3.42 \(\int \sqrt [4]{b x+c x^2} \, dx\)

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]

((b + 2*c*x)*(b*x + c*x^2)^(1/4))/(3*c) - (b^3*(-((c*(b*x + c*x^2))/b^2))^(3/4)*
EllipticF[ArcSin[1 + (2*c*x)/b]/2, 2])/(3*Sqrt[2]*c^2*(b*x + c*x^2)^(3/4))

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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]

((b + 2*c*x)*(b*x + c*x^2)^(1/4))/(3*c) - (b^3*(-((c*(b*x + c*x^2))/b^2))^(3/4)*
EllipticF[ArcSin[1 + (2*c*x)/b]/2, 2])/(3*Sqrt[2]*c^2*(b*x + c*x^2)^(3/4))

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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]

-sqrt(2)*b**3*(c*(-b*x - c*x**2)/b**2)**(3/4)*elliptic_f(asin(1 + 2*c*x/b)/2, 2)
/(6*c**2*(b*x + c*x**2)**(3/4)) + (b + 2*c*x)*(b*x + c*x**2)**(1/4)/(3*c)

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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]

(x*(b^2 + 3*b*c*x + 2*c^2*x^2 - b^2*(1 + (c*x)/b)^(3/4)*Hypergeometric2F1[1/4, 3
/4, 5/4, -((c*x)/b)]))/(3*c*(x*(b + c*x))^(3/4))

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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]

int((c*x^2+b*x)^(1/4),x)

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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]

integrate((c*x^2 + b*x)^(1/4), x)

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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]

integral((c*x^2 + b*x)^(1/4), x)

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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]

Integral((b*x + c*x**2)**(1/4), x)

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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]

integrate((c*x^2 + b*x)^(1/4), x)