3.2856 \(\int \frac{1}{a+b (c+d x)^3} \, dx\)

Optimal. Leaf size=140 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} \sqrt [3]{b} d}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b} d}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b} d} \]

[Out]

-(ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))]/(Sqrt[3]*a^(2/3)*b^(
1/3)*d)) + Log[a^(1/3) + b^(1/3)*(c + d*x)]/(3*a^(2/3)*b^(1/3)*d) - Log[a^(2/3)
- a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2]/(6*a^(2/3)*b^(1/3)*d)

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Rubi [A]  time = 0.249906, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} \sqrt [3]{b} d}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b} d}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b} d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*(c + d*x)^3)^(-1),x]

[Out]

-(ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))]/(Sqrt[3]*a^(2/3)*b^(
1/3)*d)) + Log[a^(1/3) + b^(1/3)*(c + d*x)]/(3*a^(2/3)*b^(1/3)*d) - Log[a^(2/3)
- a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2]/(6*a^(2/3)*b^(1/3)*d)

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Rubi in Sympy [A]  time = 30.4873, size = 134, normalized size = 0.96 \[ \frac{\log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{3 a^{\frac{2}{3}} \sqrt [3]{b} d} - \frac{\log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{6 a^{\frac{2}{3}} \sqrt [3]{b} d} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} \sqrt [3]{b} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a**(1/3) + b**(1/3)*(c + d*x))/(3*a**(2/3)*b**(1/3)*d) - log(a**(2/3) + a**(
1/3)*b**(1/3)*(-c - d*x) + b**(2/3)*(c + d*x)**2)/(6*a**(2/3)*b**(1/3)*d) - sqrt
(3)*atan(sqrt(3)*(a**(1/3)/3 + b**(1/3)*(-2*c/3 - 2*d*x/3))/a**(1/3))/(3*a**(2/3
)*b**(1/3)*d)

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Mathematica [A]  time = 0.0218152, size = 116, normalized size = 0.83 \[ \frac{-\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )+2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{6 a^{2/3} \sqrt [3]{b} d} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*(c + d*x)^3)^(-1),x]

[Out]

(2*Sqrt[3]*ArcTan[(-a^(1/3) + 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))] + 2*Log[a^
(1/3) + b^(1/3)*(c + d*x)] - Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(
c + d*x)^2])/(6*a^(2/3)*b^(1/3)*d)

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Maple [C]  time = 0.002, size = 71, normalized size = 0.5 \[{\frac{1}{3\,bd}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*(d*x+c)^3),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((d*x + c)^3*b + a),x, algorithm="maxima")

[Out]

integrate(1/((d*x + c)^3*b + a), x)

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Fricas [A]  time = 0.210618, size = 161, normalized size = 1.15 \[ -\frac{\sqrt{3}{\left (\sqrt{3} \log \left (a^{2} +{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \left (a^{2} b\right )^{\frac{2}{3}} - \left (a^{2} b\right )^{\frac{1}{3}}{\left (a d x + a c\right )}\right ) - 2 \, \sqrt{3} \log \left (\left (a^{2} b\right )^{\frac{1}{3}}{\left (d x + c\right )} + a\right ) - 6 \, \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}}{\left (d x + c\right )} - \sqrt{3} a}{3 \, a}\right )\right )}}{18 \, \left (a^{2} b\right )^{\frac{1}{3}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((d*x + c)^3*b + a),x, algorithm="fricas")

[Out]

-1/18*sqrt(3)*(sqrt(3)*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)*log((a^2*b)^(1/3)*(d*x + c) + a) - 6*arctan(
1/3*(2*sqrt(3)*(a^2*b)^(1/3)*(d*x + c) - sqrt(3)*a)/a))/((a^2*b)^(1/3)*d)

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Sympy [A]  time = 0.714936, size = 26, normalized size = 0.19 \[ \frac{\operatorname{RootSum}{\left (27 t^{3} a^{2} b - 1, \left ( t \mapsto t \log{\left (x + \frac{3 t a + c}{d} \right )} \right )\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b*(d*x+c)**3),x)

[Out]

RootSum(27*_t**3*a**2*b - 1, Lambda(_t, _t*log(x + (3*_t*a + c)/d)))/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((d*x + c)^3*b + a),x, algorithm="giac")

[Out]

integrate(1/((d*x + c)^3*b + a), x)