Optimal. Leaf size=132 \[ \frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 a^2 (b c-a d)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right )}{2 a^2 \sqrt{c}}+\frac{b \sqrt{c+d x^4}}{4 a \left (a+b x^4\right ) (b c-a d)} \]
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Rubi [A] time = 0.384089, antiderivative size = 132, 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 \[ \frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 a^2 (b c-a d)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right )}{2 a^2 \sqrt{c}}+\frac{b \sqrt{c+d x^4}}{4 a \left (a+b x^4\right ) (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[1/(x*(a + b*x^4)^2*Sqrt[c + d*x^4]),x]
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Rubi in Sympy [A] time = 47.7179, size = 114, normalized size = 0.86 \[ - \frac{b \sqrt{c + d x^{4}}}{4 a \left (a + b x^{4}\right ) \left (a d - b c\right )} - \frac{\sqrt{b} \left (\frac{3 a d}{2} - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{4}}}{\sqrt{a d - b c}} \right )}}{2 a^{2} \left (a d - b c\right )^{\frac{3}{2}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{4}}}{\sqrt{c}} \right )}}{2 a^{2} \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)
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Mathematica [C] time = 0.442581, size = 396, normalized size = 3. \[ \frac{b \left (\frac{6 c d x^4 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{x^4 \left (2 b c F_1\left (2;\frac{1}{2},2;3;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (2;\frac{3}{2},1;3;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )-4 a c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}+\frac{5 d x^4 \left (2 a d+b \left (c+3 d x^4\right )\right ) F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )-3 \left (c+d x^4\right ) \left (2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )\right )}{a \left (-5 b d x^4 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^4},-\frac{a}{b x^4}\right )\right )}\right )}{12 \left (a+b x^4\right ) \sqrt{c+d x^4} (a d-b c)} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x*(a + b*x^4)^2*Sqrt[c + d*x^4]),x]
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Maple [B] time = 0.02, size = 880, normalized size = 6.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(b*x^4+a)^2/(d*x^4+c)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{2} \sqrt{d x^{4} + c} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^2*sqrt(d*x^4 + c)*x),x, algorithm="maxima")
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Fricas [A] time = 0.286573, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^2*sqrt(d*x^4 + c)*x),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)
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GIAC/XCAS [A] time = 0.215774, size = 207, normalized size = 1.57 \[ -\frac{1}{4} \, d^{2}{\left (\frac{{\left (2 \, b^{2} c - 3 \, a b d\right )} \arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{-b^{2} c + a b d}} - \frac{\sqrt{d x^{4} + c} b}{{\left (a b c d - a^{2} d^{2}\right )}{\left ({\left (d x^{4} + c\right )} b - b c + a d\right )}} - \frac{2 \, \arctan \left (\frac{\sqrt{d x^{4} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} d^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^2*sqrt(d*x^4 + c)*x),x, algorithm="giac")
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