3.2874 \(\int \frac{1}{\left (a+b (c+d x)^3\right )^3} \, dx\)

Optimal. Leaf size=198 \[ -\frac{5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{8/3} \sqrt [3]{b} d}+\frac{5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{8/3} \sqrt [3]{b} d}-\frac{5 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3} \sqrt [3]{b} d}+\frac{5 (c+d x)}{18 a^2 d \left (a+b (c+d x)^3\right )}+\frac{c+d x}{6 a d \left (a+b (c+d x)^3\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.360539, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ -\frac{5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{8/3} \sqrt [3]{b} d}+\frac{5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{8/3} \sqrt [3]{b} d}-\frac{5 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3} \sqrt [3]{b} d}+\frac{5 (c+d x)}{18 a^2 d \left (a+b (c+d x)^3\right )}+\frac{c+d x}{6 a d \left (a+b (c+d x)^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*(c + d*x)^3)^(-3),x]

[Out]

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

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Rubi in Sympy [A]  time = 39.9692, size = 187, normalized size = 0.94 \[ \frac{c + d x}{6 a d \left (a + b \left (c + d x\right )^{3}\right )^{2}} + \frac{5 \left (c + d x\right )}{18 a^{2} d \left (a + b \left (c + d x\right )^{3}\right )} + \frac{5 \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{27 a^{\frac{8}{3}} \sqrt [3]{b} d} - \frac{5 \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{54 a^{\frac{8}{3}} \sqrt [3]{b} d} - \frac{5 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{8}{3}} \sqrt [3]{b} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.176096, size = 176, normalized size = 0.89 \[ \frac{-\frac{5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{\sqrt [3]{b}}+\frac{9 a^{5/3} (c+d x)}{\left (a+b (c+d x)^3\right )^2}+\frac{15 a^{2/3} (c+d x)}{a+b (c+d x)^3}+\frac{10 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{\sqrt [3]{b}}+\frac{10 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{54 a^{8/3} d} \]

Antiderivative was successfully verified.

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

[Out]

((9*a^(5/3)*(c + d*x))/(a + b*(c + d*x)^3)^2 + (15*a^(2/3)*(c + d*x))/(a + b*(c
+ d*x)^3) + (10*Sqrt[3]*ArcTan[(-a^(1/3) + 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3)
)])/b^(1/3) + (10*Log[a^(1/3) + b^(1/3)*(c + d*x)])/b^(1/3) - (5*Log[a^(2/3) - a
^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2])/b^(1/3))/(54*a^(8/3)*d)

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Maple [C]  time = 0.023, size = 185, normalized size = 0.9 \[{\frac{1}{ \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}} \left ({\frac{5\,b{d}^{3}{x}^{4}}{18\,{a}^{2}}}+{\frac{10\,bc{d}^{2}{x}^{3}}{9\,{a}^{2}}}+{\frac{5\,b{c}^{2}d{x}^{2}}{3\,{a}^{2}}}+{\frac{ \left ( 10\,b{c}^{3}+4\,a \right ) x}{9\,{a}^{2}}}+{\frac{c \left ( 5\,b{c}^{3}+8\,a \right ) }{18\,{a}^{2}d}} \right ) }+{\frac{5}{27\,{a}^{2}bd}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*(d*x+c)^3)^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*(5*b*d^4*x^4 + 20*b*c*d^3*x^3 + 30*b*c^2*d^2*x^2 + 5*b*c^4 + 4*(5*b*c^3 + 2
*a)*d*x + 8*a*c)/(a^2*b^2*d^7*x^6 + 6*a^2*b^2*c*d^6*x^5 + 15*a^2*b^2*c^2*d^5*x^4
+ 2*(10*a^2*b^2*c^3 + a^3*b)*d^4*x^3 + 3*(5*a^2*b^2*c^4 + 2*a^3*b*c)*d^3*x^2 +
6*(a^2*b^2*c^5 + a^3*b*c^2)*d^2*x + (a^2*b^2*c^6 + 2*a^3*b*c^3 + a^4)*d) + 5/9*i
ntegrate(1/(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a), x)/a^2

