3.91 \(\int \frac{e+f x}{(c+d x) \sqrt{c^3-8 d^3 x^3}} \, dx\)

Optimal. Leaf size=221 \[ -\frac{2 (d e-c f) \tanh ^{-1}\left (\frac{(c-2 d x)^2}{3 \sqrt{c} \sqrt{c^3-8 d^3 x^3}}\right )}{9 c^{3/2} d^2}-\frac{\sqrt{2+\sqrt{3}} (c-2 d x) \sqrt{\frac{c^2+2 c d x+4 d^2 x^2}{\left (\left (1+\sqrt{3}\right ) c-2 d x\right )^2}} (c f+2 d e) F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) c-2 d x}{\left (1+\sqrt{3}\right ) c-2 d x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} c d^2 \sqrt{\frac{c (c-2 d x)}{\left (\left (1+\sqrt{3}\right ) c-2 d x\right )^2}} \sqrt{c^3-8 d^3 x^3}} \]

[Out]

(-2*(d*e - c*f)*ArcTanh[(c - 2*d*x)^2/(3*Sqrt[c]*Sqrt[c^3 - 8*d^3*x^3])])/(9*c^(3/2)*d^2) - (Sqrt[2 + Sqrt[3]]
*(2*d*e + c*f)*(c - 2*d*x)*Sqrt[(c^2 + 2*c*d*x + 4*d^2*x^2)/((1 + Sqrt[3])*c - 2*d*x)^2]*EllipticF[ArcSin[((1
- Sqrt[3])*c - 2*d*x)/((1 + Sqrt[3])*c - 2*d*x)], -7 - 4*Sqrt[3]])/(3*3^(1/4)*c*d^2*Sqrt[(c*(c - 2*d*x))/((1 +
Sqrt[3])*c - 2*d*x)^2]*Sqrt[c^3 - 8*d^3*x^3])

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Rubi [A]  time = 0.284123, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {2139, 218, 2138, 206} \[ -\frac{2 (d e-c f) \tanh ^{-1}\left (\frac{(c-2 d x)^2}{3 \sqrt{c} \sqrt{c^3-8 d^3 x^3}}\right )}{9 c^{3/2} d^2}-\frac{\sqrt{2+\sqrt{3}} (c-2 d x) \sqrt{\frac{c^2+2 c d x+4 d^2 x^2}{\left (\left (1+\sqrt{3}\right ) c-2 d x\right )^2}} (c f+2 d e) F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) c-2 d x}{\left (1+\sqrt{3}\right ) c-2 d x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} c d^2 \sqrt{\frac{c (c-2 d x)}{\left (\left (1+\sqrt{3}\right ) c-2 d x\right )^2}} \sqrt{c^3-8 d^3 x^3}} \]

Antiderivative was successfully verified.

[In]

Int[(e + f*x)/((c + d*x)*Sqrt[c^3 - 8*d^3*x^3]),x]

[Out]

(-2*(d*e - c*f)*ArcTanh[(c - 2*d*x)^2/(3*Sqrt[c]*Sqrt[c^3 - 8*d^3*x^3])])/(9*c^(3/2)*d^2) - (Sqrt[2 + Sqrt[3]]
*(2*d*e + c*f)*(c - 2*d*x)*Sqrt[(c^2 + 2*c*d*x + 4*d^2*x^2)/((1 + Sqrt[3])*c - 2*d*x)^2]*EllipticF[ArcSin[((1
- Sqrt[3])*c - 2*d*x)/((1 + Sqrt[3])*c - 2*d*x)], -7 - 4*Sqrt[3]])/(3*3^(1/4)*c*d^2*Sqrt[(c*(c - 2*d*x))/((1 +
Sqrt[3])*c - 2*d*x)^2]*Sqrt[c^3 - 8*d^3*x^3])

Rule 2139

Int[((e_.) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[(2*d*e + c*f)/(3*c
*d), Int[1/Sqrt[a + b*x^3], x], x] + Dist[(d*e - c*f)/(3*c*d), Int[(c - 2*d*x)/((c + d*x)*Sqrt[a + b*x^3]), x]
, 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[b*c^3 + 8*a*d^3,
0]) && NeQ[2*d*e + c*f, 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 2138

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[(-2*e)/d, Subst[Int
[1/(9 - a*x^2), x], x, (1 + (f*x)/e)^2/Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f,
0] && EqQ[b*c^3 + 8*a*d^3, 0] && EqQ[2*d*e + c*f, 0]

