3.2864 \(\int \frac{c+d x}{\left (a+b (c+d x)^3\right )^2} \, dx\)

Optimal. Leaf size=172 \[ -\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{4/3} b^{2/3} d}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{4/3} b^{2/3} d}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{4/3} b^{2/3} d}+\frac{(c+d x)^2}{3 a d \left (a+b (c+d x)^3\right )} \]

[Out]

(c + d*x)^2/(3*a*d*(a + b*(c + d*x)^3)) - ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))
/(Sqrt[3]*a^(1/3))]/(3*Sqrt[3]*a^(4/3)*b^(2/3)*d) - Log[a^(1/3) + b^(1/3)*(c + d
*x)]/(9*a^(4/3)*b^(2/3)*d) + Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(
c + d*x)^2]/(18*a^(4/3)*b^(2/3)*d)

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Rubi [A]  time = 0.314308, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{4/3} b^{2/3} d}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{4/3} b^{2/3} d}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{4/3} b^{2/3} d}+\frac{(c+d x)^2}{3 a d \left (a+b (c+d x)^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(c + d*x)^2/(3*a*d*(a + b*(c + d*x)^3)) - ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))
/(Sqrt[3]*a^(1/3))]/(3*Sqrt[3]*a^(4/3)*b^(2/3)*d) - Log[a^(1/3) + b^(1/3)*(c + d
*x)]/(9*a^(4/3)*b^(2/3)*d) + Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(
c + d*x)^2]/(18*a^(4/3)*b^(2/3)*d)

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Rubi in Sympy [A]  time = 37.9802, size = 156, normalized size = 0.91 \[ \frac{\left (c + d x\right )^{2}}{3 a d \left (a + b \left (c + d x\right )^{3}\right )} - \frac{\log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{9 a^{\frac{4}{3}} b^{\frac{2}{3}} d} + \frac{\log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{18 a^{\frac{4}{3}} b^{\frac{2}{3}} d} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{4}{3}} b^{\frac{2}{3}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c + d*x)**2/(3*a*d*(a + b*(c + d*x)**3)) - log(a**(1/3) + b**(1/3)*(c + d*x))/(
9*a**(4/3)*b**(2/3)*d) + log(a**(2/3) + a**(1/3)*b**(1/3)*(-c - d*x) + b**(2/3)*
(c + d*x)**2)/(18*a**(4/3)*b**(2/3)*d) - sqrt(3)*atan(sqrt(3)*(a**(1/3)/3 + b**(
1/3)*(-2*c/3 - 2*d*x/3))/a**(1/3))/(9*a**(4/3)*b**(2/3)*d)

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Mathematica [A]  time = 0.150289, size = 152, normalized size = 0.88 \[ \frac{\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{b^{2/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{b^{2/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{b^{2/3}}+\frac{6 \sqrt [3]{a} (c+d x)^2}{a+b (c+d x)^3}}{18 a^{4/3} d} \]

Antiderivative was successfully verified.

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

[Out]

((6*a^(1/3)*(c + d*x)^2)/(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^(2/3) - (2*Log[a^(1/3) + b^(1/3)*(c + d*
x)])/b^(2/3) + Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2]/b^
(2/3))/(18*a^(4/3)*d)

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Maple [C]  time = 0.016, size = 144, normalized size = 0.8 \[{\frac{1}{b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a} \left ({\frac{d{x}^{2}}{3\,a}}+{\frac{2\,cx}{3\,a}}+{\frac{{c}^{2}}{3\,ad}} \right ) }+{\frac{1}{9\,abd}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{ \left ({\it \_R}\,d+c \right ) \ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(d^2*x^2 + 2*c*d*x + c^2)/(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) + 1/3*integrate((d*x + c)/(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 = 0.219591, size = 408, normalized size = 2.37 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}}{\left (d x + c\right )}\right ) - \sqrt{3}{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} \log \left (-a b + \left (-a b^{2}\right )^{\frac{2}{3}}{\left (d x + c\right )} +{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right ) - 6 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}}{\left (d x + c\right )}}{3 \, a b}\right ) + 6 \, \sqrt{3}{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}}{54 \,{\left (a b d^{4} x^{3} + 3 \, a b c d^{3} x^{2} + 3 \, a b c^{2} d^{2} x +{\left (a b c^{3} + a^{2}\right )} d\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/54*sqrt(3)*(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)*lo
g(a*b + (-a*b^2)^(2/3)*(d*x + c)) - 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*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*(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a
)*arctan(-1/3*(sqrt(3)*a*b - 2*sqrt(3)*(-a*b^2)^(2/3)*(d*x + c))/(a*b)) + 6*sqrt
(3)*(d^2*x^2 + 2*c*d*x + c^2)*(-a*b^2)^(1/3))/((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*b^2)^(1/3))

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Sympy [A]  time = 6.10617, size = 105, normalized size = 0.61 \[ \frac{c^{2} + 2 c d x + d^{2} x^{2}}{3 a^{2} d + 3 a b c^{3} d + 9 a b c^{2} d^{2} x + 9 a b c d^{3} x^{2} + 3 a b d^{4} x^{3}} + \frac{\operatorname{RootSum}{\left (729 t^{3} a^{4} b^{2} + 1, \left ( t \mapsto t \log{\left (x + \frac{81 t^{2} a^{3} b + c}{d} \right )} \right )\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c**2 + 2*c*d*x + d**2*x**2)/(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**4*b**2 + 1, Lambda(_t, _
t*log(x + (81*_t**2*a**3*b + c)/d)))/d

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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