Optimal. Leaf size=65 \[ \frac{\sqrt{c+d x^2}}{b}-\frac{\sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{3/2}} \]
[Out]
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Rubi [A] time = 0.147658, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\sqrt{c+d x^2}}{b}-\frac{\sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(x*Sqrt[c + d*x^2])/(a + b*x^2),x]
[Out]
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Rubi in Sympy [A] time = 20.0182, size = 53, normalized size = 0.82 \[ \frac{\sqrt{c + d x^{2}}}{b} - \frac{\sqrt{a d - b c} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(d*x**2+c)**(1/2)/(b*x**2+a),x)
[Out]
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Mathematica [A] time = 0.0516501, size = 65, normalized size = 1. \[ \frac{\sqrt{c+d x^2}}{b}-\frac{\sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*Sqrt[c + d*x^2])/(a + b*x^2),x]
[Out]
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Maple [B] time = 0.015, size = 936, normalized size = 14.4 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(d*x^2+c)^(1/2)/(b*x^2+a),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(sqrt(d*x^2 + c)*x/(b*x^2 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.258703, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{\frac{b c - a d}{b}} \log \left (\frac{b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \,{\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt{d x^{2} + c} \sqrt{\frac{b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, \sqrt{d x^{2} + c}}{4 \, b}, -\frac{\sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{b d x^{2} + 2 \, b c - a d}{2 \, \sqrt{d x^{2} + c} b \sqrt{-\frac{b c - a d}{b}}}\right ) - 2 \, \sqrt{d x^{2} + c}}{2 \, b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 + c)*x/(b*x^2 + a),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x \sqrt{c + d x^{2}}}{a + b x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(d*x**2+c)**(1/2)/(b*x**2+a),x)
[Out]
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GIAC/XCAS [A] time = 0.23859, size = 86, normalized size = 1.32 \[ \frac{{\left (b c - a d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b} + \frac{\sqrt{d x^{2} + c}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 + c)*x/(b*x^2 + a),x, algorithm="giac")
[Out]