Optimal. Leaf size=70 \[ \frac{\sqrt{c+d x^4}}{2 b}-\frac{\sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 b^{3/2}} \]
[Out]
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Rubi [A] time = 0.182226, antiderivative size = 70, 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 \[ \frac{\sqrt{c+d x^4}}{2 b}-\frac{\sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 b^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(x^3*Sqrt[c + d*x^4])/(a + b*x^4),x]
[Out]
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Rubi in Sympy [A] time = 18.9067, size = 56, normalized size = 0.8 \[ \frac{\sqrt{c + d x^{4}}}{2 b} - \frac{\sqrt{a d - b c} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{4}}}{\sqrt{a d - b c}} \right )}}{2 b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(d*x**4+c)**(1/2)/(b*x**4+a),x)
[Out]
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Mathematica [A] time = 0.0640862, size = 70, normalized size = 1. \[ \frac{\sqrt{c+d x^4}}{2 b}-\frac{\sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 b^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*Sqrt[c + d*x^4])/(a + b*x^4),x]
[Out]
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Maple [B] time = 0.01, size = 988, normalized size = 14.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(d*x^4+c)^(1/2)/(b*x^4+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^4 + c)*x^3/(b*x^4 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.217246, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{4} + 2 \, b c - a d - 2 \, \sqrt{d x^{4} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{4} + a}\right ) + 2 \, \sqrt{d x^{4} + c}}{4 \, b}, -\frac{\sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x^{4} + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) - \sqrt{d x^{4} + c}}{2 \, b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^4 + c)*x^3/(b*x^4 + a),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} \sqrt{c + d x^{4}}}{a + b x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(d*x**4+c)**(1/2)/(b*x**4+a),x)
[Out]
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GIAC/XCAS [A] time = 0.211631, size = 89, normalized size = 1.27 \[ \frac{{\left (b c - a d\right )} \arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{2 \, \sqrt{-b^{2} c + a b d} b} + \frac{\sqrt{d x^{4} + c}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^4 + c)*x^3/(b*x^4 + a),x, algorithm="giac")
[Out]