Optimal. Leaf size=76 \[ -\frac{\sqrt{b c-a d} \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 a^{3/2}}-\frac{\sqrt{c+d x^4}}{2 a x^2} \]
[Out]
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Rubi [A] time = 0.253304, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{\sqrt{b c-a d} \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 a^{3/2}}-\frac{\sqrt{c+d x^4}}{2 a x^2} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c + d*x^4]/(x^3*(a + b*x^4)),x]
[Out]
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Rubi in Sympy [A] time = 28.6608, size = 63, normalized size = 0.83 \[ - \frac{\sqrt{c + d x^{4}}}{2 a x^{2}} + \frac{\sqrt{a d - b c} \operatorname{atanh}{\left (\frac{x^{2} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{4}}} \right )}}{2 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**4+c)**(1/2)/x**3/(b*x**4+a),x)
[Out]
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Mathematica [A] time = 1.21772, size = 136, normalized size = 1.79 \[ \frac{\sqrt{c+d x^4} \left (-\frac{x^4 (b c-a d) \sin ^{-1}\left (\frac{\sqrt{x^4 \left (\frac{b}{a}-\frac{d}{c}\right )}}{\sqrt{\frac{b x^4}{a}+1}}\right )}{c \sqrt{\frac{b x^4}{a}+1} \sqrt{x^4 \left (\frac{b}{a}-\frac{d}{c}\right )} \sqrt{\frac{a \left (c+d x^4\right )}{c \left (a+b x^4\right )}}}-a\right )}{2 a^2 x^2} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c + d*x^4]/(x^3*(a + b*x^4)),x]
[Out]
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Maple [B] time = 0.021, size = 1075, normalized size = 14.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^4+c)^(1/2)/x^3/(b*x^4+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{4} + c}}{{\left (b x^{4} + a\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.233045, size = 1, normalized size = 0.01 \[ \left [\frac{x^{2} \sqrt{-\frac{b c - a d}{a}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} - 4 \,{\left ({\left (a b c - 2 \, a^{2} d\right )} x^{6} - a^{2} c x^{2}\right )} \sqrt{d x^{4} + c} \sqrt{-\frac{b c - a d}{a}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right ) - 4 \, \sqrt{d x^{4} + c}}{8 \, a x^{2}}, \frac{x^{2} \sqrt{\frac{b c - a d}{a}} \arctan \left (-\frac{{\left (b c - 2 \, a d\right )} x^{4} - a c}{2 \, \sqrt{d x^{4} + c} a x^{2} \sqrt{\frac{b c - a d}{a}}}\right ) - 2 \, \sqrt{d x^{4} + c}}{4 \, a x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x^{4}}}{x^{3} \left (a + b x^{4}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**4+c)**(1/2)/x**3/(b*x**4+a),x)
[Out]
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GIAC/XCAS [A] time = 0.218513, size = 89, normalized size = 1.17 \[ \frac{{\left (b c - a d\right )} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{4}}}}{\sqrt{a b c - a^{2} d}}\right )}{2 \, \sqrt{a b c - a^{2} d} a} - \frac{\sqrt{d + \frac{c}{x^{4}}}}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^3),x, algorithm="giac")
[Out]