Optimal. Leaf size=416 \[ -\frac{\sqrt [4]{c} d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}+\frac{\sqrt [4]{c} d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}-\frac{\sqrt [4]{c} d \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}+\frac{\sqrt [4]{c} d \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}-\frac{e^3 \log \left (a+c x^4\right )}{4 \left (a e^4+c d^4\right )}+\frac{e^3 \log (d+e x)}{a e^4+c d^4}-\frac{\sqrt{c} d^2 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \left (a e^4+c d^4\right )} \]
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Rubi [A] time = 0.945997, antiderivative size = 416, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 11, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.647 \[ -\frac{\sqrt [4]{c} d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}+\frac{\sqrt [4]{c} d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}-\frac{\sqrt [4]{c} d \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}+\frac{\sqrt [4]{c} d \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}-\frac{e^3 \log \left (a+c x^4\right )}{4 \left (a e^4+c d^4\right )}+\frac{e^3 \log (d+e x)}{a e^4+c d^4}-\frac{\sqrt{c} d^2 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \left (a e^4+c d^4\right )} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)*(a + c*x^4)),x]
[Out]
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Rubi in Sympy [A] time = 133.876, size = 379, normalized size = 0.91 \[ - \frac{e^{3} \log{\left (a + c x^{4} \right )}}{4 \left (a e^{4} + c d^{4}\right )} + \frac{e^{3} \log{\left (d + e x \right )}}{a e^{4} + c d^{4}} - \frac{\sqrt{c} d^{2} e \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{a} \left (a e^{4} + c d^{4}\right )} + \frac{\sqrt{2} \sqrt [4]{c} d \left (\sqrt{a} e^{2} - \sqrt{c} d^{2}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} x + \sqrt{a} \sqrt{c} + c x^{2} \right )}}{8 a^{\frac{3}{4}} \left (a e^{4} + c d^{4}\right )} - \frac{\sqrt{2} \sqrt [4]{c} d \left (\sqrt{a} e^{2} - \sqrt{c} d^{2}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} x + \sqrt{a} \sqrt{c} + c x^{2} \right )}}{8 a^{\frac{3}{4}} \left (a e^{4} + c d^{4}\right )} - \frac{\sqrt{2} \sqrt [4]{c} d \left (\sqrt{a} e^{2} + \sqrt{c} d^{2}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} \left (a e^{4} + c d^{4}\right )} + \frac{\sqrt{2} \sqrt [4]{c} d \left (\sqrt{a} e^{2} + \sqrt{c} d^{2}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} \left (a e^{4} + c d^{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)/(c*x**4+a),x)
[Out]
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Mathematica [A] time = 0.285521, size = 404, normalized size = 0.97 \[ \frac{-2 a^{3/4} e^3 \log \left (a+c x^4\right )+8 a^{3/4} e^3 \log (d+e x)-\sqrt{2} c^{3/4} d^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+\sqrt{2} c^{3/4} d^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-2 \sqrt [4]{c} d \left (-2 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt{2} \sqrt{a} e^2+\sqrt{2} \sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \sqrt [4]{c} d \left (2 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt{2} \sqrt{a} e^2+\sqrt{2} \sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )+\sqrt{2} \sqrt{a} \sqrt [4]{c} d e^2 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-\sqrt{2} \sqrt{a} \sqrt [4]{c} d e^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{8 a^{3/4} \left (a e^4+c d^4\right )} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)*(a + c*x^4)),x]
[Out]
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Maple [A] time = 0.012, size = 433, normalized size = 1. \[{\frac{c{d}^{3}\sqrt{2}}{ \left ( 8\,a{e}^{4}+8\,c{d}^{4} \right ) a}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{c{d}^{3}\sqrt{2}}{ \left ( 4\,a{e}^{4}+4\,c{d}^{4} \right ) a}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{c{d}^{3}\sqrt{2}}{ \left ( 4\,a{e}^{4}+4\,c{d}^{4} \right ) a}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }-{\frac{{d}^{2}ec}{2\,a{e}^{4}+2\,c{d}^{4}}\arctan \left ({x}^{2}\sqrt{{\frac{c}{a}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{{e}^{2}d\sqrt{2}}{8\,a{e}^{4}+8\,c{d}^{4}}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{{e}^{2}d\sqrt{2}}{4\,a{e}^{4}+4\,c{d}^{4}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{{e}^{2}d\sqrt{2}}{4\,a{e}^{4}+4\,c{d}^{4}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{{e}^{3}\ln \left ( c{x}^{4}+a \right ) }{4\,a{e}^{4}+4\,c{d}^{4}}}+{\frac{{e}^{3}\ln \left ( ex+d \right ) }{a{e}^{4}+c{d}^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)/(c*x^4+a),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(1/((c*x^4 + a)*(e*x + d)),x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^4 + a)*(e*x + d)),x, algorithm="fricas")
[Out]
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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/(e*x+d)/(c*x**4+a),x)
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GIAC/XCAS [A] time = 0.292374, size = 520, normalized size = 1.25 \[ \frac{\left (a c^{3}\right )^{\frac{1}{4}} c^{2} d \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \,{\left (\sqrt{2} a c^{3} d^{2} - 2 \, \left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d e - \sqrt{2} \sqrt{a c} a c^{2} e^{2}\right )}} + \frac{\left (a c^{3}\right )^{\frac{1}{4}} c^{2} d \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \,{\left (\sqrt{2} a c^{3} d^{2} + 2 \, \left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d e - \sqrt{2} \sqrt{a c} a c^{2} e^{2}\right )}} + \frac{{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{3} - \left (a c^{3}\right )^{\frac{3}{4}} d e^{2}\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{4 \,{\left (\sqrt{2} a c^{3} d^{4} + \sqrt{2} a^{2} c^{2} e^{4}\right )}} - \frac{{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{3} - \left (a c^{3}\right )^{\frac{3}{4}} d e^{2}\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{4 \,{\left (\sqrt{2} a c^{3} d^{4} + \sqrt{2} a^{2} c^{2} e^{4}\right )}} - \frac{e^{3}{\rm ln}\left ({\left | c x^{4} + a \right |}\right )}{4 \,{\left (c d^{4} + a e^{4}\right )}} + \frac{e^{4}{\rm ln}\left ({\left | x e + d \right |}\right )}{c d^{4} e + a e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^4 + a)*(e*x + d)),x, algorithm="giac")
[Out]