Optimal. Leaf size=111 \[ \frac{\left (\sqrt{a}+\sqrt{b} (c+d x)^2\right ) \sqrt{\frac{a+b (c+d x)^4}{\left (\sqrt{a}+\sqrt{b} (c+d x)^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} d \sqrt{a+b (c+d x)^4}} \]
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Rubi [A] time = 0.206459, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\left (\sqrt{a}+\sqrt{b} (c+d x)^2\right ) \sqrt{\frac{a+b (c+d x)^4}{\left (\sqrt{a}+\sqrt{b} (c+d x)^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} d \sqrt{a+b (c+d x)^4}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[a + b*(c + d*x)^4],x]
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Rubi in Sympy [A] time = 6.97208, size = 97, normalized size = 0.87 \[ \frac{\sqrt{\frac{a + b \left (c + d x\right )^{4}}{\left (\sqrt{a} + \sqrt{b} \left (c + d x\right )^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} \left (c + d x\right )^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \left (c + d x\right )}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} d \sqrt{a + b \left (c + d x\right )^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b*(d*x+c)**4)**(1/2),x)
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Mathematica [C] time = 0.0934418, size = 90, normalized size = 0.81 \[ -\frac{i \sqrt{\frac{a+b (c+d x)^4}{a}} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} (c+d x)\right )\right |-1\right )}{d \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b (c+d x)^4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[a + b*(c + d*x)^4],x]
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Maple [C] time = 0.345, size = 1036, normalized size = 9.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b*(d*x+c)^4)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{{\left (d x + c\right )}^{4} b + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt((d*x + c)^4*b + a),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt((d*x + c)^4*b + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.99818, size = 46, normalized size = 0.41 \[ \frac{\left (\frac{c}{d} + x\right ) \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b d^{4} \left (\frac{c}{d} + x\right )^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b*(d*x+c)**4)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{{\left (d x + c\right )}^{4} b + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt((d*x + c)^4*b + a),x, algorithm="giac")
[Out]