3.14 \(\int \frac{1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3} \, dx\)

Optimal. Leaf size=305 \[ \frac{5 c^2 \left (\frac{b}{c}+x\right )}{18 b^2 \left (b^2-3 a c\right )^2 \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}-\frac{c \left (\frac{b}{c}+x\right )}{6 b \left (b^2-3 a c\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2}+\frac{5 c^2 \log \left (-\sqrt [3]{b} \sqrt [3]{b^2-3 a c}+b+c x\right )}{27 b^{8/3} \left (b^2-3 a c\right )^{8/3}}-\frac{5 c^2 \log \left (\sqrt [3]{b} c \sqrt [3]{b^2-3 a c} \left (\frac{b}{c}+x\right )+b^{2/3} \left (b^2-3 a c\right )^{2/3}+c^2 \left (\frac{b}{c}+x\right )^2\right )}{54 b^{8/3} \left (b^2-3 a c\right )^{8/3}}-\frac{5 c^2 \tan ^{-1}\left (\frac{\frac{2 (b+c x)}{\sqrt [3]{b^2-3 a c}}+\sqrt [3]{b}}{\sqrt{3} \sqrt [3]{b}}\right )}{9 \sqrt{3} b^{8/3} \left (b^2-3 a c\right )^{8/3}} \]

[Out]

-(c*(b/c + x))/(6*b*(b^2 - 3*a*c)*(3*a*b + 3*b^2*x + 3*b*c*x^2 + c^2*x^3)^2) + (
5*c^2*(b/c + x))/(18*b^2*(b^2 - 3*a*c)^2*(3*a*b + 3*b^2*x + 3*b*c*x^2 + c^2*x^3)
) - (5*c^2*ArcTan[(b^(1/3) + (2*(b + c*x))/(b^2 - 3*a*c)^(1/3))/(Sqrt[3]*b^(1/3)
)])/(9*Sqrt[3]*b^(8/3)*(b^2 - 3*a*c)^(8/3)) + (5*c^2*Log[b - b^(1/3)*(b^2 - 3*a*
c)^(1/3) + c*x])/(27*b^(8/3)*(b^2 - 3*a*c)^(8/3)) - (5*c^2*Log[b^(2/3)*(b^2 - 3*
a*c)^(2/3) + b^(1/3)*c*(b^2 - 3*a*c)^(1/3)*(b/c + x) + c^2*(b/c + x)^2])/(54*b^(
8/3)*(b^2 - 3*a*c)^(8/3))

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Rubi [A]  time = 0.759839, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296 \[ \frac{5 c (b+c x)}{18 b^2 \left (b^2-3 a c\right )^2 \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}-\frac{b+c x}{6 b \left (b^2-3 a c\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2}+\frac{5 c^2 \log \left (\sqrt [3]{b} \left (b^{2/3}-\sqrt [3]{b^2-3 a c}\right )+c x\right )}{27 b^{8/3} \left (b^2-3 a c\right )^{8/3}}-\frac{5 c^2 \log \left (\sqrt [3]{b} \sqrt [3]{b^2-3 a c} (b+c x)+b^{2/3} \left (b^2-3 a c\right )^{2/3}+(b+c x)^2\right )}{54 b^{8/3} \left (b^2-3 a c\right )^{8/3}}-\frac{5 c^2 \tan ^{-1}\left (\frac{\frac{2 (b+c x)}{\sqrt [3]{b^2-3 a c}}+\sqrt [3]{b}}{\sqrt{3} \sqrt [3]{b}}\right )}{9 \sqrt{3} b^{8/3} \left (b^2-3 a c\right )^{8/3}} \]

Antiderivative was successfully verified.

[In]  Int[(3*a*b + 3*b^2*x + 3*b*c*x^2 + c^2*x^3)^(-3),x]

[Out]

-(b + c*x)/(6*b*(b^2 - 3*a*c)*(3*a*b + 3*b^2*x + 3*b*c*x^2 + c^2*x^3)^2) + (5*c*
(b + c*x))/(18*b^2*(b^2 - 3*a*c)^2*(3*a*b + 3*b^2*x + 3*b*c*x^2 + c^2*x^3)) - (5
*c^2*ArcTan[(b^(1/3) + (2*(b + c*x))/(b^2 - 3*a*c)^(1/3))/(Sqrt[3]*b^(1/3))])/(9
*Sqrt[3]*b^(8/3)*(b^2 - 3*a*c)^(8/3)) + (5*c^2*Log[b^(1/3)*(b^(2/3) - (b^2 - 3*a
*c)^(1/3)) + c*x])/(27*b^(8/3)*(b^2 - 3*a*c)^(8/3)) - (5*c^2*Log[b^(2/3)*(b^2 -
3*a*c)^(2/3) + b^(1/3)*(b^2 - 3*a*c)^(1/3)*(b + c*x) + (b + c*x)^2])/(54*b^(8/3)
*(b^2 - 3*a*c)^(8/3))