Rule 206

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

Rubi steps

\begin{align*} \int \frac{e+f x}{(c+d x) \sqrt{c^3-8 d^3 x^3}} \, dx &=\frac{(d e-c f) \int \frac{c-2 d x}{(c+d x) \sqrt{c^3-8 d^3 x^3}} \, dx}{3 c d}+\frac{(2 d e+c f) \int \frac{1}{\sqrt{c^3-8 d^3 x^3}} \, dx}{3 c d}\\ &=-\frac{\sqrt{2+\sqrt{3}} (2 d e+c f) (c-2 d x) \sqrt{\frac{c^2+2 c d x+4 d^2 x^2}{\left (\left (1+\sqrt{3}\right ) c-2 d x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) c-2 d x}{\left (1+\sqrt{3}\right ) c-2 d x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} c d^2 \sqrt{\frac{c (c-2 d x)}{\left (\left (1+\sqrt{3}\right ) c-2 d x\right )^2}} \sqrt{c^3-8 d^3 x^3}}-\frac{(2 (d e-c f)) \operatorname{Subst}\left (\int \frac{1}{9-c^3 x^2} \, dx,x,\frac{\left (1-\frac{2 d x}{c}\right )^2}{\sqrt{c^3-8 d^3 x^3}}\right )}{3 d^2}\\ &=-\frac{2 (d e-c f) \tanh ^{-1}\left (\frac{(c-2 d x)^2}{3 \sqrt{c} \sqrt{c^3-8 d^3 x^3}}\right )}{9 c^{3/2} d^2}-\frac{\sqrt{2+\sqrt{3}} (2 d e+c f) (c-2 d x) \sqrt{\frac{c^2+2 c d x+4 d^2 x^2}{\left (\left (1+\sqrt{3}\right ) c-2 d x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) c-2 d x}{\left (1+\sqrt{3}\right ) c-2 d x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} c d^2 \sqrt{\frac{c (c-2 d x)}{\left (\left (1+\sqrt{3}\right ) c-2 d x\right )^2}} \sqrt{c^3-8 d^3 x^3}}\\ \end{align*}

Mathematica [C]  time = 1.1169, size = 384, normalized size = 1.74 \[ -\frac{i \sqrt{\frac{c-2 d x}{\left (1+\sqrt [3]{-1}\right ) c}} \left (4 \sqrt{2} \sqrt{\frac{i c+\sqrt{3} d x+i d x}{-\sqrt{3} c+3 i c}} \sqrt{\frac{c^2+2 c d x+4 d^2 x^2}{c^2}} (d e-c f) \Pi \left (\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left (\sqrt{2} \sqrt{\frac{i c+\sqrt{3} d x+i d x}{3 i c-\sqrt{3} c}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )+f \sqrt{\frac{\left (\sqrt{3}-i\right ) c+2 \left (\sqrt{3}+i\right ) d x}{\left (\sqrt{3}-3 i\right ) c}} \left (\left (\sqrt{3}-3 i\right ) c-2 \left (\sqrt{3}+3 i\right ) d x\right ) F\left (\sin ^{-1}\left (\sqrt{2} \sqrt{\frac{i c+\sqrt{3} d x+i d x}{3 i c-\sqrt{3} c}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )\right )}{2 \left (\sqrt [3]{-1}-2\right ) d^2 \sqrt{\frac{c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt{c^3-8 d^3 x^3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(e + f*x)/((c + d*x)*Sqrt[c^3 - 8*d^3*x^3]),x]

[Out]

((-I/2)*Sqrt[(c - 2*d*x)/((1 + (-1)^(1/3))*c)]*(f*Sqrt[((-I + Sqrt[3])*c + 2*(I + Sqrt[3])*d*x)/((-3*I + Sqrt[
3])*c)]*((-3*I + Sqrt[3])*c - 2*(3*I + Sqrt[3])*d*x)*EllipticF[ArcSin[Sqrt[2]*Sqrt[(I*c + I*d*x + Sqrt[3]*d*x)
/((3*I)*c - Sqrt[3]*c)]], (1 + I*Sqrt[3])/2] + 4*Sqrt[2]*(d*e - c*f)*Sqrt[(I*c + I*d*x + Sqrt[3]*d*x)/((3*I)*c
- Sqrt[3]*c)]*Sqrt[(c^2 + 2*c*d*x + 4*d^2*x^2)/c^2]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[2]*Sq
rt[(I*c + I*d*x + Sqrt[3]*d*x)/((3*I)*c - Sqrt[3]*c)]], (1 + I*Sqrt[3])/2]))/((-2 + (-1)^(1/3))*d^2*Sqrt[(c -
2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]*Sqrt[c^3 - 8*d^3*x^3])

