3.2490 \(\int \frac{1}{(2+3 x)^2 \sqrt [3]{52-54 x+27 x^2}} \, dx\)

Optimal. Leaf size=719 \[ -\frac{\left (27 x^2-54 x+52\right )^{2/3}}{300 (3 x+2)}+\frac{\log \left (-27 \sqrt [3]{10} \sqrt [3]{27 x^2-54 x+52}-81 x+216\right )}{60\ 10^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (8-3 x)}{\sqrt{3} \sqrt [3]{5} \sqrt [3]{27 x^2-54 x+52}}+\frac{1}{\sqrt{3}}\right )}{30 \sqrt{3} 10^{2/3}}+\frac{9 (1-x)}{10\ 5^{2/3} \left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )}-\frac{\log (3 x+2)}{60\ 10^{2/3}}+\frac{\left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} F\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{5400\ 3^{3/4} \sqrt [6]{5} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}-\frac{\sqrt{2+\sqrt{3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{10800 \sqrt{2} \sqrt [4]{3} \sqrt [6]{5} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)} \]

[Out]

-(52 - 54*x + 27*x^2)^(2/3)/(300*(2 + 3*x)) + (9*(1 - x))/(10*5^(2/3)*(30*(1 - S
qrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))) - ArcTan[1/Sqrt[3] + (2^(2/3)
*(8 - 3*x))/(Sqrt[3]*5^(1/3)*(52 - 54*x + 27*x^2)^(1/3))]/(30*Sqrt[3]*10^(2/3))
- (Sqrt[2 + Sqrt[3]]*(30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))*Sqrt[(900 + 3
0*10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3) + 10^(2/3)*(2700 + (-54 + 54*x)^2)^(2/3
))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))^2]*EllipticE[ArcS
in[(30*(1 + Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(1 - Sqrt[3])
- 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))], -7 + 4*Sqrt[3]])/(10800*Sqrt[2]*3^(
1/4)*5^(1/6)*(1 - x)*Sqrt[-((30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(1
- Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))^2)]) + ((30 - 10^(1/3)*(27
00 + (-54 + 54*x)^2)^(1/3))*Sqrt[(900 + 30*10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3
) + 10^(2/3)*(2700 + (-54 + 54*x)^2)^(2/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 +
(-54 + 54*x)^2)^(1/3))^2]*EllipticF[ArcSin[(30*(1 + Sqrt[3]) - 10^(1/3)*(2700 +
(-54 + 54*x)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/
3))], -7 + 4*Sqrt[3]])/(5400*3^(3/4)*5^(1/6)*(1 - x)*Sqrt[-((30 - 10^(1/3)*(2700
+ (-54 + 54*x)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(
1/3))^2)]) - Log[2 + 3*x]/(60*10^(2/3)) + Log[216 - 81*x - 27*10^(1/3)*(52 - 54*
x + 27*x^2)^(1/3)]/(60*10^(2/3))

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Rubi [A]  time = 1.22182, antiderivative size = 719, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{\left (27 x^2-54 x+52\right )^{2/3}}{300 (3 x+2)}+\frac{\log \left (-27 \sqrt [3]{10} \sqrt [3]{27 x^2-54 x+52}-81 x+216\right )}{60\ 10^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (8-3 x)}{\sqrt{3} \sqrt [3]{5} \sqrt [3]{27 x^2-54 x+52}}+\frac{1}{\sqrt{3}}\right )}{30 \sqrt{3} 10^{2/3}}+\frac{9 (1-x)}{10\ 5^{2/3} \left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )}-\frac{\log (3 x+2)}{60\ 10^{2/3}}+\frac{\left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} F\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{5400\ 3^{3/4} \sqrt [6]{5} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}-\frac{\sqrt{2+\sqrt{3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt{\frac{10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\sin ^{-1}\left (\frac{30 \left (1+\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt{3}\right )}{10800 \sqrt{2} \sqrt [4]{3} \sqrt [6]{5} \sqrt{-\frac{30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt{3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)} \]

Warning: Unable to verify antiderivative.

