Optimal. Leaf size=737 \[ \frac{\sqrt{c} d^2 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{2 e^3}-\frac{\sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\frac{\sqrt{c} d^2}{\sqrt{a}}+e^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 e^4 \sqrt{a+c x^4}}-\frac{d \sqrt{-\frac{a e^4+c d^4}{d^2 e^2}} \tan ^{-1}\left (\frac{x \sqrt{-\frac{a e^4+c d^4}{d^2 e^2}}}{\sqrt{a+c x^4}}\right )}{2 e^2}+\frac{\sqrt [4]{c} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (a e^4+c d^4\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} e^4 \sqrt{a+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right )}-\frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \left (a e^4+c d^4\right ) \Pi \left (\frac{\left (\sqrt{c} d^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d e^4 \sqrt{a+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right )}-\frac{\sqrt{a e^4+c d^4} \tanh ^{-1}\left (\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right )}{2 e^3}-\frac{\sqrt{c} d x \sqrt{a+c x^4}}{e^2 \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{\sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{e^2 \sqrt{a+c x^4}}+\frac{\sqrt{a+c x^4}}{2 e} \]
[Out]
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Rubi [A] time = 1.39254, antiderivative size = 737, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 12, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.632 \[ \frac{\sqrt{c} d^2 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{2 e^3}-\frac{\sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\frac{\sqrt{c} d^2}{\sqrt{a}}+e^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 e^4 \sqrt{a+c x^4}}-\frac{d \sqrt{-\frac{a e^4+c d^4}{d^2 e^2}} \tan ^{-1}\left (\frac{x \sqrt{-\frac{a e^4+c d^4}{d^2 e^2}}}{\sqrt{a+c x^4}}\right )}{2 e^2}+\frac{\sqrt [4]{c} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (a e^4+c d^4\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} e^4 \sqrt{a+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right )}-\frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \left (a e^4+c d^4\right ) \Pi \left (\frac{\left (\sqrt{c} d^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d e^4 \sqrt{a+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right )}-\frac{\sqrt{a e^4+c d^4} \tanh ^{-1}\left (\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right )}{2 e^3}-\frac{\sqrt{c} d x \sqrt{a+c x^4}}{e^2 \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{\sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{e^2 \sqrt{a+c x^4}}+\frac{\sqrt{a+c x^4}}{2 e} \]
Warning: Unable to verify antiderivative.
[In] Int[Sqrt[a + c*x^4]/(d + e*x),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + c x^{4}}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**4+a)**(1/2)/(e*x+d),x)
[Out]
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Mathematica [C] time = 1.63642, size = 451, normalized size = 0.61 \[ \frac{2 c^{3/4} d^2 \sqrt{\frac{c x^4}{a}+1} \left (\sqrt{a} e^2+i \sqrt{c} d^2\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )-2 \sqrt{a} c^{3/4} d^2 e^2 \sqrt{\frac{c x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \left (\sqrt [4]{c} d e \left (\sqrt{c} d^2 \sqrt{a+c x^4} \log \left (\sqrt{c} \sqrt{a+c x^4}+c x^2\right )+\sqrt{a+c x^4} \sqrt{a e^4+c d^4} \log \left (e^2 x^2-d^2\right )-\sqrt{a+c x^4} \sqrt{a e^4+c d^4} \log \left (\sqrt{a+c x^4} \sqrt{a e^4+c d^4}+a e^2+c d^2 x^2\right )+a e^2+c e^2 x^4\right )-2 \sqrt [4]{-1} \sqrt [4]{a} \sqrt{\frac{c x^4}{a}+1} \left (a e^4+c d^4\right ) \Pi \left (\frac{i \sqrt{a} e^2}{\sqrt{c} d^2};\left .\sin ^{-1}\left (\frac{(-1)^{3/4} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )\right )}{2 \sqrt [4]{c} d e^4 \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + c*x^4]/(d + e*x),x]
[Out]
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Maple [C] time = 0.021, size = 565, normalized size = 0.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^4+a)^(1/2)/(e*x+d),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{4} + a}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + a)/(e*x + d),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(sqrt(c*x^4 + a)/(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + c x^{4}}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**4+a)**(1/2)/(e*x+d),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{4} + a}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + a)/(e*x + d),x, algorithm="giac")
[Out]