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

Optimal. Leaf size=246 \[ \frac{\sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (c+2^{2/3} d x\right ) \sqrt{\frac{c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) c+2^{2/3} d x}{\left (1+\sqrt{3}\right ) c+2^{2/3} d x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} d^2 \sqrt{\frac{c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} \sqrt{c^3+4 d^3 x^3}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{c} (c+2 d x)}{\sqrt{c^3+4 d^3 x^3}}\right )}{3 \sqrt{3} \sqrt{c} d^2} \]

[Out]

(-2*ArcTan[(Sqrt[3]*Sqrt[c]*(c + 2*d*x))/Sqrt[c^3 + 4*d^3*x^3]])/(3*Sqrt[3]*Sqrt
[c]*d^2) + (2^(1/3)*Sqrt[2 + Sqrt[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]*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)*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 = 0.527489, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{\sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (c+2^{2/3} d x\right ) \sqrt{\frac{c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) c+2^{2/3} d x}{\left (1+\sqrt{3}\right ) c+2^{2/3} d x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} d^2 \sqrt{\frac{c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} \sqrt{c^3+4 d^3 x^3}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{c} (c+2 d x)}{\sqrt{c^3+4 d^3 x^3}}\right )}{3 \sqrt{3} \sqrt{c} d^2} \]

Antiderivative was successfully verified.

[In]  Int[x/((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]*Sqrt
[c]*d^2) + (2^(1/3)*Sqrt[2 + Sqrt[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]*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)*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 in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(d*x+c)/(4*d**3*x**3+c**3)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 1.89997, size = 372, normalized size = 1.51 \[ \frac{\sqrt [6]{2} \sqrt{\frac{\sqrt [3]{2} c+2 d x}{\left (1+\sqrt [3]{-1}\right ) c}} \left (-\sqrt{\frac{\sqrt [3]{-2} c-2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \left (\sqrt [3]{-1} \left (2+\sqrt [3]{-2}\right ) c-2 \left (\sqrt [3]{-1}+2^{2/3}\right ) d x\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}}}{\sqrt [6]{2}}\right )|\sqrt [3]{-1}\right )+\frac{\sqrt [3]{-1} 2^{2/3} \left (1+\sqrt [3]{-1}\right ) c \sqrt{\frac{\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt{\frac{4 d^2 x^2}{c^2}-\frac{2 \sqrt [3]{2} d x}{c}+2^{2/3}} \Pi \left (\frac{i \sqrt [3]{2} \sqrt{3}}{2+\sqrt [3]{-2}};\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}}}{\sqrt [6]{2}}\right )|\sqrt [3]{-1}\right )}{\sqrt{3}}\right )}{\left (2+\sqrt [3]{-2}\right ) d^2 \sqrt{\frac{\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt{c^3+4 d^3 x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[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)]*(-(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))*
c*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 = 0.012, size = 892, normalized size = 3.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(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)*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/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. \[ \int \frac{x}{\sqrt{4 \, d^{3} x^{3} + c^{3}}{\left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(sqrt(4*d^3*x^3 + c^3)*(d*x + c)),x, algorithm="maxima")

[Out]

integrate(x/(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. \[{\rm integral}\left (\frac{x}{\sqrt{4 \, d^{3} x^{3} + c^{3}}{\left (d x + c\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(sqrt(4*d^3*x^3 + c^3)*(d*x + c)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{4 \, d^{3} x^{3} + c^{3}}{\left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(sqrt(4*d^3*x^3 + c^3)*(d*x + c)),x, algorithm="giac")

[Out]

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