Optimal. Leaf size=52 \[ \frac{x^2}{b \sqrt{d x^2}}-\frac{\sqrt{a} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2} \sqrt{d x^2}} \]
[Out]
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Rubi [A] time = 0.0446219, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{x^2}{b \sqrt{d x^2}}-\frac{\sqrt{a} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2} \sqrt{d x^2}} \]
Antiderivative was successfully verified.
[In] Int[x^3/(Sqrt[d*x^2]*(a + b*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 17.4194, size = 51, normalized size = 0.98 \[ - \frac{\sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d x^{2}}}{\sqrt{a} \sqrt{d}} \right )}}{b^{\frac{3}{2}} \sqrt{d}} + \frac{\sqrt{d x^{2}}}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(b*x**2+a)/(d*x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0257385, size = 44, normalized size = 0.85 \[ \frac{x \left (\sqrt{b} x-\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right )}{b^{3/2} \sqrt{d x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/(Sqrt[d*x^2]*(a + b*x^2)),x]
[Out]
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Maple [A] time = 0.008, size = 38, normalized size = 0.7 \[{\frac{x}{b} \left ( x\sqrt{ab}-a\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ) \right ){\frac{1}{\sqrt{d{x}^{2}}}}{\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(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^3/((b*x^2 + a)*sqrt(d*x^2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222269, size = 1, normalized size = 0.02 \[ \left [\frac{d \sqrt{-\frac{a}{b d}} \log \left (-\frac{2 \, b d x^{2} \sqrt{-\frac{a}{b d}} -{\left (b x^{2} - a\right )} \sqrt{d x^{2}}}{b x^{3} + a x}\right ) + 2 \, \sqrt{d x^{2}}}{2 \, b d}, -\frac{d \sqrt{\frac{a}{b d}} \arctan \left (\frac{\sqrt{d x^{2}}}{d \sqrt{\frac{a}{b d}}}\right ) - \sqrt{d x^{2}}}{b d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((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. \[ \int \frac{x^{3}}{\sqrt{d x^{2}} \left (a + b x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(b*x**2+a)/(d*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.236749, size = 62, normalized size = 1.19 \[ -\frac{\frac{a d \arctan \left (\frac{\sqrt{d x^{2}} b}{\sqrt{a b d}}\right )}{\sqrt{a b d} b} - \frac{\sqrt{d x^{2}}}{b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x^2 + a)*sqrt(d*x^2)),x, algorithm="giac")
[Out]