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Fricas [A]  time = 0.242507, size = 914, normalized size = 4.62 \[ -\frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + 2 \,{\left (10 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + 2 \, a b c^{3} + 3 \,{\left (5 \, b^{2} c^{4} + 2 \, a b c\right )} d^{2} x^{2} + 6 \,{\left (b^{2} c^{5} + a b c^{2}\right )} d x + a^{2}\right )} \log \left (a^{2} +{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \left (a^{2} b\right )^{\frac{2}{3}} - \left (a^{2} b\right )^{\frac{1}{3}}{\left (a d x + a c\right )}\right ) - 10 \, \sqrt{3}{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + 2 \,{\left (10 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + 2 \, a b c^{3} + 3 \,{\left (5 \, b^{2} c^{4} + 2 \, a b c\right )} d^{2} x^{2} + 6 \,{\left (b^{2} c^{5} + a b c^{2}\right )} d x + a^{2}\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}}{\left (d x + c\right )} + a\right ) - 30 \,{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + 2 \,{\left (10 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + 2 \, a b c^{3} + 3 \,{\left (5 \, b^{2} c^{4} + 2 \, a b c\right )} d^{2} x^{2} + 6 \,{\left (b^{2} c^{5} + a b c^{2}\right )} d x + a^{2}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}}{\left (d x + c\right )} - \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (5 \, b d^{4} x^{4} + 20 \, b c d^{3} x^{3} + 30 \, b c^{2} d^{2} x^{2} + 5 \, b c^{4} + 4 \,{\left (5 \, b c^{3} + 2 \, a\right )} d x + 8 \, a c\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{162 \,{\left (a^{2} b^{2} d^{7} x^{6} + 6 \, a^{2} b^{2} c d^{6} x^{5} + 15 \, a^{2} b^{2} c^{2} d^{5} x^{4} + 2 \,{\left (10 \, a^{2} b^{2} c^{3} + a^{3} b\right )} d^{4} x^{3} + 3 \,{\left (5 \, a^{2} b^{2} c^{4} + 2 \, a^{3} b c\right )} d^{3} x^{2} + 6 \,{\left (a^{2} b^{2} c^{5} + a^{3} b c^{2}\right )} d^{2} x +{\left (a^{2} b^{2} c^{6} + 2 \, a^{3} b c^{3} + a^{4}\right )} d\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/162*sqrt(3)*(5*sqrt(3)*(b^2*d^6*x^6 + 6*b^2*c*d^5*x^5 + 15*b^2*c^2*d^4*x^4 +
b^2*c^6 + 2*(10*b^2*c^3 + a*b)*d^3*x^3 + 2*a*b*c^3 + 3*(5*b^2*c^4 + 2*a*b*c)*d^2
*x^2 + 6*(b^2*c^5 + a*b*c^2)*d*x + a^2)*log(a^2 + (d^2*x^2 + 2*c*d*x + c^2)*(a^2
*b)^(2/3) - (a^2*b)^(1/3)*(a*d*x + a*c)) - 10*sqrt(3)*(b^2*d^6*x^6 + 6*b^2*c*d^5
*x^5 + 15*b^2*c^2*d^4*x^4 + b^2*c^6 + 2*(10*b^2*c^3 + a*b)*d^3*x^3 + 2*a*b*c^3 +
3*(5*b^2*c^4 + 2*a*b*c)*d^2*x^2 + 6*(b^2*c^5 + a*b*c^2)*d*x + a^2)*log((a^2*b)^
(1/3)*(d*x + c) + a) - 30*(b^2*d^6*x^6 + 6*b^2*c*d^5*x^5 + 15*b^2*c^2*d^4*x^4 +
b^2*c^6 + 2*(10*b^2*c^3 + a*b)*d^3*x^3 + 2*a*b*c^3 + 3*(5*b^2*c^4 + 2*a*b*c)*d^2
*x^2 + 6*(b^2*c^5 + a*b*c^2)*d*x + a^2)*arctan(1/3*(2*sqrt(3)*(a^2*b)^(1/3)*(d*x
+ c) - sqrt(3)*a)/a) - 3*sqrt(3)*(5*b*d^4*x^4 + 20*b*c*d^3*x^3 + 30*b*c^2*d^2*x
^2 + 5*b*c^4 + 4*(5*b*c^3 + 2*a)*d*x + 8*a*c)*(a^2*b)^(1/3))/((a^2*b^2*d^7*x^6 +
6*a^2*b^2*c*d^6*x^5 + 15*a^2*b^2*c^2*d^5*x^4 + 2*(10*a^2*b^2*c^3 + a^3*b)*d^4*x
^3 + 3*(5*a^2*b^2*c^4 + 2*a^3*b*c)*d^3*x^2 + 6*(a^2*b^2*c^5 + a^3*b*c^2)*d^2*x +
(a^2*b^2*c^6 + 2*a^3*b*c^3 + a^4)*d)*(a^2*b)^(1/3))

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Sympy [A]  time = 54.1086, size = 267, normalized size = 1.35 \[ \frac{8 a c + 5 b c^{4} + 30 b c^{2} d^{2} x^{2} + 20 b c d^{3} x^{3} + 5 b d^{4} x^{4} + x \left (8 a d + 20 b c^{3} d\right )}{18 a^{4} d + 36 a^{3} b c^{3} d + 18 a^{2} b^{2} c^{6} d + 270 a^{2} b^{2} c^{2} d^{5} x^{4} + 108 a^{2} b^{2} c d^{6} x^{5} + 18 a^{2} b^{2} d^{7} x^{6} + x^{3} \left (36 a^{3} b d^{4} + 360 a^{2} b^{2} c^{3} d^{4}\right ) + x^{2} \left (108 a^{3} b c d^{3} + 270 a^{2} b^{2} c^{4} d^{3}\right ) + x \left (108 a^{3} b c^{2} d^{2} + 108 a^{2} b^{2} c^{5} d^{2}\right )} + \frac{\operatorname{RootSum}{\left (19683 t^{3} a^{8} b - 125, \left ( t \mapsto t \log{\left (x + \frac{27 t a^{3} + 5 c}{5 d} \right )} \right )\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(8*a*c + 5*b*c**4 + 30*b*c**2*d**2*x**2 + 20*b*c*d**3*x**3 + 5*b*d**4*x**4 + x*(
8*a*d + 20*b*c**3*d))/(18*a**4*d + 36*a**3*b*c**3*d + 18*a**2*b**2*c**6*d + 270*
a**2*b**2*c**2*d**5*x**4 + 108*a**2*b**2*c*d**6*x**5 + 18*a**2*b**2*d**7*x**6 +
x**3*(36*a**3*b*d**4 + 360*a**2*b**2*c**3*d**4) + x**2*(108*a**3*b*c*d**3 + 270*
a**2*b**2*c**4*d**3) + x*(108*a**3*b*c**2*d**2 + 108*a**2*b**2*c**5*d**2)) + Roo
tSum(19683*_t**3*a**8*b - 125, Lambda(_t, _t*log(x + (27*_t*a**3 + 5*c)/(5*d))))
/d

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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