^2}} \]
Antiderivative was successfully verified.
[In] Int[x/(Sqrt[d*x^2]*(a + b*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 12.2338, size = 39, normalized size = 1.15 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d x^{2}}}{\sqrt{a} \sqrt{d}} \right )}}{\sqrt{a} \sqrt{b} \sqrt{d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(b*x**2+a)/(d*x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.012135, size = 34, normalized size = 1. \[ \frac{x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \sqrt{d x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[x/(Sqrt[d*x^2]*(a + b*x^2)),x]
[Out]
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Maple [A] time = 0.005, size = 24, normalized size = 0.7 \[{x\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{d{x}^{2}}}}{\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(b*x^2+a)/(d*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/((b*x^2 + a)*sqrt(d*x^2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.225072, size = 1, normalized size = 0.03 \[ \left [-\frac{\sqrt{-a b d} \log \left (\frac{2 \, a b d x^{2} + \sqrt{-a b d}{\left (b x^{2} - a\right )} \sqrt{d x^{2}}}{b x^{3} + a x}\right )}{2 \, a b d}, \frac{\sqrt{a b d} \arctan \left (\frac{\sqrt{a b d} \sqrt{d x^{2}}}{a d}\right )}{a b d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x^2 + a)*sqrt(d*x^2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{d x^{2}} \left (a + b x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x**2+a)/(d*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.23842, size = 31, normalized size = 0.91 \[ \frac{\arctan \left (\frac{\sqrt{d x^{2}} b}{\sqrt{a b d}}\right )}{\sqrt{a b d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x^2 + a)*sqrt(d*x^2)),x, algorithm="giac")
[Out]