Optimal. Leaf size=72 \[ \frac{a^{3/2} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2} \sqrt{d x^2}}-\frac{a x^2}{b^2 \sqrt{d x^2}}+\frac{x^4}{3 b \sqrt{d x^2}} \]
[Out]
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Rubi [A] time = 0.066319, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{a^{3/2} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2} \sqrt{d x^2}}-\frac{a x^2}{b^2 \sqrt{d x^2}}+\frac{x^4}{3 b \sqrt{d x^2}} \]
Antiderivative was successfully verified.
[In] Int[x^5/(Sqrt[d*x^2]*(a + b*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 22.3482, size = 70, normalized size = 0.97 \[ \frac{a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d x^{2}}}{\sqrt{a} \sqrt{d}} \right )}}{b^{\frac{5}{2}} \sqrt{d}} - \frac{a \sqrt{d x^{2}}}{b^{2} d} + \frac{\left (d x^{2}\right )^{\frac{3}{2}}}{3 b d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5/(b*x**2+a)/(d*x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.040234, size = 56, normalized size = 0.78 \[ \frac{x \left (3 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+\sqrt{b} x \left (b x^2-3 a\right )\right )}{3 b^{5/2} \sqrt{d x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^5/(Sqrt[d*x^2]*(a + b*x^2)),x]
[Out]
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Maple [A] time = 0.01, size = 53, normalized size = 0.7 \[{\frac{x}{3\,{b}^{2}} \left ( \sqrt{ab}{x}^{3}b-3\,\sqrt{ab}xa+3\,{a}^{2}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) \right ){\frac{1}{\sqrt{d{x}^{2}}}}{\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5/(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^5/((b*x^2 + a)*sqrt(d*x^2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.220001, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a d \sqrt{-\frac{a}{b d}} \log \left (\frac{2 \, b d x^{2} \sqrt{-\frac{a}{b d}} +{\left (b x^{2} - a\right )} \sqrt{d x^{2}}}{b x^{3} + a x}\right ) + 2 \,{\left (b x^{2} - 3 \, a\right )} \sqrt{d x^{2}}}{6 \, b^{2} d}, \frac{3 \, a d \sqrt{\frac{a}{b d}} \arctan \left (\frac{\sqrt{d x^{2}}}{d \sqrt{\frac{a}{b d}}}\right ) +{\left (b x^{2} - 3 \, a\right )} \sqrt{d x^{2}}}{3 \, b^{2} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/((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. \[ \int \frac{x^{5}}{\sqrt{d x^{2}} \left (a + b x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5/(b*x**2+a)/(d*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.236454, size = 95, normalized size = 1.32 \[ \frac{a^{2} \arctan \left (\frac{\sqrt{d x^{2}} b}{\sqrt{a b d}}\right )}{\sqrt{a b d} b^{2}} + \frac{\sqrt{d x^{2}} b^{2} d^{5} x^{2} - 3 \, \sqrt{d x^{2}} a b d^{5}}{3 \, b^{3} d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/((b*x^2 + a)*sqrt(d*x^2)),x, algorithm="giac")
[Out]