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Rubi in Sympy [A]  time = 112.989, size = 280, normalized size = 0.92 \[ - \frac{c^{2} \left (b + c x\right )}{6 b \left (- 3 a c + b^{2}\right ) \left (b \left (3 a c - b^{2}\right ) + \left (b + c x\right )^{3}\right )^{2}} + \frac{5 c^{2} \left (b + c x\right )}{18 b^{2} \left (- 3 a c + b^{2}\right )^{2} \left (b \left (3 a c - b^{2}\right ) + \left (b + c x\right )^{3}\right )} + \frac{5 c^{2} \log{\left (\sqrt [3]{b} \sqrt [3]{- 3 a c + b^{2}} - b - c x \right )}}{27 b^{\frac{8}{3}} \left (- 3 a c + b^{2}\right )^{\frac{8}{3}}} - \frac{5 c^{2} \log{\left (9 b^{\frac{8}{3}} \left (- 3 a c + b^{2}\right )^{\frac{2}{3}} + b^{\frac{7}{3}} \left (9 b + 9 c x\right ) \sqrt [3]{- 3 a c + b^{2}} + 9 b^{2} \left (b + c x\right )^{2} \right )}}{54 b^{\frac{8}{3}} \left (- 3 a c + b^{2}\right )^{\frac{8}{3}}} - \frac{5 \sqrt{3} c^{2} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{b}}{3} + \frac{\frac{2 b}{3} + \frac{2 c x}{3}}{\sqrt [3]{- 3 a c + b^{2}}}\right )}{\sqrt [3]{b}} \right )}}{27 b^{\frac{8}{3}} \left (- 3 a c + b^{2}\right )^{\frac{8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c**2*x**3+3*b*c*x**2+3*b**2*x+3*a*b)**3,x)

[Out]

-c**2*(b + c*x)/(6*b*(-3*a*c + b**2)*(b*(3*a*c - b**2) + (b + c*x)**3)**2) + 5*c
**2*(b + c*x)/(18*b**2*(-3*a*c + b**2)**2*(b*(3*a*c - b**2) + (b + c*x)**3)) + 5
*c**2*log(b**(1/3)*(-3*a*c + b**2)**(1/3) - b - c*x)/(27*b**(8/3)*(-3*a*c + b**2
)**(8/3)) - 5*c**2*log(9*b**(8/3)*(-3*a*c + b**2)**(2/3) + b**(7/3)*(9*b + 9*c*x
)*(-3*a*c + b**2)**(1/3) + 9*b**2*(b + c*x)**2)/(54*b**(8/3)*(-3*a*c + b**2)**(8
/3)) - 5*sqrt(3)*c**2*atan(sqrt(3)*(b**(1/3)/3 + (2*b/3 + 2*c*x/3)/(-3*a*c + b**
2)**(1/3))/b**(1/3))/(27*b**(8/3)*(-3*a*c + b**2)**(8/3))

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Mathematica [C]  time = 0.142461, size = 149, normalized size = 0.49 \[ \frac{10 c^2 \text{RootSum}\left [\text{$\#$1}^3 c^2+3 \text{$\#$1}^2 b c+3 \text{$\#$1} b^2+3 a b\&,\frac{\log (x-\text{$\#$1})}{\text{$\#$1}^2 c^2+2 \text{$\#$1} b c+b^2}\&\right ]-\frac{3 (b+c x) \left (-3 b c \left (8 a+5 c x^2\right )+3 b^3-15 b^2 c x-5 c^3 x^3\right )}{\left (3 a b+x \left (3 b^2+3 b c x+c^2 x^2\right )\right )^2}}{54 \left (b^3-3 a b c\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(3*a*b + 3*b^2*x + 3*b*c*x^2 + c^2*x^3)^(-3),x]

[Out]

((-3*(b + c*x)*(3*b^3 - 15*b^2*c*x - 5*c^3*x^3 - 3*b*c*(8*a + 5*c*x^2)))/(3*a*b
+ x*(3*b^2 + 3*b*c*x + c^2*x^2))^2 + 10*c^2*RootSum[3*a*b + 3*b^2*#1 + 3*b*c*#1^
2 + c^2*#1^3 & , Log[x - #1]/(b^2 + 2*b*c*#1 + c^2*#1^2) & ])/(54*(b^3 - 3*a*b*c
)^2)

