Optimal. Leaf size=218 \[ \frac{\sqrt [4]{a} \sqrt{1-\frac{c x^4}{a}} (e f-d g) \Pi \left (\frac{\sqrt{a} e^2}{\sqrt{c} d^2};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d e \sqrt{c x^4-a}}+\frac{(e f-d g) \tanh ^{-1}\left (\frac{a e^2-c d^2 x^2}{\sqrt{c x^4-a} \sqrt{c d^4-a e^4}}\right )}{2 \sqrt{c d^4-a e^4}}+\frac{\sqrt [4]{a} g \sqrt{1-\frac{c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} e \sqrt{c x^4-a}} \]
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Rubi [A] time = 0.60376, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346 \[ \frac{\sqrt [4]{a} \sqrt{1-\frac{c x^4}{a}} (e f-d g) \Pi \left (\frac{\sqrt{a} e^2}{\sqrt{c} d^2};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d e \sqrt{c x^4-a}}+\frac{(e f-d g) \tanh ^{-1}\left (\frac{a e^2-c d^2 x^2}{\sqrt{c x^4-a} \sqrt{c d^4-a e^4}}\right )}{2 \sqrt{c d^4-a e^4}}+\frac{\sqrt [4]{a} g \sqrt{1-\frac{c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} e \sqrt{c x^4-a}} \]
Antiderivative was successfully verified.
[In] Int[(f + g*x)/((d + e*x)*Sqrt[-a + c*x^4]),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((g*x+f)/(e*x+d)/(c*x**4-a)**(1/2),x)
[Out]
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Mathematica [C] time = 2.00761, size = 719, normalized size = 3.3 \[ \frac{\frac{i f \sqrt{-\frac{(1-i) \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [4]{c} x+i \sqrt [4]{a}}} \sqrt{\frac{(1+i) \left (\sqrt [4]{a}+i \sqrt [4]{c} x\right ) \left (\sqrt [4]{a}+\sqrt [4]{c} x\right )}{\left (\sqrt [4]{a}-i \sqrt [4]{c} x\right )^2}} \left (\sqrt [4]{a}-i \sqrt [4]{c} x\right )^2 \left (\left (\sqrt [4]{a} e-\sqrt [4]{c} d\right ) F\left (\left .\sin ^{-1}\left (\sqrt{\frac{(1+i) \left (\sqrt [4]{c} x+\sqrt [4]{a}\right )}{2 \sqrt [4]{c} x+2 i \sqrt [4]{a}}}\right )\right |2\right )-(1-i) \sqrt [4]{a} e \Pi \left (\frac{(1-i) \left (\sqrt [4]{c} d-i \sqrt [4]{a} e\right )}{\sqrt [4]{c} d-\sqrt [4]{a} e};\left .\sin ^{-1}\left (\sqrt{\frac{(1+i) \left (\sqrt [4]{c} x+\sqrt [4]{a}\right )}{2 \sqrt [4]{c} x+2 i \sqrt [4]{a}}}\right )\right |2\right )\right )}{\sqrt [4]{a} \left (\sqrt [4]{a} e-\sqrt [4]{c} d\right ) \left (\sqrt [4]{a} e+i \sqrt [4]{c} d\right )}+\frac{d g \sqrt{-\frac{(1-i) \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [4]{c} x+i \sqrt [4]{a}}} \sqrt{\frac{(1+i) \left (\sqrt [4]{a}+i \sqrt [4]{c} x\right ) \left (\sqrt [4]{a}+\sqrt [4]{c} x\right )}{\left (\sqrt [4]{a}-i \sqrt [4]{c} x\right )^2}} \left (\sqrt [4]{a}-i \sqrt [4]{c} x\right )^2 \left (i \left (\sqrt [4]{c} d-\sqrt [4]{a} e\right ) F\left (\left .\sin ^{-1}\left (\sqrt{\frac{(1+i) \left (\sqrt [4]{c} x+\sqrt [4]{a}\right )}{2 \sqrt [4]{c} x+2 i \sqrt [4]{a}}}\right )\right |2\right )+(1+i) \sqrt [4]{a} e \Pi \left (\frac{(1-i) \left (\sqrt [4]{c} d-i \sqrt [4]{a} e\right )}{\sqrt [4]{c} d-\sqrt [4]{a} e};\left .\sin ^{-1}\left (\sqrt{\frac{(1+i) \left (\sqrt [4]{c} x+\sqrt [4]{a}\right )}{2 \sqrt [4]{c} x+2 i \sqrt [4]{a}}}\right )\right |2\right )\right )}{\sqrt [4]{a} e \left (\sqrt [4]{a} e-\sqrt [4]{c} d\right ) \left (\sqrt [4]{a} e+i \sqrt [4]{c} d\right )}-\frac{i g \sqrt{1-\frac{c x^4}{a}} F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{e \sqrt{-\frac{\sqrt{c}}{\sqrt{a}}}}}{\sqrt{c x^4-a}} \]
Antiderivative was successfully verified.
[In] Integrate[(f + g*x)/((d + e*x)*Sqrt[-a + c*x^4]),x]
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Maple [A] time = 0.039, size = 247, normalized size = 1.1 \[{\frac{g}{e}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}-a}}}}+{\frac{-dg+ef}{{e}^{2}} \left ( -{\frac{1}{2}{\it Artanh} \left ({\frac{1}{2} \left ( 2\,{\frac{c{d}^{2}{x}^{2}}{{e}^{2}}}-2\,a \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}-a}}}{\frac{1}{\sqrt{c{x}^{4}-a}}}} \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}-a}}}}+{\frac{e}{d}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticPi} \left ( x\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}},-{\frac{{e}^{2}}{{d}^{2}}\sqrt{a}{\frac{1}{\sqrt{c}}}},{1\sqrt{{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\frac{1}{\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}} \right ){\frac{1}{\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}-a}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((g*x+f)/(e*x+d)/(c*x^4-a)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{g x + f}{\sqrt{c x^{4} - a}{\left (e x + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x + f)/(sqrt(c*x^4 - a)*(e*x + d)),x, algorithm="maxima")
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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((g*x + f)/(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. \[ \int \frac{f + g x}{\sqrt{- a + c x^{4}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x+f)/(e*x+d)/(c*x**4-a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{g x + f}{\sqrt{c x^{4} - a}{\left (e x + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x + f)/(sqrt(c*x^4 - a)*(e*x + d)),x, algorithm="giac")
[Out]