Optimal. Leaf size=85 \[ \frac{\sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 a \sqrt{b}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right )}{2 a} \]
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Rubi [A] time = 0.223052, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{\sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 a \sqrt{b}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right )}{2 a} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c + d*x^4]/(x*(a + b*x^4)),x]
[Out]
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Rubi in Sympy [A] time = 23.5395, size = 70, normalized size = 0.82 \[ - \frac{\sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{4}}}{\sqrt{c}} \right )}}{2 a} + \frac{\sqrt{a d - b c} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{4}}}{\sqrt{a d - b c}} \right )}}{2 a \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**4+c)**(1/2)/x/(b*x**4+a),x)
[Out]
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Mathematica [C] time = 0.287604, size = 162, normalized size = 1.91 \[ -\frac{3 b d x^4 \sqrt{c+d x^4} F_1\left (\frac{1}{2};-\frac{1}{2},1;\frac{3}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )}{2 \left (a+b x^4\right ) \left (3 b d x^4 F_1\left (\frac{1}{2};-\frac{1}{2},1;\frac{3}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )-2 a d F_1\left (\frac{3}{2};-\frac{1}{2},2;\frac{5}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )+b c F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[Sqrt[c + d*x^4]/(x*(a + b*x^4)),x]
[Out]
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Maple [B] time = 0.022, size = 1037, normalized size = 12.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^4+c)^(1/2)/x/(b*x^4+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{4} + c}}{{\left (b x^{4} + a\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x),x, algorithm="maxima")
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Fricas [A] time = 0.235215, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{4} + 2 \, b c - a d + 2 \, \sqrt{d x^{4} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{4} + a}\right ) + \sqrt{c} \log \left (\frac{d x^{4} - 2 \, \sqrt{d x^{4} + c} \sqrt{c} + 2 \, c}{x^{4}}\right )}{4 \, a}, \frac{2 \, \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x^{4} + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) + \sqrt{c} \log \left (\frac{d x^{4} - 2 \, \sqrt{d x^{4} + c} \sqrt{c} + 2 \, c}{x^{4}}\right )}{4 \, a}, -\frac{2 \, \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{4} + c}}{\sqrt{-c}}\right ) - \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{4} + 2 \, b c - a d + 2 \, \sqrt{d x^{4} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{4} + a}\right )}{4 \, a}, -\frac{\sqrt{-c} \arctan \left (\frac{\sqrt{d x^{4} + c}}{\sqrt{-c}}\right ) - \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x^{4} + c}}{\sqrt{-\frac{b c - a d}{b}}}\right )}{2 \, a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x^{4}}}{x \left (a + b x^{4}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**4+c)**(1/2)/x/(b*x**4+a),x)
[Out]
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GIAC/XCAS [A] time = 0.223032, size = 117, normalized size = 1.38 \[ -\frac{1}{2} \, d{\left (\frac{{\left (b c - a d\right )} \arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a d} - \frac{c \arctan \left (\frac{\sqrt{d x^{4} + c}}{\sqrt{-c}}\right )}{a \sqrt{-c} d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x),x, algorithm="giac")
[Out]