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Maple [C]  time = 0.023, size = 276, normalized size = 0.9 \[{\frac{1}{ \left ({c}^{2}{x}^{3}+3\,bc{x}^{2}+3\,{b}^{2}x+3\,ab \right ) ^{2}} \left ({\frac{5\,{c}^{4}{x}^{4}}{18\,{b}^{2} \left ( 9\,{a}^{2}{c}^{2}-6\,a{b}^{2}c+{b}^{4} \right ) }}+{\frac{10\,{c}^{3}{x}^{3}}{9\,b \left ( 9\,{a}^{2}{c}^{2}-6\,a{b}^{2}c+{b}^{4} \right ) }}+{\frac{5\,{c}^{2}{x}^{2}}{27\,{a}^{2}{c}^{2}-18\,a{b}^{2}c+3\,{b}^{4}}}+{\frac{ \left ( 4\,ac+2\,{b}^{2} \right ) cx}{3\,b \left ( 9\,{a}^{2}{c}^{2}-6\,a{b}^{2}c+{b}^{4} \right ) }}+{\frac{8\,ac-{b}^{2}}{54\,{a}^{2}{c}^{2}-36\,a{b}^{2}c+6\,{b}^{4}}} \right ) }+{\frac{5\,{c}^{2}}{27\,{b}^{2} \left ( 9\,{a}^{2}{c}^{2}-6\,a{b}^{2}c+{b}^{4} \right ) }\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}{c}^{2}+3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,{b}^{2}+3\,ab \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{2}{c}^{2}+2\,{\it \_R}\,bc+{b}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c^2*x^3+3*b*c*x^2+3*b^2*x+3*a*b)^3,x)

[Out]

(5/18*c^4/b^2/(9*a^2*c^2-6*a*b^2*c+b^4)*x^4+10/9/b*c^3/(9*a^2*c^2-6*a*b^2*c+b^4)
*x^3+5/3*c^2/(9*a^2*c^2-6*a*b^2*c+b^4)*x^2+2/3/b*(2*a*c+b^2)*c/(9*a^2*c^2-6*a*b^
2*c+b^4)*x+1/6*(8*a*c-b^2)/(9*a^2*c^2-6*a*b^2*c+b^4))/(c^2*x^3+3*b*c*x^2+3*b^2*x
+3*a*b)^2+5/27*c^2/b^2/(9*a^2*c^2-6*a*b^2*c+b^4)*sum(1/(_R^2*c^2+2*_R*b*c+b^2)*l
n(x-_R),_R=RootOf(_Z^3*c^2+3*_Z^2*b*c+3*_Z*b^2+3*a*b))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{5 \, c^{2} \int \frac{1}{c^{2} x^{3} + 3 \, b c x^{2} + 3 \, b^{2} x + 3 \, a b}\,{d x}}{9 \,{\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}} + \frac{5 \, c^{4} x^{4} + 20 \, b c^{3} x^{3} + 30 \, b^{2} c^{2} x^{2} - 3 \, b^{4} + 24 \, a b^{2} c + 12 \,{\left (b^{3} c + 2 \, a b c^{2}\right )} x}{18 \,{\left (9 \, a^{2} b^{8} - 54 \, a^{3} b^{6} c + 81 \, a^{4} b^{4} c^{2} +{\left (b^{6} c^{4} - 6 \, a b^{4} c^{5} + 9 \, a^{2} b^{2} c^{6}\right )} x^{6} + 6 \,{\left (b^{7} c^{3} - 6 \, a b^{5} c^{4} + 9 \, a^{2} b^{3} c^{5}\right )} x^{5} + 15 \,{\left (b^{8} c^{2} - 6 \, a b^{6} c^{3} + 9 \, a^{2} b^{4} c^{4}\right )} x^{4} + 6 \,{\left (3 \, b^{9} c - 17 \, a b^{7} c^{2} + 21 \, a^{2} b^{5} c^{3} + 9 \, a^{3} b^{3} c^{4}\right )} x^{3} + 9 \,{\left (b^{10} - 4 \, a b^{8} c - 3 \, a^{2} b^{6} c^{2} + 18 \, a^{3} b^{4} c^{3}\right )} x^{2} + 18 \,{\left (a b^{9} - 6 \, a^{2} b^{7} c + 9 \, a^{3} b^{5} c^{2}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c^2*x^3 + 3*b*c*x^2 + 3*b^2*x + 3*a*b)^(-3),x, algorithm="maxima")

