3.2890 \(\int \frac{c e+d e x}{\left (a+b (c+d x)^3\right )^2} \, dx\)

Optimal. Leaf size=176 \[ -\frac{e \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{4/3} b^{2/3} d}+\frac{e \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{4/3} b^{2/3} d}-\frac{e \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{4/3} b^{2/3} d}+\frac{e (c+d x)^2}{3 a d \left (a+b (c+d x)^3\right )} \]

[Out]

(e*(c + d*x)^2)/(3*a*d*(a + b*(c + d*x)^3)) - (e*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) - (e*Log[a^(1/3) + b^(
1/3)*(c + d*x)])/(9*a^(4/3)*b^(2/3)*d) + (e*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 = 0.350857, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{e \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{4/3} b^{2/3} d}+\frac{e \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{4/3} b^{2/3} d}-\frac{e \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{4/3} b^{2/3} d}+\frac{e (c+d x)^2}{3 a d \left (a+b (c+d x)^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(e*(c + d*x)^2)/(3*a*d*(a + b*(c + d*x)^3)) - (e*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) - (e*Log[a^(1/3) + b^(
1/3)*(c + d*x)])/(9*a^(4/3)*b^(2/3)*d) + (e*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 = 39.9313, size = 163, normalized size = 0.93 \[ \frac{e \left (c + d x\right )^{2}}{3 a d \left (a + b \left (c + d x\right )^{3}\right )} - \frac{e \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{9 a^{\frac{4}{3}} b^{\frac{2}{3}} d} + \frac{e \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{18 a^{\frac{4}{3}} b^{\frac{2}{3}} d} - \frac{\sqrt{3} e \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{4}{3}} b^{\frac{2}{3}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*(c + d*x)**2/(3*a*d*(a + b*(c + d*x)**3)) - e*log(a**(1/3) + b**(1/3)*(c + d*x
))/(9*a**(4/3)*b**(2/3)*d) + e*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)*e*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 = 0.155548, size = 153, normalized size = 0.87 \[ \frac{e \left (\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{b^{2/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{b^{2/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{b^{2/3}}+\frac{6 \sqrt [3]{a} (c+d x)^2}{a+b (c+d x)^3}\right )}{18 a^{4/3} d} \]

Antiderivative was successfully verified.

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

[Out]

(e*((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 = 0.007, size = 216, normalized size = 1.2 \[{\frac{de{x}^{2}}{ \left ( 3\,b{d}^{3}{x}^{3}+9\,bc{d}^{2}{x}^{2}+9\,b{c}^{2}dx+3\,b{c}^{3}+3\,a \right ) a}}+{\frac{2\,cex}{ \left ( 3\,b{d}^{3}{x}^{3}+9\,bc{d}^{2}{x}^{2}+9\,b{c}^{2}dx+3\,b{c}^{3}+3\,a \right ) a}}+{\frac{e{c}^{2}}{ \left ( 3\,b{d}^{3}{x}^{3}+9\,bc{d}^{2}{x}^{2}+9\,b{c}^{2}dx+3\,b{c}^{3}+3\,a \right ) da}}+{\frac{e}{9\,abd}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{ \left ({\it \_R}\,d+c \right ) \ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*e/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)*d/a*x^2+2/3*e/(b*d^3*x^3+3*b
*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)/a*c*x+1/3*e/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x
+b*c^3+a)*c^2/d/a+1/9*e/a/b/d*sum((_R*d+c)/(_R^2*d^2+2*_R*c*d+c^2)*ln(x-_R),_R=R
ootOf(_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. \[ \frac{e \int \frac{d x + c}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}\,{d x}}{3 \, a} + \frac{d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{3 \,{\left (a b d^{4} x^{3} + 3 \, a b c d^{3} x^{2} + 3 \, a b c^{2} d^{2} x +{\left (a b c^{3} + a^{2}\right )} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*e*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 + 1/3*(d^2*e*x^2 + 2*c*d*e*x + c^2*e)/(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)

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Fricas [A]  time = 0.22132, size = 437, normalized size = 2.48 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (b d^{3} e x^{3} + 3 \, b c d^{2} e x^{2} + 3 \, b c^{2} d e x +{\left (b c^{3} + a\right )} e\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}}{\left (d x + c\right )}\right ) - \sqrt{3}{\left (b d^{3} e x^{3} + 3 \, b c d^{2} e x^{2} + 3 \, b c^{2} d e x +{\left (b c^{3} + a\right )} e\right )} \log \left (-a b + \left (-a b^{2}\right )^{\frac{2}{3}}{\left (d x + c\right )} +{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right ) - 6 \,{\left (b d^{3} e x^{3} + 3 \, b c d^{2} e x^{2} + 3 \, b c^{2} d e x +{\left (b c^{3} + a\right )} e\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}}{\left (d x + c\right )}}{3 \, a b}\right ) + 6 \, \sqrt{3}{\left (d^{2} e x^{2} + 2 \, c d e x + c^{2} e\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}}{54 \,{\left (a b d^{4} x^{3} + 3 \, a b c d^{3} x^{2} + 3 \, a b c^{2} d^{2} x +{\left (a b c^{3} + a^{2}\right )} d\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/54*sqrt(3)*(2*sqrt(3)*(b*d^3*e*x^3 + 3*b*c*d^2*e*x^2 + 3*b*c^2*d*e*x + (b*c^3
+ a)*e)*log(a*b + (-a*b^2)^(2/3)*(d*x + c)) - sqrt(3)*(b*d^3*e*x^3 + 3*b*c*d^2*e
*x^2 + 3*b*c^2*d*e*x + (b*c^3 + a)*e)*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*e*x^3 + 3*b*c*d^2*e*x^2 +
3*b*c^2*d*e*x + (b*c^3 + a)*e)*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*e*x^2 + 2*c*d*e*x + c^2*e)*(-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 = 6.6052, size = 122, normalized size = 0.69 \[ \frac{c^{2} e + 2 c d e x + d^{2} e x^{2}}{3 a^{2} d + 3 a b c^{3} d + 9 a b c^{2} d^{2} x + 9 a b c d^{3} x^{2} + 3 a b d^{4} x^{3}} + \frac{e \operatorname{RootSum}{\left (729 t^{3} a^{4} b^{2} + 1, \left ( t \mapsto t \log{\left (x + \frac{81 t^{2} a^{3} b e^{2} + c e^{2}}{d e^{2}} \right )} \right )\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c**2*e + 2*c*d*e*x + d**2*e*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) + e*RootSum(729*_t**3*a**4*b**2 + 1, Lamb
da(_t, _t*log(x + (81*_t**2*a**3*b*e**2 + c*e**2)/(d*e**2))))/d

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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