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

Optimal. Leaf size=224 \[ -\frac{\left (2 \sqrt [3]{a}-\sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} c+b^{2/3} c^2\right )}+\frac{\sqrt [3]{b} c \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} c+b^{2/3} c^2\right )}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} c\right )}+\frac{\log (x)}{a+b c^3} \]

[Out]

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

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Rubi [A]  time = 0.609398, antiderivative size = 238, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529 \[ -\frac{\sqrt [3]{b} c \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} \left (a+b c^3\right )}+\frac{\sqrt [3]{b} c \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \left (a+b c^3\right )}+\frac{\sqrt [3]{b} c \left (\sqrt [3]{a}+\sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \left (a+b c^3\right )}-\frac{\log \left (a+b (c+d x)^3\right )}{3 \left (a+b c^3\right )}+\frac{\log (x)}{a+b c^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 74.278, size = 219, normalized size = 0.98 \[ \frac{\log{\left (- d x \right )}}{a + b c^{3}} - \frac{\log{\left (a + b \left (c + d x\right )^{3} \right )}}{3 \left (a + b c^{3}\right )} + \frac{\sqrt [3]{b} c \left (\sqrt [3]{a} - \sqrt [3]{b} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{3 a^{\frac{2}{3}} \left (a + b c^{3}\right )} - \frac{\sqrt [3]{b} c \left (\sqrt [3]{a} - \sqrt [3]{b} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} \left (c + d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{6 a^{\frac{2}{3}} \left (a + b c^{3}\right )} + \frac{\sqrt{3} \sqrt [3]{b} c \left (\sqrt [3]{a} + \sqrt [3]{b} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \left (c + d x\right )}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} \left (a + b c^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0765582, size = 119, normalized size = 0.53 \[ -\frac{\text{RootSum}\left [\text{$\#$1}^3 b d^3+3 \text{$\#$1}^2 b c d^2+3 \text{$\#$1} b c^2 d+a+b c^3\&,\frac{\text{$\#$1}^2 d^2 \log (x-\text{$\#$1})+3 c^2 \log (x-\text{$\#$1})+3 \text{$\#$1} c d \log (x-\text{$\#$1})}{\text{$\#$1}^2 d^2+2 \text{$\#$1} c d+c^2}\&\right ]-3 \log (x)}{3 \left (a+b c^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(-3*Log[x] + RootSum[a + b*c^3 + 3*b*c^2*d*#1 + 3*b*c*d^2*#1^2 + b*d^3*#1^3 & ,
(3*c^2*Log[x - #1] + 3*c*d*Log[x - #1]*#1 + d^2*Log[x - #1]*#1^2)/(c^2 + 2*c*d*
#1 + d^2*#1^2) & ])/(3*(a + b*c^3))

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Maple [C]  time = 0.009, size = 105, normalized size = 0.5 \[{\frac{\ln \left ( x \right ) }{b{c}^{3}+a}}-{\frac{1}{3\,b{c}^{3}+3\,a}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{ \left ({d}^{2}{{\it \_R}}^{2}+3\,cd{\it \_R}+3\,{c}^{2} \right ) \ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b*(d*x+c)^3),x)

[Out]

