3.64 \(\int \frac{e+f x}{(c+d x) \sqrt{c^3+4 d^3 x^3}} \, dx\)

Optimal. Leaf size=265 \[ \frac{2 (d e-c f) \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{c} (c+2 d x)}{\sqrt{c^3+4 d^3 x^3}}\right )}{3 \sqrt{3} c^{3/2} d^2}+\frac{\sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (c+2^{2/3} d x\right ) \sqrt{\frac{c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} (c f+2 d e) F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) c+2^{2/3} d x}{\left (1+\sqrt{3}\right ) c+2^{2/3} d x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} c d^2 \sqrt{\frac{c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} \sqrt{c^3+4 d^3 x^3}} \]

[Out]

(2*(d*e - c*f)*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 + Sqrt[3]]*(2*d*e + c*f)*(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]*EllipticF[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^2*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 = 0.296453, antiderivative size = 265, 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, 2137, 203} \[ \frac{2 (d e-c f) \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{c} (c+2 d x)}{\sqrt{c^3+4 d^3 x^3}}\right )}{3 \sqrt{3} c^{3/2} d^2}+\frac{\sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (c+2^{2/3} d x\right ) \sqrt{\frac{c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} (c f+2 d e) F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) c+2^{2/3} d x}{\left (1+\sqrt{3}\right ) c+2^{2/3} d x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} c d^2 \sqrt{\frac{c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} \sqrt{c^3+4 d^3 x^3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d*e - c*f)*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 + Sqrt[3]]*(2*d*e + c*f)*(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]*EllipticF[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^2*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 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 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{e+f x}{(c+d x) \sqrt{c^3+4 d^3 x^3}} \, dx &=\frac{(d e-c f) \int \frac{c-2 d x}{(c+d x) \sqrt{c^3+4 d^3 x^3}} \, dx}{3 c d}+\frac{(2 d e+c f) \int \frac{1}{\sqrt{c^3+4 d^3 x^3}} \, dx}{3 c d}\\ &=\frac{\sqrt [3]{2} \sqrt{2+\sqrt{3}} (2 d e+c f) \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^2 \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 (d e-c f)) \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^2}\\ &=\frac{2 (d e-c f) \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^2}+\frac{\sqrt [3]{2} \sqrt{2+\sqrt{3}} (2 d e+c f) \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^2 \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 = 1.40619, size = 380, normalized size = 1.43 \[ \frac{\sqrt [6]{2} \sqrt{\frac{\sqrt [3]{2} c+2 d x}{\left (1+\sqrt [3]{-1}\right ) c}} \left (-f \sqrt{\frac{\sqrt [3]{-2} c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \left (\sqrt [3]{-1} \left (2+\sqrt [3]{-2}\right ) c-2 \left (\sqrt [3]{-1}+2^{2/3}\right ) d x\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}}}{\sqrt [6]{2}}\right )|\sqrt [3]{-1}\right )+\frac{\sqrt [3]{-1} 2^{2/3} \left (1+\sqrt [3]{-1}\right ) \sqrt{\frac{\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt{\frac{4 d^2 x^2}{c^2}-\frac{2 \sqrt [3]{2} d x}{c}+2^{2/3}} (c f-d e) \Pi \left (\frac{i \sqrt [3]{2} \sqrt{3}}{2+\sqrt [3]{-2}};\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}}}{\sqrt [6]{2}}\right )|\sqrt [3]{-1}\right )}{\sqrt{3}}\right )}{\left (2+\sqrt [3]{-2}\right ) d^2 \sqrt{\frac{\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt{c^3+4 d^3 x^3}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2^(1/6)*Sqrt[(2^(1/3)*c + 2*d*x)/((1 + (-1)^(1/3))*c)]*(-(f*Sqrt[((-2)^(1/3)*c - 2*(-1)^(2/3)*d*x)/((1 + (-1)
^(1/3))*c)]*((-1)^(1/3)*(2 + (-2)^(1/3))*c - 2*((-1)^(1/3) + 2^(2/3))*d*x)*EllipticF[ArcSin[Sqrt[(2^(1/3)*c +
2*(-1)^(2/3)*d*x)/((1 + (-1)^(1/3))*c)]/2^(1/6)], (-1)^(1/3)]) + ((-1)^(1/3)*2^(2/3)*(1 + (-1)^(1/3))*(-(d*e)
+ c*f)*Sqrt[(2^(1/3)*c + 2*(-1)^(2/3)*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)])/Sqrt[3]))/((2 + (-2)^(1/3))*d^2*Sqrt[(2^(1/3)*c + 2*(-1)^(2/3)*d*x)/((1 + (-
1)^(1/3))*c)]*Sqrt[c^3 + 4*d^3*x^3])

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Maple [B]  time = 0.007, size = 900, normalized size = 3.4 \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)/(4*d^3*x^3+c^3)^(1/2),x)

[Out]

2*f/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)*EllipticF(((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+1/2*2^(1/3)*c/d))^(1/2))+2*(-c*f+d*e)/
d^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)*((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{f x + e}{\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((f*x+e)/(d*x+c)/(4*d^3*x^3+c^3)^(1/2),x, algorithm="maxima")

[Out]

integrate((f*x + e)/(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}}{\left (f x + e\right )}}{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((f*x+e)/(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)*(f*x + e)/(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{e + f x}{\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((f*x+e)/(d*x+c)/(4*d**3*x**3+c**3)**(1/2),x)

[Out]

Integral((e + f*x)/((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{f x + e}{\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((f*x+e)/(d*x+c)/(4*d^3*x^3+c^3)^(1/2),x, algorithm="giac")

[Out]

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