3.2855 \(\int \frac{c+d x}{a+b (c+d x)^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.263957, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 \sqrt [3]{a} b^{2/3} d}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3} d}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} b^{2/3} d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 31.8789, size = 134, normalized size = 0.96 \[ - \frac{\log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{3 \sqrt [3]{a} b^{\frac{2}{3}} d} + \frac{\log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{6 \sqrt [3]{a} b^{\frac{2}{3}} d} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{3 \sqrt [3]{a} b^{\frac{2}{3}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0235431, size = 114, normalized size = 0.81 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{6 \sqrt [3]{a} b^{2/3} d} \]

Antiderivative was successfully verified.

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

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Maple [C]  time = 0.002, size = 76, normalized size = 0.5 \[{\frac{1}{3\,bd}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{ \left ({\it \_R}\,d+c \right ) \ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)/(a+b*(d*x+c)^3),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.215159, size = 177, normalized size = 1.26 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}}{\left (d x + c\right )}\right ) - \sqrt{3} \log \left (-a b + \left (-a b^{2}\right )^{\frac{2}{3}}{\left (d x + c\right )} +{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right ) - 6 \, \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}}{\left (d x + c\right )}}{3 \, a b}\right )\right )}}{18 \, \left (-a b^{2}\right )^{\frac{1}{3}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.719889, size = 29, normalized size = 0.21 \[ \frac{\operatorname{RootSum}{\left (27 t^{3} a b^{2} + 1, \left ( t \mapsto t \log{\left (x + \frac{9 t^{2} a b + c}{d} \right )} \right )\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(27*_t**3*a*b**2 + 1, Lambda(_t, _t*log(x + (9*_t**2*a*b + c)/d)))/d

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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