3.9 \(\int \frac{1}{(c+d x) \sqrt{c^3+4 d^3 x^3}} \, dx\)

Optimal. Leaf size=249 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{c} (c+2 d x)}{\sqrt{c^3+4 d^3 x^3}}\right )}{3 \sqrt{3} c^{3/2} d}+\frac{2 \sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (c+2^{2/3} d x\right ) \sqrt{\frac{c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) c+2^{2/3} d x}{\left (1+\sqrt{3}\right ) c+2^{2/3} d x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} c d \sqrt{\frac{c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} \sqrt{c^3+4 d^3 x^3}} \]

[Out]

(2*ArcTan[(Sqrt[3]*Sqrt[c]*(c + 2*d*x))/Sqrt[c^3 + 4*d^3*x^3]])/(3*Sqrt[3]*c^(3/2)*d) + (2*2^(1/3)*Sqrt[2 + Sq
rt[3]]*(c + 2^(2/3)*d*x)*Sqrt[(c^2 - 2^(2/3)*c*d*x + 2*2^(1/3)*d^2*x^2)/((1 + Sqrt[3])*c + 2^(2/3)*d*x)^2]*Ell
ipticF[ArcSin[((1 - Sqrt[3])*c + 2^(2/3)*d*x)/((1 + Sqrt[3])*c + 2^(2/3)*d*x)], -7 - 4*Sqrt[3]])/(3*3^(1/4)*c*
d*Sqrt[(c*(c + 2^(2/3)*d*x))/((1 + Sqrt[3])*c + 2^(2/3)*d*x)^2]*Sqrt[c^3 + 4*d^3*x^3])

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Rubi [A]  time = 0.286599, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2134, 218, 2137, 203} \[ \frac{2 \sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (c+2^{2/3} d x\right ) \sqrt{\frac{c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) c+2^{2/3} d x}{\left (1+\sqrt{3}\right ) c+2^{2/3} d x}\right ),-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} c d \sqrt{\frac{c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} \sqrt{c^3+4 d^3 x^3}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{c} (c+2 d x)}{\sqrt{c^3+4 d^3 x^3}}\right )}{3 \sqrt{3} c^{3/2} d} \]

Antiderivative was successfully verified.

[In]

Int[1/((c + d*x)*Sqrt[c^3 + 4*d^3*x^3]),x]

[Out]

(2*ArcTan[(Sqrt[3]*Sqrt[c]*(c + 2*d*x))/Sqrt[c^3 + 4*d^3*x^3]])/(3*Sqrt[3]*c^(3/2)*d) + (2*2^(1/3)*Sqrt[2 + Sq
rt[3]]*(c + 2^(2/3)*d*x)*Sqrt[(c^2 - 2^(2/3)*c*d*x + 2*2^(1/3)*d^2*x^2)/((1 + Sqrt[3])*c + 2^(2/3)*d*x)^2]*Ell
ipticF[ArcSin[((1 - Sqrt[3])*c + 2^(2/3)*d*x)/((1 + Sqrt[3])*c + 2^(2/3)*d*x)], -7 - 4*Sqrt[3]])/(3*3^(1/4)*c*
d*Sqrt[(c*(c + 2^(2/3)*d*x))/((1 + Sqrt[3])*c + 2^(2/3)*d*x)^2]*Sqrt[c^3 + 4*d^3*x^3])

Rule 2134

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[2/(3*c), Int[1/Sqrt[a + b*x^3], x], x
] + Dist[1/(3*c), Int[(c - 2*d*x)/((c + d*x)*Sqrt[a + b*x^3]), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 -
4*a*d^3, 0]

Rule 218

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3
])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 2137

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[(2*e)/d, Subst[Int[
1/(1 + 3*a*x^2), x], x, (1 + (2*d*x)/c)/Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f,
0] && EqQ[b*c^3 - 4*a*d^3, 0] && EqQ[2*d*e + c*f, 0]

Rule 203

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

Rubi steps

\begin{align*} \int \frac{1}{(c+d x) \sqrt{c^3+4 d^3 x^3}} \, dx &=\frac{\int \frac{c-2 d x}{(c+d x) \sqrt{c^3+4 d^3 x^3}} \, dx}{3 c}+\frac{2 \int \frac{1}{\sqrt{c^3+4 d^3 x^3}} \, dx}{3 c}\\ &=\frac{2 \sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (c+2^{2/3} d x\right ) \sqrt{\frac{c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) c+2^{2/3} d x}{\left (1+\sqrt{3}\right ) c+2^{2/3} d x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} c d \sqrt{\frac{c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} \sqrt{c^3+4 d^3 x^3}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{1+3 c^3 x^2} \, dx,x,\frac{1+\frac{2 d x}{c}}{\sqrt{c^3+4 d^3 x^3}}\right )}{3 d}\\ &=\frac{2 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{c} (c+2 d x)}{\sqrt{c^3+4 d^3 x^3}}\right )}{3 \sqrt{3} c^{3/2} d}+\frac{2 \sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (c+2^{2/3} d x\right ) \sqrt{\frac{c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) c+2^{2/3} d x}{\left (1+\sqrt{3}\right ) c+2^{2/3} d x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} c d \sqrt{\frac{c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} \sqrt{c^3+4 d^3 x^3}}\\ \end{align*}

Mathematica [C]  time = 0.206626, size = 169, normalized size = 0.68 \[ -\frac{i 2^{5/6} \sqrt{\frac{\sqrt [3]{2} c+2 d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt{\frac{4 d^2 x^2}{c^2}-\frac{2 \sqrt [3]{2} d x}{c}+2^{2/3}} \Pi \left (\frac{i \sqrt [3]{2} \sqrt{3}}{2+\sqrt [3]{-2}};\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}}}{\sqrt [6]{2}}\right )|\sqrt [3]{-1}\right )}{\left (2+\sqrt [3]{-2}\right ) d \sqrt{c^3+4 d^3 x^3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((c + d*x)*Sqrt[c^3 + 4*d^3*x^3]),x]

