3.43 \(\int \frac{1}{\sqrt [4]{b x+c x^2}} \, dx\)

Optimal. Leaf size=58 \[ \frac{\sqrt{2} b \sqrt [4]{-\frac{c \left (b x+c x^2\right )}{b^2}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{2 c x}{b}+1\right )\right |2\right )}{c \sqrt [4]{b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0507608, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{\sqrt{2} b \sqrt [4]{-\frac{c \left (b x+c x^2\right )}{b^2}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{2 c x}{b}+1\right )\right |2\right )}{c \sqrt [4]{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(-1/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.8577, size = 51, normalized size = 0.88 \[ \frac{\sqrt{2} b \sqrt [4]{\frac{c \left (- b x - c x^{2}\right )}{b^{2}}} E\left (\frac{\operatorname{asin}{\left (1 + \frac{2 c x}{b} \right )}}{2}\middle | 2\right )}{c \sqrt [4]{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x**2+b*x)**(1/4),x)

[Out]

sqrt(2)*b*(c*(-b*x - c*x**2)/b**2)**(1/4)*elliptic_e(asin(1 + 2*c*x/b)/2, 2)/(c*
(b*x + c*x**2)**(1/4))

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Mathematica [C]  time = 0.0287287, size = 46, normalized size = 0.79 \[ \frac{4 x \sqrt [4]{\frac{b+c x}{b}} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{c x}{b}\right )}{3 \sqrt [4]{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(-1/4),x]

[Out]

(4*x*((b + c*x)/b)^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, -((c*x)/b)])/(3*(x*(b
+ c*x))^(1/4))

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt [4]{c{x}^{2}+bx}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+b*x)^(1/4),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{1}{4}}}\,{d x} \]

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. \[{\rm integral}\left (\frac{1}{{\left (c x^{2} + b x\right )}^{\frac{1}{4}}}, x\right ) \]

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. \[ \int \frac{1}{\sqrt [4]{b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(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. \[ \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{1}{4}}}\,{d x} \]

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)