3.2865 \(\int \frac{1}{\left (a+b (c+d x)^3\right )^2} \, dx\)

Optimal. Leaf size=170 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{9 a^{5/3} \sqrt [3]{b} d}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{5/3} \sqrt [3]{b} d}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} \sqrt [3]{b} d}+\frac{c+d x}{3 a d \left (a+b (c+d x)^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.291087, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{9 a^{5/3} \sqrt [3]{b} d}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{5/3} \sqrt [3]{b} d}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} \sqrt [3]{b} d}+\frac{c+d x}{3 a d \left (a+b (c+d x)^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*(c + d*x)^3)^(-2),x]

[Out]

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

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Rubi in Sympy [A]  time = 35.4552, size = 158, normalized size = 0.93 \[ \frac{c + d x}{3 a d \left (a + b \left (c + d x\right )^{3}\right )} + \frac{2 \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{b} d} - \frac{\log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{b} d} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{b} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c + d*x)/(3*a*d*(a + b*(c + d*x)**3)) + 2*log(a**(1/3) + b**(1/3)*(c + d*x))/(9
*a**(5/3)*b**(1/3)*d) - log(a**(2/3) + a**(1/3)*b**(1/3)*(-c - d*x) + b**(2/3)*(
c + d*x)**2)/(9*a**(5/3)*b**(1/3)*d) - 2*sqrt(3)*atan(sqrt(3)*(a**(1/3)/3 + b**(
1/3)*(-2*c/3 - 2*d*x/3))/a**(1/3))/(9*a**(5/3)*b**(1/3)*d)

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Mathematica [A]  time = 0.0977292, size = 151, normalized size = 0.89 \[ \frac{-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{\sqrt [3]{b}}+\frac{3 a^{2/3} (c+d x)}{a+b (c+d x)^3}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{\sqrt [3]{b}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{9 a^{5/3} d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.013, size = 127, normalized size = 0.8 \[{\frac{1}{b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a} \left ({\frac{x}{3\,a}}+{\frac{c}{3\,ad}} \right ) }+{\frac{2}{9\,abd}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*(d*x+c)^3)^2,x)

[Out]

(1/3*x/a+1/3*c/d/a)/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)+2/9/a/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. \[ \frac{d x + c}{3 \,{\left (a b d^{4} x^{3} + 3 \, a b c d^{3} x^{2} + 3 \, a b c^{2} d^{2} x +{\left (a b c^{3} + a^{2}\right )} d\right )}} + \frac{2 \, \int \frac{1}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}\,{d x}}{3 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(d*x + c)/(a*b*d^4*x^3 + 3*a*b*c*d^3*x^2 + 3*a*b*c^2*d^2*x + (a*b*c^3 + a^2)
*d) + 2/3*integrate(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

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Fricas [A]  time = 0.218236, size = 375, normalized size = 2.21 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} \log \left (a^{2} +{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \left (a^{2} b\right )^{\frac{2}{3}} - \left (a^{2} b\right )^{\frac{1}{3}}{\left (a d x + a c\right )}\right ) - 2 \, \sqrt{3}{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}}{\left (d x + c\right )} + a\right ) - 6 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}}{\left (d x + c\right )} - \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}}{\left (d x + c\right )}\right )}}{27 \,{\left (a b d^{4} x^{3} + 3 \, a b c d^{3} x^{2} + 3 \, a b c^{2} d^{2} x +{\left (a b c^{3} + a^{2}\right )} d\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*sqrt(3)*(sqrt(3)*(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)*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)) -
2*sqrt(3)*(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)*log((a^2*b)^(1/3
)*(d*x + c) + a) - 6*(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)*arcta
n(1/3*(2*sqrt(3)*(a^2*b)^(1/3)*(d*x + c) - sqrt(3)*a)/a) - 3*sqrt(3)*(a^2*b)^(1/
3)*(d*x + c))/((a*b*d^4*x^3 + 3*a*b*c*d^3*x^2 + 3*a*b*c^2*d^2*x + (a*b*c^3 + a^2
)*d)*(a^2*b)^(1/3))

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Sympy [A]  time = 4.9404, size = 92, normalized size = 0.54 \[ \frac{c + d x}{3 a^{2} d + 3 a b c^{3} d + 9 a b c^{2} d^{2} x + 9 a b c d^{3} x^{2} + 3 a b d^{4} x^{3}} + \frac{\operatorname{RootSum}{\left (729 t^{3} a^{5} b - 8, \left ( t \mapsto t \log{\left (x + \frac{9 t a^{2} + 2 c}{2 d} \right )} \right )\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c + d*x)/(3*a**2*d + 3*a*b*c**3*d + 9*a*b*c**2*d**2*x + 9*a*b*c*d**3*x**2 + 3*a
*b*d**4*x**3) + RootSum(729*_t**3*a**5*b - 8, Lambda(_t, _t*log(x + (9*_t*a**2 +
2*c)/(2*d))))/d

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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