3.1.43 \(\int \frac {1}{\sqrt [4]{b x+c x^2}} \, dx\) [43]

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

[Out]

b*(-c*(c*x^2+b*x)/b^2)^(1/4)*(cos(1/2*arcsin(1+2*c*x/b))^2)^(1/2)/cos(1/2*arcsin(1+2*c*x/b))*EllipticE(sin(1/2
*arcsin(1+2*c*x/b)),2^(1/2))*2^(1/2)/c/(c*x^2+b*x)^(1/4)

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Rubi [A]
time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, 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 = {636, 633, 234} \begin {gather*} \frac {\sqrt {2} b \sqrt [4]{-\frac {c \left (b x+c x^2\right )}{b^2}} E\left (\left .\frac {1}{2} \text {ArcSin}\left (\frac {2 c x}{b}+1\right )\right |2\right )}{c \sqrt [4]{b x+c x^2}} \end {gather*}

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 234

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

Rule 633

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 636

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

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 ) \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.02, size = 45, normalized size = 0.78 \begin {gather*} \frac {4 x \sqrt [4]{1+\frac {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)}} \end {gather*}

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))

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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (c \,x^{2}+b x \right )^{\frac {1}{4}}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{b x + c x^{2}}}\, dx \end {gather*}

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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 0.20, size = 36, normalized size = 0.62 \begin {gather*} \frac {4\,x\,{\left (\frac {c\,x}{b}+1\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {3}{4};\ \frac {7}{4};\ -\frac {c\,x}{b}\right )}{3\,{\left (c\,x^2+b\,x\right )}^{1/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x + c*x^2)^(1/4),x)

[Out]

(4*x*((c*x)/b + 1)^(1/4)*hypergeom([1/4, 3/4], 7/4, -(c*x)/b))/(3*(b*x + c*x^2)^(1/4))

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