[In]  Int[1/((2 + 3*x)^2*(52 - 54*x + 27*x^2)^(1/3)),x]

[Out]

-(52 - 54*x + 27*x^2)^(2/3)/(300*(2 + 3*x)) + (9*(1 - x))/(10*5^(2/3)*(30*(1 - S
qrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))) - ArcTan[1/Sqrt[3] + (2^(2/3)
*(8 - 3*x))/(Sqrt[3]*5^(1/3)*(52 - 54*x + 27*x^2)^(1/3))]/(30*Sqrt[3]*10^(2/3))
- (Sqrt[2 + Sqrt[3]]*(30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))*Sqrt[(900 + 3
0*10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3) + 10^(2/3)*(2700 + (-54 + 54*x)^2)^(2/3
))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))^2]*EllipticE[ArcS
in[(30*(1 + Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(1 - Sqrt[3])
- 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))], -7 + 4*Sqrt[3]])/(10800*Sqrt[2]*3^(
1/4)*5^(1/6)*(1 - x)*Sqrt[-((30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(1
- Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))^2)]) + ((30 - 10^(1/3)*(27
00 + (-54 + 54*x)^2)^(1/3))*Sqrt[(900 + 30*10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3
) + 10^(2/3)*(2700 + (-54 + 54*x)^2)^(2/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 +
(-54 + 54*x)^2)^(1/3))^2]*EllipticF[ArcSin[(30*(1 + Sqrt[3]) - 10^(1/3)*(2700 +
(-54 + 54*x)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/
3))], -7 + 4*Sqrt[3]])/(5400*3^(3/4)*5^(1/6)*(1 - x)*Sqrt[-((30 - 10^(1/3)*(2700
+ (-54 + 54*x)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(
1/3))^2)]) - Log[2 + 3*x]/(60*10^(2/3)) + Log[216 - 81*x - 27*10^(1/3)*(52 - 54*
x + 27*x^2)^(1/3)]/(60*10^(2/3))

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Rubi in Sympy [A]  time = 39.8552, size = 524, normalized size = 0.73 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2+3*x)**2/(27*x**2-54*x+52)**(1/3),x)

[Out]

5**(1/3)*(-54*x + 54)/(9000*(-((54*x - 54)**2/2700 + 1)**(1/3) - sqrt(3) + 1)) -
10**(1/3)*log(3*x + 2)/600 + 10**(1/3)*log(-81*x - 27*10**(1/3)*(27*x**2 - 54*x
+ 52)**(1/3) + 216)/600 - 10**(1/3)*sqrt(3)*atan(10**(2/3)*sqrt(3)*(-81*x + 216
)/(405*(27*x**2 - 54*x + 52)**(1/3)) + sqrt(3)/3)/900 - (27*x**2 - 54*x + 52)**(
2/3)/(300*(3*x + 2)) - 3*3**(1/4)*5**(1/3)*sqrt((((54*x - 54)**2/2700 + 1)**(2/3
) + ((54*x - 54)**2/2700 + 1)**(1/3) + 1)/(-((54*x - 54)**2/2700 + 1)**(1/3) - s
qrt(3) + 1)**2)*sqrt(sqrt(3) + 2)*(-((54*x - 54)**2/2700 + 1)**(1/3) + 1)*ellipt
ic_e(asin((-((54*x - 54)**2/2700 + 1)**(1/3) + 1 + sqrt(3))/(-((54*x - 54)**2/27
00 + 1)**(1/3) - sqrt(3) + 1)), -7 + 4*sqrt(3))/(20*sqrt((((54*x - 54)**2/2700 +
1)**(1/3) - 1)/(-((54*x - 54)**2/2700 + 1)**(1/3) - sqrt(3) + 1)**2)*(-54*x + 5
4)) + sqrt(2)*3**(3/4)*5**(1/3)*sqrt((((54*x - 54)**2/2700 + 1)**(2/3) + ((54*x
- 54)**2/2700 + 1)**(1/3) + 1)/(-((54*x - 54)**2/2700 + 1)**(1/3) - sqrt(3) + 1)
**2)*(-((54*x - 54)**2/2700 + 1)**(1/3) + 1)*elliptic_f(asin((-((54*x - 54)**2/2
700 + 1)**(1/3) + 1 + sqrt(3))/(-((54*x - 54)**2/2700 + 1)**(1/3) - sqrt(3) + 1)
), -7 + 4*sqrt(3))/(10*sqrt((((54*x - 54)**2/2700 + 1)**(1/3) - 1)/(-((54*x - 54
)**2/2700 + 1)**(1/3) - sqrt(3) + 1)**2)*(-54*x + 54))

