Optimal. Leaf size=114 \[ \frac{d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{5/4} \sqrt [4]{c} \sqrt{a+c x^4}}+\frac{x (d+e x)}{2 a \sqrt{a+c x^4}} \]
[Out]
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Rubi [A] time = 0.110956, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{5/4} \sqrt [4]{c} \sqrt{a+c x^4}}+\frac{x (d+e x)}{2 a \sqrt{a+c x^4}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)/(a + c*x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 12.695, size = 100, normalized size = 0.88 \[ \frac{x \left (d + e x\right )}{2 a \sqrt{a + c x^{4}}} + \frac{d \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{4 a^{\frac{5}{4}} \sqrt [4]{c} \sqrt{a + c x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)/(c*x**4+a)**(3/2),x)
[Out]
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Mathematica [C] time = 0.182713, size = 90, normalized size = 0.79 \[ \frac{x (d+e x)-\frac{i d \sqrt{\frac{c x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}}}{2 a \sqrt{a+c x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)/(a + c*x^4)^(3/2),x]
[Out]
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Maple [C] time = 0.006, size = 115, normalized size = 1. \[ d \left ({\frac{x}{2\,a}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{c}} \right ) c}}}}+{\frac{1}{2\,a}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \right ) +{\frac{e{x}^{2}}{2\,a}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)/(c*x^4+a)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{e x + d}{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(c*x^4 + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{e x + d}{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(c*x^4 + a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 24.2276, size = 61, normalized size = 0.54 \[ \frac{d x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} \Gamma \left (\frac{5}{4}\right )} + \frac{e x^{2}}{2 a^{\frac{3}{2}} \sqrt{1 + \frac{c x^{4}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)/(c*x**4+a)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{e x + d}{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(c*x^4 + a)^(3/2),x, algorithm="giac")
[Out]