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Maple [B]  time = 0.008, size = 661, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)/(d*x+c)/(-8*d^3*x^3+c^3)^(1/2),x)

[Out]

2*f/d*(1/2*(-1/2+1/2*I*3^(1/2))*c/d-1/2*(-1/2-1/2*I*3^(1/2))*c/d)*((x-1/2*(-1/2-1/2*I*3^(1/2))*c/d)/(1/2*(-1/2
+1/2*I*3^(1/2))*c/d-1/2*(-1/2-1/2*I*3^(1/2))*c/d))^(1/2)*((x-1/2*c/d)/(1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*c/d))^
(1/2)*((x-1/2*(-1/2+1/2*I*3^(1/2))*c/d)/(1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*(-1/2+1/2*I*3^(1/2))*c/d))^(1/2)/(-8
*d^3*x^3+c^3)^(1/2)*EllipticF(((x-1/2*(-1/2-1/2*I*3^(1/2))*c/d)/(1/2*(-1/2+1/2*I*3^(1/2))*c/d-1/2*(-1/2-1/2*I*
3^(1/2))*c/d))^(1/2),((1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*(-1/2+1/2*I*3^(1/2))*c/d)/(1/2*(-1/2-1/2*I*3^(1/2))*c/
d-1/2*c/d))^(1/2))+2*(-c*f+d*e)/d^2*(1/2*(-1/2+1/2*I*3^(1/2))*c/d-1/2*(-1/2-1/2*I*3^(1/2))*c/d)*((x-1/2*(-1/2-
1/2*I*3^(1/2))*c/d)/(1/2*(-1/2+1/2*I*3^(1/2))*c/d-1/2*(-1/2-1/2*I*3^(1/2))*c/d))^(1/2)*((x-1/2*c/d)/(1/2*(-1/2
-1/2*I*3^(1/2))*c/d-1/2*c/d))^(1/2)*((x-1/2*(-1/2+1/2*I*3^(1/2))*c/d)/(1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*(-1/2+
1/2*I*3^(1/2))*c/d))^(1/2)/(-8*d^3*x^3+c^3)^(1/2)/(1/2*(-1/2-1/2*I*3^(1/2))*c/d+c/d)*EllipticPi(((x-1/2*(-1/2-
1/2*I*3^(1/2))*c/d)/(1/2*(-1/2+1/2*I*3^(1/2))*c/d-1/2*(-1/2-1/2*I*3^(1/2))*c/d))^(1/2),(1/2*(-1/2-1/2*I*3^(1/2
))*c/d-1/2*(-1/2+1/2*I*3^(1/2))*c/d)/(1/2*(-1/2-1/2*I*3^(1/2))*c/d+c/d),((1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*(-1
/2+1/2*I*3^(1/2))*c/d)/(1/2*(-1/2-1/2*I*3^(1/2))*c/d-1/2*c/d))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f x + e}{\sqrt{-8 \, 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((f*x+e)/(d*x+c)/(-8*d^3*x^3+c^3)^(1/2),x, algorithm="maxima")

[Out]

integrate((f*x + e)/(sqrt(-8*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{-8 \, d^{3} x^{3} + c^{3}}{\left (f x + e\right )}}{8 \, d^{4} x^{4} + 8 \, 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((f*x+e)/(d*x+c)/(-8*d^3*x^3+c^3)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-8*d^3*x^3 + c^3)*(f*x + e)/(8*d^4*x^4 + 8*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{e + f x}{\sqrt{- \left (- c + 2 d x\right ) \left (c^{2} + 2 c d x + 4 d^{2} x^{2}\right )} \left (c + d x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)/(d*x+c)/(-8*d**3*x**3+c**3)**(1/2),x)

[Out]

Integral((e + f*x)/(sqrt(-(-c + 2*d*x)*(c**2 + 2*c*d*x + 4*d**2*x**2))*(c + d*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f x + e}{\sqrt{-8 \, 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((f*x+e)/(d*x+c)/(-8*d^3*x^3+c^3)^(1/2),x, algorithm="giac")

[Out]

integrate((f*x + e)/(sqrt(-8*d^3*x^3 + c^3)*(d*x + c)), x)