3.2888 \(\int \frac{(c e+d e x)^3}{\left (a+b (c+d x)^3\right )^2} \, dx\)

Optimal. Leaf size=182 \[ \frac{e^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{2/3} b^{4/3} d}-\frac{e^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{2/3} b^{4/3} d}-\frac{e^3 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3} b^{4/3} d}-\frac{e^3 (c+d x)}{3 b d \left (a+b (c+d x)^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.351053, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{e^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{2/3} b^{4/3} d}-\frac{e^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{2/3} b^{4/3} d}-\frac{e^3 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3} b^{4/3} d}-\frac{e^3 (c+d x)}{3 b d \left (a+b (c+d x)^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 41.5447, size = 168, normalized size = 0.92 \[ - \frac{e^{3} \left (c + d x\right )}{3 b d \left (a + b \left (c + d x\right )^{3}\right )} + \frac{e^{3} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{9 a^{\frac{2}{3}} b^{\frac{4}{3}} d} - \frac{e^{3} \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{18 a^{\frac{2}{3}} b^{\frac{4}{3}} d} - \frac{\sqrt{3} e^{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{2}{3}} b^{\frac{4}{3}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.124237, size = 154, normalized size = 0.85 \[ \frac{e^3 \left (-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{a^{2/3}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{a^{2/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{2/3}}-\frac{6 \sqrt [3]{b} (c+d x)}{a+b (c+d x)^3}\right )}{18 b^{4/3} d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.007, size = 166, normalized size = 0.9 \[ -{\frac{{e}^{3}x}{ \left ( 3\,b{d}^{3}{x}^{3}+9\,bc{d}^{2}{x}^{2}+9\,b{c}^{2}dx+3\,b{c}^{3}+3\,a \right ) b}}-{\frac{{e}^{3}c}{ \left ( 3\,b{d}^{3}{x}^{3}+9\,bc{d}^{2}{x}^{2}+9\,b{c}^{2}dx+3\,b{c}^{3}+3\,a \right ) db}}+{\frac{{e}^{3}}{9\,{b}^{2}d}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*e^3*integrate(1/(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a), x)/b
- 1/3*(d*e^3*x + c*e^3)/(b^2*d^4*x^3 + 3*b^2*c*d^3*x^2 + 3*b^2*c^2*d^2*x + (b^2*
c^3 + a*b)*d)

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Fricas [A]  time = 0.221119, size = 447, normalized size = 2.46 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left (b d^{3} e^{3} x^{3} + 3 \, b c d^{2} e^{3} x^{2} + 3 \, b c^{2} d e^{3} x +{\left (b c^{3} + a\right )} e^{3}\right )} \log \left (a^{2} +{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \left (a^{2} b\right )^{\frac{2}{3}} - \left (a^{2} b\right )^{\frac{1}{3}}{\left (a d x + a c\right )}\right ) - 2 \, \sqrt{3}{\left (b d^{3} e^{3} x^{3} + 3 \, b c d^{2} e^{3} x^{2} + 3 \, b c^{2} d e^{3} x +{\left (b c^{3} + a\right )} e^{3}\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}}{\left (d x + c\right )} + a\right ) - 6 \,{\left (b d^{3} e^{3} x^{3} + 3 \, b c d^{2} e^{3} x^{2} + 3 \, b c^{2} d e^{3} x +{\left (b c^{3} + a\right )} e^{3}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}}{\left (d x + c\right )} - \sqrt{3} a}{3 \, a}\right ) + 6 \, \sqrt{3}{\left (d e^{3} x + c e^{3}\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{54 \,{\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x +{\left (b^{2} c^{3} + a b\right )} d\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/54*sqrt(3)*(sqrt(3)*(b*d^3*e^3*x^3 + 3*b*c*d^2*e^3*x^2 + 3*b*c^2*d*e^3*x + (b
*c^3 + a)*e^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)*(b*d^3*e^3*x^3 + 3*b*c*d^2*e^3*x^2 + 3*b*c^2*d*e^3*x
+ (b*c^3 + a)*e^3)*log((a^2*b)^(1/3)*(d*x + c) + a) - 6*(b*d^3*e^3*x^3 + 3*b*c*
d^2*e^3*x^2 + 3*b*c^2*d*e^3*x + (b*c^3 + a)*e^3)*arctan(1/3*(2*sqrt(3)*(a^2*b)^(
1/3)*(d*x + c) - sqrt(3)*a)/a) + 6*sqrt(3)*(d*e^3*x + c*e^3)*(a^2*b)^(1/3))/((b^
2*d^4*x^3 + 3*b^2*c*d^3*x^2 + 3*b^2*c^2*d^2*x + (b^2*c^3 + a*b)*d)*(a^2*b)^(1/3)
)

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Sympy [A]  time = 8.93972, size = 110, normalized size = 0.6 \[ - \frac{c e^{3} + d e^{3} x}{3 a b d + 3 b^{2} c^{3} d + 9 b^{2} c^{2} d^{2} x + 9 b^{2} c d^{3} x^{2} + 3 b^{2} d^{4} x^{3}} + \frac{e^{3} \operatorname{RootSum}{\left (729 t^{3} a^{2} b^{4} - 1, \left ( t \mapsto t \log{\left (x + \frac{9 t a b e^{3} + c e^{3}}{d e^{3}} \right )} \right )\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(c*e**3 + d*e**3*x)/(3*a*b*d + 3*b**2*c**3*d + 9*b**2*c**2*d**2*x + 9*b**2*c*d*
*3*x**2 + 3*b**2*d**4*x**3) + e**3*RootSum(729*_t**3*a**2*b**4 - 1, Lambda(_t, _
t*log(x + (9*_t*a*b*e**3 + c*e**3)/(d*e**3))))/d

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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