Optimal. Leaf size=223 \[ \frac{(b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{3/4} d^{5/4}}-\frac{(b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{3/4} d^{5/4}}+\frac{(b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{3/4} d^{5/4}}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{2 \sqrt{2} c^{3/4} d^{5/4}}+\frac{b x}{d} \]
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Rubi [A] time = 0.328354, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412 \[ \frac{(b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{3/4} d^{5/4}}-\frac{(b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{4 \sqrt{2} c^{3/4} d^{5/4}}+\frac{(b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt{2} c^{3/4} d^{5/4}}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{2 \sqrt{2} c^{3/4} d^{5/4}}+\frac{b x}{d} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^4)/(c + d*x^4),x]
[Out]
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Rubi in Sympy [A] time = 57.1622, size = 204, normalized size = 0.91 \[ \frac{b x}{d} - \frac{\sqrt{2} \left (a d - b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x + \sqrt{c} + \sqrt{d} x^{2} \right )}}{8 c^{\frac{3}{4}} d^{\frac{5}{4}}} + \frac{\sqrt{2} \left (a d - b c\right ) \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x + \sqrt{c} + \sqrt{d} x^{2} \right )}}{8 c^{\frac{3}{4}} d^{\frac{5}{4}}} - \frac{\sqrt{2} \left (a d - b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}} \right )}}{4 c^{\frac{3}{4}} d^{\frac{5}{4}}} + \frac{\sqrt{2} \left (a d - b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}} \right )}}{4 c^{\frac{3}{4}} d^{\frac{5}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**4+a)/(d*x**4+c),x)
[Out]
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Mathematica [A] time = 0.227625, size = 196, normalized size = 0.88 \[ \frac{\sqrt{2} (b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )-\sqrt{2} (b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )+2 \sqrt{2} (b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )-2 \sqrt{2} (b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )+8 b c^{3/4} \sqrt [4]{d} x}{8 c^{3/4} d^{5/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^4)/(c + d*x^4),x]
[Out]
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Maple [A] time = 0.01, size = 266, normalized size = 1.2 \[{\frac{bx}{d}}+{\frac{\sqrt{2}a}{4\,c}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }-{\frac{\sqrt{2}b}{4\,d}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{\sqrt{2}a}{8\,c}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}b}{8\,d}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}a}{4\,c}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }-{\frac{\sqrt{2}b}{4\,d}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^4+a)/(d*x^4+c),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((b*x^4 + a)/(d*x^4 + c),x, algorithm="maxima")
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Fricas [A] time = 0.239847, size = 743, normalized size = 3.33 \[ -\frac{4 \, d \left (-\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5}}\right )^{\frac{1}{4}} \arctan \left (-\frac{c d \left (-\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5}}\right )^{\frac{1}{4}}}{{\left (b c - a d\right )} x +{\left (b c - a d\right )} \sqrt{\frac{c^{2} d^{2} \sqrt{-\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5}}} +{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}}}\right ) - d \left (-\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5}}\right )^{\frac{1}{4}} \log \left (c d \left (-\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5}}\right )^{\frac{1}{4}} -{\left (b c - a d\right )} x\right ) + d \left (-\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5}}\right )^{\frac{1}{4}} \log \left (-c d \left (-\frac{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5}}\right )^{\frac{1}{4}} -{\left (b c - a d\right )} x\right ) - 4 \, b x}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)/(d*x^4 + c),x, algorithm="fricas")
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Sympy [A] time = 2.23477, size = 87, normalized size = 0.39 \[ \frac{b x}{d} + \operatorname{RootSum}{\left (256 t^{4} c^{3} d^{5} + a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}, \left ( t \mapsto t \log{\left (\frac{4 t c d}{a d - b c} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**4+a)/(d*x**4+c),x)
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GIAC/XCAS [A] time = 0.220027, size = 331, normalized size = 1.48 \[ \frac{b x}{d} - \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b c - \left (c d^{3}\right )^{\frac{1}{4}} a d\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{4 \, c d^{2}} - \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b c - \left (c d^{3}\right )^{\frac{1}{4}} a d\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{4 \, c d^{2}} - \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b c - \left (c d^{3}\right )^{\frac{1}{4}} a d\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{c}{d}\right )^{\frac{1}{4}} + \sqrt{\frac{c}{d}}\right )}{8 \, c d^{2}} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b c - \left (c d^{3}\right )^{\frac{1}{4}} a d\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{c}{d}\right )^{\frac{1}{4}} + \sqrt{\frac{c}{d}}\right )}{8 \, c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)/(d*x^4 + c),x, algorithm="giac")
[Out]