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Mathematica [C]  time = 0.583211, size = 402, normalized size = 0.56 \[ \frac{-\frac{900 (3 x+2) \left (9 x-5 i \sqrt{3}-9\right ) \left (9 x+5 i \sqrt{3}-9\right ) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{15-5 i \sqrt{3}}{9 x+6},\frac{15+5 i \sqrt{3}}{9 x+6}\right )}{(9 x+6) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{15-5 i \sqrt{3}}{9 x+6},\frac{15+5 i \sqrt{3}}{9 x+6}\right )+\left (3+i \sqrt{3}\right ) F_1\left (\frac{5}{3};\frac{1}{3},\frac{4}{3};\frac{8}{3};\frac{15-5 i \sqrt{3}}{9 x+6},\frac{15+5 i \sqrt{3}}{9 x+6}\right )+\left (3-i \sqrt{3}\right ) F_1\left (\frac{5}{3};\frac{4}{3},\frac{1}{3};\frac{8}{3};\frac{15-5 i \sqrt{3}}{9 x+6},\frac{15+5 i \sqrt{3}}{9 x+6}\right )}+3^{5/6} 10^{2/3} \sqrt [3]{-9 i x+5 \sqrt{3}+9 i} \left (9 x-5 i \sqrt{3}-9\right ) \left (27 x^2-54 x+52\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{9 i x+5 \sqrt{3}-9 i}{10 \sqrt{3}}\right )-\frac{60 \left (27 x^2-54 x+52\right )^2}{3 x+2}}{18000 \left (27 x^2-54 x+52\right )^{4/3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((2 + 3*x)^2*(52 - 54*x + 27*x^2)^(1/3)),x]

[Out]

((-60*(52 - 54*x + 27*x^2)^2)/(2 + 3*x) - (900*(2 + 3*x)*(-9 - (5*I)*Sqrt[3] + 9
*x)*(-9 + (5*I)*Sqrt[3] + 9*x)*AppellF1[2/3, 1/3, 1/3, 5/3, (15 - (5*I)*Sqrt[3])
/(6 + 9*x), (15 + (5*I)*Sqrt[3])/(6 + 9*x)])/((6 + 9*x)*AppellF1[2/3, 1/3, 1/3,
5/3, (15 - (5*I)*Sqrt[3])/(6 + 9*x), (15 + (5*I)*Sqrt[3])/(6 + 9*x)] + (3 + I*Sq
rt[3])*AppellF1[5/3, 1/3, 4/3, 8/3, (15 - (5*I)*Sqrt[3])/(6 + 9*x), (15 + (5*I)*
Sqrt[3])/(6 + 9*x)] + (3 - I*Sqrt[3])*AppellF1[5/3, 4/3, 1/3, 8/3, (15 - (5*I)*S
qrt[3])/(6 + 9*x), (15 + (5*I)*Sqrt[3])/(6 + 9*x)]) + 3^(5/6)*10^(2/3)*(9*I + 5*
Sqrt[3] - (9*I)*x)^(1/3)*(-9 - (5*I)*Sqrt[3] + 9*x)*(52 - 54*x + 27*x^2)*Hyperge
ometric2F1[1/3, 2/3, 5/3, (-9*I + 5*Sqrt[3] + (9*I)*x)/(10*Sqrt[3])])/(18000*(52
- 54*x + 27*x^2)^(4/3))

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Maple [F]  time = 0.119, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( 2+3\,x \right ) ^{2}}{\frac{1}{\sqrt [3]{27\,{x}^{2}-54\,x+52}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2+3*x)^2/(27*x^2-54*x+52)^(1/3),x)

[Out]

int(1/(2+3*x)^2/(27*x^2-54*x+52)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac{1}{3}}{\left (3 \, x + 2\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((27*x^2 - 54*x + 52)^(1/3)*(3*x + 2)^2),x, algorithm="maxima")

[Out]

integrate(1/((27*x^2 - 54*x + 52)^(1/3)*(3*x + 2)^2), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((27*x^2 - 54*x + 52)^(1/3)*(3*x + 2)^2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (3 x + 2\right )^{2} \sqrt [3]{27 x^{2} - 54 x + 52}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2+3*x)**2/(27*x**2-54*x+52)**(1/3),x)

[Out]

Integral(1/((3*x + 2)**2*(27*x**2 - 54*x + 52)**(1/3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac{1}{3}}{\left (3 \, x + 2\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((27*x^2 - 54*x + 52)^(1/3)*(3*x + 2)^2),x, algorithm="giac")

[Out]

integrate(1/((27*x^2 - 54*x + 52)^(1/3)*(3*x + 2)^2), x)