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))

________________________________________________________________________________________

Rubi [A]  time = 0.0226108, 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, Rules used = {622, 619, 228} \[ \frac{\sqrt{2} b \sqrt [4]{-\frac{c \left (b x+c x^2\right )}{b^2}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{2 c x}{b}+1\right )\right |2\right )}{c \sqrt [4]{b x+c x^2}} \]

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))

Rule 622

Int[((b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(b*x + c*x^2)^p/(-((c*(b*x + c*x^2))/b^2))^p, Int[(-((
c*x)/b) - (c^2*x^2)/b^2)^p, x], x] /; FreeQ[{b, c}, x] && RationalQ[p] && 3 <= Denominator[p] <= 4

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 228

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(2*EllipticE[(1*ArcSin[Rt[-(b/a), 2]*x])/2, 2])/(a^(1/4)*R
t[-(b/a), 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b/a]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt [4]{b x+c x^2}} \, dx &=\frac{\sqrt [4]{-\frac{c \left (b x+c x^2\right )}{b^2}} \int \frac{1}{\sqrt [4]{-\frac{c x}{b}-\frac{c^2 x^2}{b^2}}} \, dx}{\sqrt [4]{b x+c x^2}}\\ &=-\frac{\left (b^2 \sqrt [4]{-\frac{c \left (b x+c x^2\right )}{b^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1-\frac{b^2 x^2}{c^2}}} \, dx,x,-\frac{c}{b}-\frac{2 c^2 x}{b^2}\right )}{\sqrt{2} c^2 \sqrt [4]{b x+c x^2}}\\ &=\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 (1+\frac{2 c x}{b}\right )\right |2\right )}{c \sqrt [4]{b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 0.0098401, size = 45, normalized size = 0.78 \[ \frac{4 x \sqrt [4]{\frac{c x}{b}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{c x}{b}\right )}{3 \sqrt [4]{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^(-1/4),x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.608, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt [4]{c{x}^{2}+bx}}}\, dx \end{align*}

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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x)^(1/4),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(-1/4), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{{\left (c x^{2} + b x\right )}^{\frac{1}{4}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x)^(1/4),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x)^(-1/4), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [4]{b x + c x^{2}}}\, dx \end{align*}

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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x)^(1/4),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(-1/4), x)