ln(x)/(b*c^3+a)-1/3/(b*c^3+a)*sum((_R^2*d^2+3*_R*c*d+3*c^2)/(_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. \[ -\frac{b d \int \frac{d^{2} x^{2} + 3 \, c d x + 3 \, c^{2}}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}\,{d x}}{b c^{3} + a} + \frac{\log \left (x\right )}{b c^{3} + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 52.1419, size = 559, normalized size = 2.5 \[ \operatorname{RootSum}{\left (t^{3} \left (27 a^{3} + 27 a^{2} b c^{3}\right ) + 27 t^{2} a^{2} + 9 t a + 1, \left ( t \mapsto t \log{\left (x + \frac{- 432 t^{3} a^{6} - 837 t^{3} a^{5} b c^{3} - 405 t^{3} a^{4} b^{2} c^{6} - 27 t^{3} a^{3} b^{3} c^{9} - 27 t^{3} a^{2} b^{4} c^{12} + 144 t^{2} a^{5} + 270 t^{2} a^{4} b c^{3} + 108 t^{2} a^{3} b^{2} c^{6} - 18 t^{2} a^{2} b^{3} c^{9} + 240 t a^{4} - 261 t a^{3} b c^{3} - 27 t a^{2} b^{2} c^{6} - 12 t a b^{3} c^{9} + 48 a^{3} + 60 a^{2} b c^{3} + 12 a b^{2} c^{6}}{64 a^{2} b c^{2} d + 11 a b^{2} c^{5} d + b^{3} c^{8} d} \right )} \right )\right )} + \frac{\log{\left (x + \frac{- \frac{432 a^{6}}{\left (a + b c^{3}\right )^{3}} - \frac{837 a^{5} b c^{3}}{\left (a + b c^{3}\right )^{3}} + \frac{144 a^{5}}{\left (a + b c^{3}\right )^{2}} - \frac{405 a^{4} b^{2} c^{6}}{\left (a + b c^{3}\right )^{3}} + \frac{270 a^{4} b c^{3}}{\left (a + b c^{3}\right )^{2}} + \frac{240 a^{4}}{a + b c^{3}} - \frac{27 a^{3} b^{3} c^{9}}{\left (a + b c^{3}\right )^{3}} + \frac{108 a^{3} b^{2} c^{6}}{\left (a + b c^{3}\right )^{2}} - \frac{261 a^{3} b c^{3}}{a + b c^{3}} + 48 a^{3} - \frac{27 a^{2} b^{4} c^{12}}{\left (a + b c^{3}\right )^{3}} - \frac{18 a^{2} b^{3} c^{9}}{\left (a + b c^{3}\right )^{2}} - \frac{27 a^{2} b^{2} c^{6}}{a + b c^{3}} + 60 a^{2} b c^{3} - \frac{12 a b^{3} c^{9}}{a + b c^{3}} + 12 a b^{2} c^{6}}{64 a^{2} b c^{2} d + 11 a b^{2} c^{5} d + b^{3} c^{8} d} \right )}}{a + b c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(_t**3*(27*a**3 + 27*a**2*b*c**3) + 27*_t**2*a**2 + 9*_t*a + 1, Lambda(_t
, _t*log(x + (-432*_t**3*a**6 - 837*_t**3*a**5*b*c**3 - 405*_t**3*a**4*b**2*c**6
- 27*_t**3*a**3*b**3*c**9 - 27*_t**3*a**2*b**4*c**12 + 144*_t**2*a**5 + 270*_t*
*2*a**4*b*c**3 + 108*_t**2*a**3*b**2*c**6 - 18*_t**2*a**2*b**3*c**9 + 240*_t*a**
4 - 261*_t*a**3*b*c**3 - 27*_t*a**2*b**2*c**6 - 12*_t*a*b**3*c**9 + 48*a**3 + 60
*a**2*b*c**3 + 12*a*b**2*c**6)/(64*a**2*b*c**2*d + 11*a*b**2*c**5*d + b**3*c**8*
d)))) + log(x + (-432*a**6/(a + b*c**3)**3 - 837*a**5*b*c**3/(a + b*c**3)**3 + 1
44*a**5/(a + b*c**3)**2 - 405*a**4*b**2*c**6/(a + b*c**3)**3 + 270*a**4*b*c**3/(
a + b*c**3)**2 + 240*a**4/(a + b*c**3) - 27*a**3*b**3*c**9/(a + b*c**3)**3 + 108
*a**3*b**2*c**6/(a + b*c**3)**2 - 261*a**3*b*c**3/(a + b*c**3) + 48*a**3 - 27*a*
*2*b**4*c**12/(a + b*c**3)**3 - 18*a**2*b**3*c**9/(a + b*c**3)**2 - 27*a**2*b**2
*c**6/(a + b*c**3) + 60*a**2*b*c**3 - 12*a*b**3*c**9/(a + b*c**3) + 12*a*b**2*c*
*6)/(64*a**2*b*c**2*d + 11*a*b**2*c**5*d + b**3*c**8*d))/(a + b*c**3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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