Optimal. Leaf size=158 \[ \frac{a^{3/4} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{c} \sqrt{a+c x^4}}+\frac{1}{3} d x \sqrt{a+c x^4}+\frac{1}{4} e x^2 \sqrt{a+c x^4}+\frac{a e \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{4 \sqrt{c}} \]
[Out]
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Rubi [A] time = 0.181423, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ \frac{a^{3/4} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{c} \sqrt{a+c x^4}}+\frac{1}{3} d x \sqrt{a+c x^4}+\frac{1}{4} e x^2 \sqrt{a+c x^4}+\frac{a e \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{4 \sqrt{c}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)*Sqrt[a + c*x^4],x]
[Out]
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Rubi in Sympy [A] time = 16.5768, size = 143, normalized size = 0.91 \[ \frac{a^{\frac{3}{4}} d \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{3 \sqrt [4]{c} \sqrt{a + c x^{4}}} + \frac{a e \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a + c x^{4}}} \right )}}{4 \sqrt{c}} + \frac{d x \sqrt{a + c x^{4}}}{3} + \frac{e x^{2} \sqrt{a + c x^{4}}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)*(c*x**4+a)**(1/2),x)
[Out]
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Mathematica [C] time = 0.553328, size = 132, normalized size = 0.84 \[ \frac{1}{12} \left (x \sqrt{a+c x^4} (4 d+3 e x)-\frac{8 i a d \sqrt{\frac{c x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4}}+\frac{3 a e \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{\sqrt{c}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)*Sqrt[a + c*x^4],x]
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Maple [C] time = 0.006, size = 127, normalized size = 0.8 \[{\frac{dx}{3}\sqrt{c{x}^{4}+a}}+{\frac{2\,ad}{3}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}}+{\frac{e{x}^{2}}{4}\sqrt{c{x}^{4}+a}}+{\frac{ae}{4}\ln \left ({x}^{2}\sqrt{c}+\sqrt{c{x}^{4}+a} \right ){\frac{1}{\sqrt{c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)*(c*x^4+a)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{4} + a}{\left (e x + d\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + a)*(e*x + d),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{c x^{4} + a}{\left (e x + d\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + a)*(e*x + d),x, algorithm="fricas")
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Sympy [A] time = 8.5021, size = 88, normalized size = 0.56 \[ \frac{\sqrt{a} d x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} + \frac{\sqrt{a} e x^{2} \sqrt{1 + \frac{c x^{4}}{a}}}{4} + \frac{a e \operatorname{asinh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{4 \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((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. \[ \int \sqrt{c x^{4} + a}{\left (e x + d\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + a)*(e*x + d),x, algorithm="giac")
[Out]