[Out]

5/9*c^2*integrate(1/(c^2*x^3 + 3*b*c*x^2 + 3*b^2*x + 3*a*b), x)/(b^6 - 6*a*b^4*c
+ 9*a^2*b^2*c^2) + 1/18*(5*c^4*x^4 + 20*b*c^3*x^3 + 30*b^2*c^2*x^2 - 3*b^4 + 24
*a*b^2*c + 12*(b^3*c + 2*a*b*c^2)*x)/(9*a^2*b^8 - 54*a^3*b^6*c + 81*a^4*b^4*c^2
+ (b^6*c^4 - 6*a*b^4*c^5 + 9*a^2*b^2*c^6)*x^6 + 6*(b^7*c^3 - 6*a*b^5*c^4 + 9*a^2
*b^3*c^5)*x^5 + 15*(b^8*c^2 - 6*a*b^6*c^3 + 9*a^2*b^4*c^4)*x^4 + 6*(3*b^9*c - 17
*a*b^7*c^2 + 21*a^2*b^5*c^3 + 9*a^3*b^3*c^4)*x^3 + 9*(b^10 - 4*a*b^8*c - 3*a^2*b
^6*c^2 + 18*a^3*b^4*c^3)*x^2 + 18*(a*b^9 - 6*a^2*b^7*c + 9*a^3*b^5*c^2)*x)

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Fricas [A]  time = 0.288221, size = 1160, normalized size = 3.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c^2*x^3 + 3*b*c*x^2 + 3*b^2*x + 3*a*b)^(-3),x, algorithm="fricas")

[Out]

-1/162*sqrt(3)*(5*sqrt(3)*(c^6*x^6 + 6*b*c^5*x^5 + 15*b^2*c^4*x^4 + 18*a*b^3*c^2
*x + 9*a^2*b^2*c^2 + 6*(3*b^3*c^3 + a*b*c^4)*x^3 + 9*(b^4*c^2 + 2*a*b^2*c^3)*x^2
)*log(b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2)^(2/3)*
(c^2*x^2 + 2*b*c*x + b^2) + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2)^(1/3)*(b^4 - 3*a*b
^2*c + (b^3*c - 3*a*b*c^2)*x)) - 10*sqrt(3)*(c^6*x^6 + 6*b*c^5*x^5 + 15*b^2*c^4*
x^4 + 18*a*b^3*c^2*x + 9*a^2*b^2*c^2 + 6*(3*b^3*c^3 + a*b*c^4)*x^3 + 9*(b^4*c^2
+ 2*a*b^2*c^3)*x^2)*log(-b^3 + 3*a*b*c + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2)^(1/3)
*(c*x + b)) - 30*(c^6*x^6 + 6*b*c^5*x^5 + 15*b^2*c^4*x^4 + 18*a*b^3*c^2*x + 9*a^
2*b^2*c^2 + 6*(3*b^3*c^3 + a*b*c^4)*x^3 + 9*(b^4*c^2 + 2*a*b^2*c^3)*x^2)*arctan(
-1/3*(2*sqrt(3)*(b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2)^(1/3)*(c*x + b) + sqrt(3)*(b^3
- 3*a*b*c))/(b^3 - 3*a*b*c)) - 3*sqrt(3)*(5*c^4*x^4 + 20*b*c^3*x^3 + 30*b^2*c^2
*x^2 - 3*b^4 + 24*a*b^2*c + 12*(b^3*c + 2*a*b*c^2)*x)*(b^6 - 6*a*b^4*c + 9*a^2*b
^2*c^2)^(1/3))/((9*a^2*b^8 - 54*a^3*b^6*c + 81*a^4*b^4*c^2 + (b^6*c^4 - 6*a*b^4*
c^5 + 9*a^2*b^2*c^6)*x^6 + 6*(b^7*c^3 - 6*a*b^5*c^4 + 9*a^2*b^3*c^5)*x^5 + 15*(b
^8*c^2 - 6*a*b^6*c^3 + 9*a^2*b^4*c^4)*x^4 + 6*(3*b^9*c - 17*a*b^7*c^2 + 21*a^2*b
^5*c^3 + 9*a^3*b^3*c^4)*x^3 + 9*(b^10 - 4*a*b^8*c - 3*a^2*b^6*c^2 + 18*a^3*b^4*c
^3)*x^2 + 18*(a*b^9 - 6*a^2*b^7*c + 9*a^3*b^5*c^2)*x)*(b^6 - 6*a*b^4*c + 9*a^2*b
^2*c^2)^(1/3))

