Optimal. Leaf size=99 \[ \frac{a \sqrt{c+d x^2}}{2 b \left (a+b x^2\right ) (b c-a d)}-\frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 b^{3/2} (b c-a d)^{3/2}} \]
[Out]
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Rubi [A] time = 0.245129, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{a \sqrt{c+d x^2}}{2 b \left (a+b x^2\right ) (b c-a d)}-\frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 b^{3/2} (b c-a d)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[x^3/((a + b*x^2)^2*Sqrt[c + d*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 26.5652, size = 78, normalized size = 0.79 \[ - \frac{a \sqrt{c + d x^{2}}}{2 b \left (a + b x^{2}\right ) \left (a d - b c\right )} + \frac{\left (\frac{a d}{2} - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{b^{\frac{3}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(b*x**2+a)**2/(d*x**2+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.126695, size = 99, normalized size = 1. \[ \frac{a \sqrt{c+d x^2}}{2 b \left (a+b x^2\right ) (b c-a d)}-\frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 b^{3/2} (b c-a d)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/((a + b*x^2)^2*Sqrt[c + d*x^2]),x]
[Out]
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Maple [B] time = 0.019, size = 807, normalized size = 8.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(b*x^2+a)^2/(d*x^2+c)^(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)^2*sqrt(d*x^2 + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.268286, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{b^{2} c - a b d} \sqrt{d x^{2} + c} a +{\left (2 \, a b c - a^{2} d +{\left (2 \, b^{2} c - a b d\right )} x^{2}\right )} \log \left (\frac{{\left (b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2}\right )} \sqrt{b^{2} c - a b d} - 4 \,{\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2} +{\left (b^{3} c d - a b^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{8 \,{\left (a b^{2} c - a^{2} b d +{\left (b^{3} c - a b^{2} d\right )} x^{2}\right )} \sqrt{b^{2} c - a b d}}, \frac{2 \, \sqrt{-b^{2} c + a b d} \sqrt{d x^{2} + c} a +{\left (2 \, a b c - a^{2} d +{\left (2 \, b^{2} c - a b d\right )} x^{2}\right )} \arctan \left (-\frac{{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt{-b^{2} c + a b d}}{2 \,{\left (b^{2} c - a b d\right )} \sqrt{d x^{2} + c}}\right )}{4 \,{\left (a b^{2} c - a^{2} b d +{\left (b^{3} c - a b^{2} d\right )} x^{2}\right )} \sqrt{-b^{2} c + a b d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x^2 + a)^2*sqrt(d*x^2 + c)),x, algorithm="fricas")
[Out]
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Sympy [F(-2)] 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)**2/(d*x**2+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.236052, size = 157, normalized size = 1.59 \[ \frac{\frac{\sqrt{d x^{2} + c} a d^{2}}{{\left (b^{2} c - a b d\right )}{\left ({\left (d x^{2} + c\right )} b - b c + a d\right )}} + \frac{{\left (2 \, b c d - a d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (b^{2} c - a b d\right )} \sqrt{-b^{2} c + a b d}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x^2 + a)^2*sqrt(d*x^2 + c)),x, algorithm="giac")
[Out]