3.2917 \(\int \frac{1}{\sqrt{a+b (c+d x)^4}} \, dx\)

Optimal. Leaf size=111 \[ \frac{\left (\sqrt{a}+\sqrt{b} (c+d x)^2\right ) \sqrt{\frac{a+b (c+d x)^4}{\left (\sqrt{a}+\sqrt{b} (c+d x)^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} d \sqrt{a+b (c+d x)^4}} \]

[Out]

((Sqrt[a] + Sqrt[b]*(c + d*x)^2)*Sqrt[(a + b*(c + d*x)^4)/(Sqrt[a] + Sqrt[b]*(c + d*x)^2)^2]*EllipticF[2*ArcTa
n[(b^(1/4)*(c + d*x))/a^(1/4)], 1/2])/(2*a^(1/4)*b^(1/4)*d*Sqrt[a + b*(c + d*x)^4])

________________________________________________________________________________________

Rubi [A]  time = 0.101165, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {247, 220} \[ \frac{\left (\sqrt{a}+\sqrt{b} (c+d x)^2\right ) \sqrt{\frac{a+b (c+d x)^4}{\left (\sqrt{a}+\sqrt{b} (c+d x)^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} d \sqrt{a+b (c+d x)^4}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + b*(c + d*x)^4],x]

[Out]

((Sqrt[a] + Sqrt[b]*(c + d*x)^2)*Sqrt[(a + b*(c + d*x)^4)/(Sqrt[a] + Sqrt[b]*(c + d*x)^2)^2]*EllipticF[2*ArcTa
n[(b^(1/4)*(c + d*x))/a^(1/4)], 1/2])/(2*a^(1/4)*b^(1/4)*d*Sqrt[a + b*(c + d*x)^4])

Rule 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rubi steps

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

Mathematica [C]  time = 0.0554436, size = 90, normalized size = 0.81 \[ -\frac{i \sqrt{\frac{a+b (c+d x)^4}{a}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} (c+d x)\right ),-1\right )}{d \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b (c+d x)^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a + b*(c + d*x)^4],x]

[Out]

((-I)*Sqrt[(a + b*(c + d*x)^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*(c + d*x)], -1])/(Sqrt[(I*Sqrt
[b])/Sqrt[a]]*d*Sqrt[a + b*(c + d*x)^4])

________________________________________________________________________________________

Maple [C]  time = 0.409, size = 1036, normalized size = 9.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*((1/b*(-a*b^3)^(1/4)-c)/d-(-I/b*(-a*b^3)^(1/4)-c)/d)*(((-I/b*(-a*b^3)^(1/4)-c)/d-(I/b*(-a*b^3)^(1/4)-c)/d)*(
x-(1/b*(-a*b^3)^(1/4)-c)/d)/((-I/b*(-a*b^3)^(1/4)-c)/d-(1/b*(-a*b^3)^(1/4)-c)/d)/(x-(I/b*(-a*b^3)^(1/4)-c)/d))
^(1/2)*(x-(I/b*(-a*b^3)^(1/4)-c)/d)^2*(((I/b*(-a*b^3)^(1/4)-c)/d-(1/b*(-a*b^3)^(1/4)-c)/d)*(x-(-1/b*(-a*b^3)^(
1/4)-c)/d)/((-1/b*(-a*b^3)^(1/4)-c)/d-(1/b*(-a*b^3)^(1/4)-c)/d)/(x-(I/b*(-a*b^3)^(1/4)-c)/d))^(1/2)*(((I/b*(-a
*b^3)^(1/4)-c)/d-(1/b*(-a*b^3)^(1/4)-c)/d)*(x-(-I/b*(-a*b^3)^(1/4)-c)/d)/((-I/b*(-a*b^3)^(1/4)-c)/d-(1/b*(-a*b
^3)^(1/4)-c)/d)/(x-(I/b*(-a*b^3)^(1/4)-c)/d))^(1/2)/((-I/b*(-a*b^3)^(1/4)-c)/d-(I/b*(-a*b^3)^(1/4)-c)/d)/((I/b
*(-a*b^3)^(1/4)-c)/d-(1/b*(-a*b^3)^(1/4)-c)/d)/(b*d^4*(x-(1/b*(-a*b^3)^(1/4)-c)/d)*(x-(I/b*(-a*b^3)^(1/4)-c)/d
)*(x-(-1/b*(-a*b^3)^(1/4)-c)/d)*(x-(-I/b*(-a*b^3)^(1/4)-c)/d))^(1/2)*EllipticF((((-I/b*(-a*b^3)^(1/4)-c)/d-(I/
b*(-a*b^3)^(1/4)-c)/d)*(x-(1/b*(-a*b^3)^(1/4)-c)/d)/((-I/b*(-a*b^3)^(1/4)-c)/d-(1/b*(-a*b^3)^(1/4)-c)/d)/(x-(I
/b*(-a*b^3)^(1/4)-c)/d))^(1/2),(((I/b*(-a*b^3)^(1/4)-c)/d-(-1/b*(-a*b^3)^(1/4)-c)/d)*((1/b*(-a*b^3)^(1/4)-c)/d
-(-I/b*(-a*b^3)^(1/4)-c)/d)/((1/b*(-a*b^3)^(1/4)-c)/d-(-1/b*(-a*b^3)^(1/4)-c)/d)/((I/b*(-a*b^3)^(1/4)-c)/d-(-I
/b*(-a*b^3)^(1/4)-c)/d))^(1/2))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt((d*x + c)^4*b + a), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sqrt{b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(b*d^4*x^4 + 4*b*c*d^3*x^3 + 6*b*c^2*d^2*x^2 + 4*b*c^3*d*x + b*c^4 + a), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*(d*x+c)**4)**(1/2),x)

[Out]

Integral(1/sqrt(a + b*(c + d*x)**4), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt((d*x + c)^4*b + a), x)