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

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 = 0.550436, 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 \[ -\frac{2 (d e-c f) \tanh ^{-1}\left (\frac{(c-2 d x)^2}{3 \sqrt{c} \sqrt{c^3-8 d^3 x^3}}\right )}{9 c^{3/2} d^2}-\frac{\sqrt{2+\sqrt{3}} (c-2 d x) \sqrt{\frac{c^2+2 c d x+4 d^2 x^2}{\left (\left (1+\sqrt{3}\right ) c-2 d x\right )^2}} (c f+2 d e) F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) c-2 d x}{\left (1+\sqrt{3}\right ) c-2 d x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} c d^2 \sqrt{\frac{c (c-2 d x)}{\left (\left (1+\sqrt{3}\right ) c-2 d x\right )^2}} \sqrt{c^3-8 d^3 x^3}} \]

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])

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Rubi in Sympy [A]  time = 154.518, size = 588, normalized size = 2.66 \[ \frac{3^{\frac{3}{4}} \sqrt{\frac{c^{2} + 2 c d x + 4 d^{2} x^{2}}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (c - 2 d x\right ) \left (c f \left (1 + \sqrt{3}\right ) + 2 d e\right ) F\left (\operatorname{asin}{\left (- \frac{- c \left (-1 + \sqrt{3}\right ) - 2 d x}{c \left (1 + \sqrt{3}\right ) - 2 d x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 c d^{2} \sqrt{\frac{c \left (c - 2 d x\right )}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}}} \left (\sqrt{3} + 3\right ) \sqrt{c^{3} - 8 d^{3} x^{3}}} + \frac{3^{\frac{3}{4}} \sqrt{\frac{c^{2} \left (1 + \frac{2 d x}{c} + \frac{4 d^{2} x^{2}}{c^{2}}\right )}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}}} \sqrt{3 + 2 \sqrt{3}} \sqrt{- \sqrt{3} + 2} \left (c - 2 d x\right ) \left (c f - d e\right ) \operatorname{atanh}{\left (\frac{\sqrt{- \frac{\left (c \left (-1 + \sqrt{3}\right ) + 2 d x\right )^{2}}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}} + 1}}{\sqrt{3 + 2 \sqrt{3}} \sqrt{\frac{\left (c \left (-1 + \sqrt{3}\right ) + 2 d x\right )^{2}}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}} - 4 \sqrt{3} + 7}} \right )}}{9 c d^{2} \sqrt{\frac{c \left (c - 2 d x\right )}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}}} \sqrt{c^{3} - 8 d^{3} x^{3}}} - \frac{4 \sqrt [4]{3} \sqrt{\frac{c^{2} \left (1 + \frac{2 d x}{c} + \frac{4 d^{2} x^{2}}{c^{2}}\right )}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (c - 2 d x\right ) \left (c f - d e\right ) \Pi \left (4 \sqrt{3} + 7; \operatorname{asin}{\left (\frac{c \left (-1 + \sqrt{3}\right ) + 2 d x}{c \left (1 + \sqrt{3}\right ) - 2 d x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{c d^{2} \sqrt{\frac{c \left (c - 2 d x\right )}{\left (c \left (1 + \sqrt{3}\right ) - 2 d x\right )^{2}}} \sqrt{- 4 \sqrt{3} + 7} \left (- \sqrt{3} + 3\right ) \left (\sqrt{3} + 3\right ) \sqrt{c^{3} - 8 d^{3} x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x+e)/(d*x+c)/(-8*d**3*x**3+c**3)**(1/2),x)

[Out]

3**(3/4)*sqrt((c**2 + 2*c*d*x + 4*d**2*x**2)/(c*(1 + sqrt(3)) - 2*d*x)**2)*sqrt(
sqrt(3) + 2)*(c - 2*d*x)*(c*f*(1 + sqrt(3)) + 2*d*e)*elliptic_f(asin(-(-c*(-1 +
sqrt(3)) - 2*d*x)/(c*(1 + sqrt(3)) - 2*d*x)), -7 - 4*sqrt(3))/(3*c*d**2*sqrt(c*(
c - 2*d*x)/(c*(1 + sqrt(3)) - 2*d*x)**2)*(sqrt(3) + 3)*sqrt(c**3 - 8*d**3*x**3))
+ 3**(3/4)*sqrt(c**2*(1 + 2*d*x/c + 4*d**2*x**2/c**2)/(c*(1 + sqrt(3)) - 2*d*x)
**2)*sqrt(3 + 2*sqrt(3))*sqrt(-sqrt(3) + 2)*(c - 2*d*x)*(c*f - d*e)*atanh(sqrt(-
(c*(-1 + sqrt(3)) + 2*d*x)**2/(c*(1 + sqrt(3)) - 2*d*x)**2 + 1)/(sqrt(3 + 2*sqrt
(3))*sqrt((c*(-1 + sqrt(3)) + 2*d*x)**2/(c*(1 + sqrt(3)) - 2*d*x)**2 - 4*sqrt(3)
+ 7)))/(9*c*d**2*sqrt(c*(c - 2*d*x)/(c*(1 + sqrt(3)) - 2*d*x)**2)*sqrt(c**3 - 8
*d**3*x**3)) - 4*3**(1/4)*sqrt(c**2*(1 + 2*d*x/c + 4*d**2*x**2/c**2)/(c*(1 + sqr
t(3)) - 2*d*x)**2)*sqrt(-sqrt(3) + 2)*(c - 2*d*x)*(c*f - d*e)*elliptic_pi(4*sqrt
(3) + 7, asin((c*(-1 + sqrt(3)) + 2*d*x)/(c*(1 + sqrt(3)) - 2*d*x)), -7 - 4*sqrt
(3))/(c*d**2*sqrt(c*(c - 2*d*x)/(c*(1 + sqrt(3)) - 2*d*x)**2)*sqrt(-4*sqrt(3) +
7)*(-sqrt(3) + 3)*(sqrt(3) + 3)*sqrt(c**3 - 8*d**3*x**3))

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Mathematica [C]  time = 2.00568, 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*Sqr
t[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[2]*Sqrt[(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 = 0.012, size = 661, normalized size = 3. \[ \text{result too large to display} \]

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/d*f*(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. \[ \int \frac{f x + e}{\sqrt{-8 \, d^{3} x^{3} + c^{3}}{\left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)/(sqrt(-8*d^3*x^3 + c^3)*(d*x + c)),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. \[{\rm integral}\left (\frac{f x + e}{\sqrt{-8 \, d^{3} x^{3} + c^{3}}{\left (d x + c\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)/(sqrt(-8*d^3*x^3 + c^3)*(d*x + c)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e + f x}{\sqrt{- \left (- c + 2 d x\right ) \left (c^{2} + 2 c d x + 4 d^{2} x^{2}\right )} \left (c + d x\right )}\, dx \]

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f x + e}{\sqrt{-8 \, d^{3} x^{3} + c^{3}}{\left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)/(sqrt(-8*d^3*x^3 + c^3)*(d*x + c)),x, algorithm="giac")

[Out]

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