Optimal. Leaf size=115 \[ \frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right )}{4 a^2 \sqrt{c}}-\frac{\sqrt{b} \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 a^2}-\frac{\sqrt{c+d x^4}}{4 a x^4} \]
[Out]
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Rubi [A] time = 0.384249, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right )}{4 a^2 \sqrt{c}}-\frac{\sqrt{b} \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 a^2}-\frac{\sqrt{c+d x^4}}{4 a x^4} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c + d*x^4]/(x^5*(a + b*x^4)),x]
[Out]
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Rubi in Sympy [A] time = 43.369, size = 100, normalized size = 0.87 \[ - \frac{\sqrt{c + d x^{4}}}{4 a x^{4}} - \frac{\sqrt{b} \sqrt{a d - b c} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{4}}}{\sqrt{a d - b c}} \right )}}{2 a^{2}} - \frac{\left (\frac{a d}{2} - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{4}}}{\sqrt{c}} \right )}}{2 a^{2} \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**4+c)**(1/2)/x**5/(b*x**4+a),x)
[Out]
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Mathematica [C] time = 0.956401, size = 407, normalized size = 3.54 \[ \frac{\frac{6 b c d x^8 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{x^4 \left (2 b c F_1\left (2;\frac{1}{2},2;3;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (2;\frac{3}{2},1;3;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )-4 a c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}+\frac{5 b d x^4 \left (3 a c+4 a d x^4+b c x^4+3 b d x^8\right ) F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )-3 \left (a+b x^4\right ) \left (c+d x^4\right ) \left (2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )\right )}{a \left (-5 b d x^4 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )\right )}}{12 x^4 \left (a+b x^4\right ) \sqrt{c+d x^4}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[Sqrt[c + d*x^4]/(x^5*(a + b*x^4)),x]
[Out]
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Maple [B] time = 0.023, size = 1107, normalized size = 9.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^4+c)^(1/2)/x^5/(b*x^4+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{4} + c}}{{\left (b x^{4} + a\right )} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.243514, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{b^{2} c - a b d} \sqrt{c} x^{4} \log \left (\frac{b d x^{4} + 2 \, b c - a d - 2 \, \sqrt{d x^{4} + c} \sqrt{b^{2} c - a b d}}{b x^{4} + a}\right ) -{\left (2 \, b c - a d\right )} x^{4} \log \left (\frac{{\left (d x^{4} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{4} + c} c}{x^{4}}\right ) - 2 \, \sqrt{d x^{4} + c} a \sqrt{c}}{8 \, a^{2} \sqrt{c} x^{4}}, \frac{4 \, \sqrt{-b^{2} c + a b d} \sqrt{c} x^{4} \arctan \left (\frac{\sqrt{-b^{2} c + a b d}}{\sqrt{d x^{4} + c} b}\right ) -{\left (2 \, b c - a d\right )} x^{4} \log \left (\frac{{\left (d x^{4} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{4} + c} c}{x^{4}}\right ) - 2 \, \sqrt{d x^{4} + c} a \sqrt{c}}{8 \, a^{2} \sqrt{c} x^{4}}, -\frac{{\left (2 \, b c - a d\right )} x^{4} \arctan \left (\frac{c}{\sqrt{d x^{4} + c} \sqrt{-c}}\right ) - \sqrt{b^{2} c - a b d} \sqrt{-c} x^{4} \log \left (\frac{b d x^{4} + 2 \, b c - a d - 2 \, \sqrt{d x^{4} + c} \sqrt{b^{2} c - a b d}}{b x^{4} + a}\right ) + \sqrt{d x^{4} + c} a \sqrt{-c}}{4 \, a^{2} \sqrt{-c} x^{4}}, -\frac{{\left (2 \, b c - a d\right )} x^{4} \arctan \left (\frac{c}{\sqrt{d x^{4} + c} \sqrt{-c}}\right ) - 2 \, \sqrt{-b^{2} c + a b d} \sqrt{-c} x^{4} \arctan \left (\frac{\sqrt{-b^{2} c + a b d}}{\sqrt{d x^{4} + c} b}\right ) + \sqrt{d x^{4} + c} a \sqrt{-c}}{4 \, a^{2} \sqrt{-c} x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^5),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x^{4}}}{x^{5} \left (a + b x^{4}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**4+c)**(1/2)/x**5/(b*x**4+a),x)
[Out]
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GIAC/XCAS [A] time = 0.215831, size = 163, normalized size = 1.42 \[ \frac{1}{4} \, d^{2}{\left (\frac{2 \,{\left (b^{2} c - a b d\right )} \arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} d^{2}} - \frac{{\left (2 \, b c - a d\right )} \arctan \left (\frac{\sqrt{d x^{4} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} d^{2}} - \frac{\sqrt{d x^{4} + c}}{a d^{2} x^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^5),x, algorithm="giac")
[Out]