Optimal. Leaf size=68 \[ \frac{b^{3/2} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} \sqrt{d x^2}}+\frac{b}{a^2 \sqrt{d x^2}}-\frac{1}{3 a x^2 \sqrt{d x^2}} \]
[Out]
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Rubi [A] time = 0.0611417, antiderivative size = 68, 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 \[ \frac{b^{3/2} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} \sqrt{d x^2}}+\frac{b}{a^2 \sqrt{d x^2}}-\frac{1}{3 a x^2 \sqrt{d x^2}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*Sqrt[d*x^2]*(a + b*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 23.9533, size = 66, normalized size = 0.97 \[ - \frac{d}{3 a \left (d x^{2}\right )^{\frac{3}{2}}} + \frac{b}{a^{2} \sqrt{d x^{2}}} + \frac{b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d x^{2}}}{\sqrt{a} \sqrt{d}} \right )}}{a^{\frac{5}{2}} \sqrt{d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(b*x**2+a)/(d*x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0358477, size = 58, normalized size = 0.85 \[ \frac{d \left (3 b^{3/2} x^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )-\sqrt{a} \left (a-3 b x^2\right )\right )}{3 a^{5/2} \left (d x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*Sqrt[d*x^2]*(a + b*x^2)),x]
[Out]
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Maple [A] time = 0.009, size = 58, normalized size = 0.9 \[{\frac{1}{3\,{a}^{2}{x}^{2}} \left ( 3\,{b}^{2}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{3}+3\,b{x}^{2}\sqrt{ab}-a\sqrt{ab} \right ){\frac{1}{\sqrt{d{x}^{2}}}}{\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/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(1/((b*x^2 + a)*sqrt(d*x^2)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.245265, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b d x^{4} \sqrt{-\frac{b}{a d}} \log \left (\frac{2 \, a d x^{2} \sqrt{-\frac{b}{a d}} +{\left (b x^{2} - a\right )} \sqrt{d x^{2}}}{b x^{3} + a x}\right ) + 2 \,{\left (3 \, b x^{2} - a\right )} \sqrt{d x^{2}}}{6 \, a^{2} d x^{4}}, \frac{3 \, b d x^{4} \sqrt{\frac{b}{a d}} \arctan \left (\frac{\sqrt{d x^{2}} b}{a d \sqrt{\frac{b}{a d}}}\right ) +{\left (3 \, b x^{2} - a\right )} \sqrt{d x^{2}}}{3 \, a^{2} d x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*sqrt(d*x^2)*x^3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \sqrt{d x^{2}} \left (a + b x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(b*x**2+a)/(d*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.232168, size = 92, normalized size = 1.35 \[ \frac{1}{3} \, d^{2}{\left (\frac{3 \, b^{2} \arctan \left (\frac{\sqrt{d x^{2}} b}{\sqrt{a b d}}\right )}{\sqrt{a b d} a^{2} d^{2}} + \frac{3 \, b d x^{2} - a d}{\sqrt{d x^{2}} a^{2} d^{3} x^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*sqrt(d*x^2)*x^3),x, algorithm="giac")
[Out]