[Out]

((-I)*2^(5/6)*Sqrt[(2^(1/3)*c + 2*d*x)/((1 + (-1)^(1/3))*c)]*Sqrt[2^(2/3) - (2*2^(1/3)*d*x)/c + (4*d^2*x^2)/c^
2]*EllipticPi[(I*2^(1/3)*Sqrt[3])/(2 + (-2)^(1/3)), ArcSin[Sqrt[(2^(1/3)*c + 2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3
))*c)]/2^(1/6)], (-1)^(1/3)])/((2 + (-2)^(1/3))*d*Sqrt[c^3 + 4*d^3*x^3])

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Maple [B]  time = 0.157, size = 495, normalized size = 2. \begin{align*} 2\,{\frac{1}{d\sqrt{4\,{d}^{3}{x}^{3}+{c}^{3}}} \left ({\frac{ \left ( 1/4\,\sqrt [3]{2}-i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}}-{\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}} \right ) \sqrt{{ \left ( x-{\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}} \right ) \left ({\frac{ \left ( 1/4\,\sqrt [3]{2}-i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}}-{\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}} \right ) ^{-1}}}\sqrt{{ \left ( x+1/2\,{\frac{\sqrt [3]{2}c}{d}} \right ) \left ({\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}}+1/2\,{\frac{\sqrt [3]{2}c}{d}} \right ) ^{-1}}}\sqrt{{ \left ( x-{\frac{ \left ( 1/4\,\sqrt [3]{2}-i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}} \right ) \left ({\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}}-{\frac{ \left ( 1/4\,\sqrt [3]{2}-i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}} \right ) ^{-1}}}{\it EllipticPi} \left ( \sqrt{{ \left ( x-{\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}} \right ) \left ({\frac{ \left ( 1/4\,\sqrt [3]{2}-i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}}-{\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}} \right ) ^{-1}}},{ \left ({\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}}-{\frac{ \left ( 1/4\,\sqrt [3]{2}-i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}} \right ) \left ({\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}}+{\frac{c}{d}} \right ) ^{-1}},\sqrt{{ \left ({\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}}-{\frac{ \left ( 1/4\,\sqrt [3]{2}-i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}} \right ) \left ({\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}}+1/2\,{\frac{\sqrt [3]{2}c}{d}} \right ) ^{-1}}} \right ) \left ({\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}}+{\frac{c}{d}} \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x+c)/(4*d^3*x^3+c^3)^(1/2),x)

[Out]

2/d*((1/4*2^(1/3)-1/4*I*3^(1/2)*2^(1/3))*c/d-(1/4*2^(1/3)+1/4*I*3^(1/2)*2^(1/3))*c/d)*((x-(1/4*2^(1/3)+1/4*I*3
^(1/2)*2^(1/3))*c/d)/((1/4*2^(1/3)-1/4*I*3^(1/2)*2^(1/3))*c/d-(1/4*2^(1/3)+1/4*I*3^(1/2)*2^(1/3))*c/d))^(1/2)*
((x+1/2*2^(1/3)*c/d)/((1/4*2^(1/3)+1/4*I*3^(1/2)*2^(1/3))*c/d+1/2*2^(1/3)*c/d))^(1/2)*((x-(1/4*2^(1/3)-1/4*I*3
^(1/2)*2^(1/3))*c/d)/((1/4*2^(1/3)+1/4*I*3^(1/2)*2^(1/3))*c/d-(1/4*2^(1/3)-1/4*I*3^(1/2)*2^(1/3))*c/d))^(1/2)/
(4*d^3*x^3+c^3)^(1/2)/((1/4*2^(1/3)+1/4*I*3^(1/2)*2^(1/3))*c/d+c/d)*EllipticPi(((x-(1/4*2^(1/3)+1/4*I*3^(1/2)*
2^(1/3))*c/d)/((1/4*2^(1/3)-1/4*I*3^(1/2)*2^(1/3))*c/d-(1/4*2^(1/3)+1/4*I*3^(1/2)*2^(1/3))*c/d))^(1/2),((1/4*2
^(1/3)+1/4*I*3^(1/2)*2^(1/3))*c/d-(1/4*2^(1/3)-1/4*I*3^(1/2)*2^(1/3))*c/d)/((1/4*2^(1/3)+1/4*I*3^(1/2)*2^(1/3)
)*c/d+c/d),(((1/4*2^(1/3)+1/4*I*3^(1/2)*2^(1/3))*c/d-(1/4*2^(1/3)-1/4*I*3^(1/2)*2^(1/3))*c/d)/((1/4*2^(1/3)+1/
4*I*3^(1/2)*2^(1/3))*c/d+1/2*2^(1/3)*c/d))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{4 \, d^{3} x^{3} + c^{3}}{\left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(4*d^3*x^3+c^3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(4*d^3*x^3 + c^3)*(d*x + c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(4*d^3*x^3+c^3)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(4*d^3*x^3 + c^3)/(4*d^4*x^4 + 4*c*d^3*x^3 + c^3*d*x + c^4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c + d x\right ) \sqrt{c^{3} + 4 d^{3} x^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(4*d**3*x**3+c**3)**(1/2),x)

[Out]

Integral(1/((c + d*x)*sqrt(c**3 + 4*d**3*x**3)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{4 \, d^{3} x^{3} + c^{3}}{\left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(4*d^3*x^3+c^3)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(4*d^3*x^3 + c^3)*(d*x + c)), x)