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Sympy [A]  time = 23.6042, size = 474, normalized size = 1.55 \[ \frac{24 a b^{2} c - 3 b^{4} + 30 b^{2} c^{2} x^{2} + 20 b c^{3} x^{3} + 5 c^{4} x^{4} + x \left (24 a b c^{2} + 12 b^{3} c\right )}{1458 a^{4} b^{4} c^{2} - 972 a^{3} b^{6} c + 162 a^{2} b^{8} + x^{6} \left (162 a^{2} b^{2} c^{6} - 108 a b^{4} c^{5} + 18 b^{6} c^{4}\right ) + x^{5} \left (972 a^{2} b^{3} c^{5} - 648 a b^{5} c^{4} + 108 b^{7} c^{3}\right ) + x^{4} \left (2430 a^{2} b^{4} c^{4} - 1620 a b^{6} c^{3} + 270 b^{8} c^{2}\right ) + x^{3} \left (972 a^{3} b^{3} c^{4} + 2268 a^{2} b^{5} c^{3} - 1836 a b^{7} c^{2} + 324 b^{9} c\right ) + x^{2} \left (2916 a^{3} b^{4} c^{3} - 486 a^{2} b^{6} c^{2} - 648 a b^{8} c + 162 b^{10}\right ) + x \left (2916 a^{3} b^{5} c^{2} - 1944 a^{2} b^{7} c + 324 a b^{9}\right )} + \operatorname{RootSum}{\left (t^{3} \left (129140163 a^{8} b^{8} c^{8} - 344373768 a^{7} b^{10} c^{7} + 401769396 a^{6} b^{12} c^{6} - 267846264 a^{5} b^{14} c^{5} + 111602610 a^{4} b^{16} c^{4} - 29760696 a^{3} b^{18} c^{3} + 4960116 a^{2} b^{20} c^{2} - 472392 a b^{22} c + 19683 b^{24}\right ) - 125 c^{6}, \left ( t \mapsto t \log{\left (x + \frac{729 t a^{3} b^{3} c^{3} - 729 t a^{2} b^{5} c^{2} + 243 t a b^{7} c - 27 t b^{9} + 5 b c^{2}}{5 c^{3}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c**2*x**3+3*b*c*x**2+3*b**2*x+3*a*b)**3,x)

[Out]

(24*a*b**2*c - 3*b**4 + 30*b**2*c**2*x**2 + 20*b*c**3*x**3 + 5*c**4*x**4 + x*(24
*a*b*c**2 + 12*b**3*c))/(1458*a**4*b**4*c**2 - 972*a**3*b**6*c + 162*a**2*b**8 +
x**6*(162*a**2*b**2*c**6 - 108*a*b**4*c**5 + 18*b**6*c**4) + x**5*(972*a**2*b**
3*c**5 - 648*a*b**5*c**4 + 108*b**7*c**3) + x**4*(2430*a**2*b**4*c**4 - 1620*a*b
**6*c**3 + 270*b**8*c**2) + x**3*(972*a**3*b**3*c**4 + 2268*a**2*b**5*c**3 - 183
6*a*b**7*c**2 + 324*b**9*c) + x**2*(2916*a**3*b**4*c**3 - 486*a**2*b**6*c**2 - 6
48*a*b**8*c + 162*b**10) + x*(2916*a**3*b**5*c**2 - 1944*a**2*b**7*c + 324*a*b**
9)) + RootSum(_t**3*(129140163*a**8*b**8*c**8 - 344373768*a**7*b**10*c**7 + 4017
69396*a**6*b**12*c**6 - 267846264*a**5*b**14*c**5 + 111602610*a**4*b**16*c**4 -
29760696*a**3*b**18*c**3 + 4960116*a**2*b**20*c**2 - 472392*a*b**22*c + 19683*b*
*24) - 125*c**6, Lambda(_t, _t*log(x + (729*_t*a**3*b**3*c**3 - 729*_t*a**2*b**5
*c**2 + 243*_t*a*b**7*c - 27*_t*b**9 + 5*b*c**2)/(5*c**3))))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c^2*x^3 + 3*b*c*x^2 + 3*b^2*x + 3*a*b)^(-3),x, algorithm="giac")

[Out]

integrate((c^2*x^3 + 3*b*c*x^2 + 3*b^2*x + 3*a*b)^(-3), x)