Optimal. Leaf size=180 \[ \frac{b x (4 b c-9 a d)}{5 a^2 \sqrt [4]{a+b x^4} (b c-a d)^2}+\frac{d^2 \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{9/4}}+\frac{d^2 \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{9/4}}+\frac{b x}{5 a \left (a+b x^4\right )^{5/4} (b c-a d)} \]
[Out]
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Rubi [A] time = 0.489959, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{b x (4 b c-9 a d)}{5 a^2 \sqrt [4]{a+b x^4} (b c-a d)^2}+\frac{d^2 \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{9/4}}+\frac{d^2 \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{9/4}}+\frac{b x}{5 a \left (a+b x^4\right )^{5/4} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^4)^(9/4)*(c + d*x^4)),x]
[Out]
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Rubi in Sympy [A] time = 80.6157, size = 158, normalized size = 0.88 \[ \frac{d^{2} \operatorname{atan}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{2 c^{\frac{3}{4}} \left (- a d + b c\right )^{\frac{9}{4}}} + \frac{d^{2} \operatorname{atanh}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{2 c^{\frac{3}{4}} \left (- a d + b c\right )^{\frac{9}{4}}} - \frac{b x}{5 a \left (a + b x^{4}\right )^{\frac{5}{4}} \left (a d - b c\right )} - \frac{b x \left (9 a d - 4 b c\right )}{5 a^{2} \sqrt [4]{a + b x^{4}} \left (a d - b c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**4+a)**(9/4)/(d*x**4+c),x)
[Out]
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Mathematica [A] time = 0.563055, size = 184, normalized size = 1.02 \[ \frac{b x \left (\left (a+b x^4\right ) (4 b c-9 a d)+a (b c-a d)\right )}{5 a^2 \left (a+b x^4\right )^{5/4} (b c-a d)^2}+\frac{d^2 \left (-\log \left (\sqrt [4]{c}-\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}\right )+\log \left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{a x^4+b}}+\sqrt [4]{c}\right )+2 \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a x^4+b}}\right )\right )}{4 c^{3/4} (b c-a d)^{9/4}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/((a + b*x^4)^(9/4)*(c + d*x^4)),x]
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Maple [F] time = 0.057, size = 0, normalized size = 0. \[ \int{\frac{1}{d{x}^{4}+c} \left ( b{x}^{4}+a \right ) ^{-{\frac{9}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^4+a)^(9/4)/(d*x^4+c),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{9}{4}}{\left (d x^{4} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(9/4)*(d*x^4 + c)),x, algorithm="maxima")
[Out]
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Fricas [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/((b*x^4 + a)^(9/4)*(d*x^4 + c)),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/(b*x**4+a)**(9/4)/(d*x**4+c),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{9}{4}}{\left (d x^{4} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(9/4)*(d*x^4 + c)),x, algorithm="giac")
[Out]