Optimal. Leaf size=90 \[ \frac{(b+2 c x) \sqrt [4]{b x+c x^2}}{3 c}-\frac{b^3 \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{2 c x}{b}+1\right )\right |2\right )}{3 \sqrt{2} c^2 \left (b x+c x^2\right )^{3/4}} \]
[Out]
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Rubi [A] time = 0.0751646, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{(b+2 c x) \sqrt [4]{b x+c x^2}}{3 c}-\frac{b^3 \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{2 c x}{b}+1\right )\right |2\right )}{3 \sqrt{2} c^2 \left (b x+c x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)^(1/4),x]
[Out]
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Rubi in Sympy [A] time = 13.8965, size = 78, normalized size = 0.87 \[ - \frac{\sqrt{2} b^{3} \left (\frac{c \left (- b x - c x^{2}\right )}{b^{2}}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (1 + \frac{2 c x}{b} \right )}}{2}\middle | 2\right )}{6 c^{2} \left (b x + c x^{2}\right )^{\frac{3}{4}}} + \frac{\left (b + 2 c x\right ) \sqrt [4]{b x + c x^{2}}}{3 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**(1/4),x)
[Out]
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Mathematica [C] time = 0.0519025, size = 70, normalized size = 0.78 \[ \frac{x \left (b^2 \left (-\left (\frac{c x}{b}+1\right )^{3/4}\right ) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{c x}{b}\right )+b^2+3 b c x+2 c^2 x^2\right )}{3 c (x (b+c x))^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)^(1/4),x]
[Out]
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Maple [F] time = 0.049, size = 0, normalized size = 0. \[ \int \sqrt [4]{c{x}^{2}+bx}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)^(1/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x\right )}^{\frac{1}{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(1/4),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + b x\right )}^{\frac{1}{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(1/4),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt [4]{b x + c x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)**(1/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x\right )}^{\frac{1}{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(1/4),x, algorithm="giac")
[Out]