3.2896 \(\int \frac{(c e+d e x)^3}{\left (a+b (c+d x)^3\right )^3} \, dx\)

Optimal. Leaf size=216 \[ \frac{e^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{5/3} b^{4/3} d}-\frac{e^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{5/3} b^{4/3} d}-\frac{e^3 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{5/3} b^{4/3} d}+\frac{e^3 (c+d x)}{18 a b d \left (a+b (c+d x)^3\right )}-\frac{e^3 (c+d x)}{6 b d \left (a+b (c+d x)^3\right )^2} \]

[Out]

-(e^3*(c + d*x))/(6*b*d*(a + b*(c + d*x)^3)^2) + (e^3*(c + d*x))/(18*a*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))]
)/(9*Sqrt[3]*a^(5/3)*b^(4/3)*d) + (e^3*Log[a^(1/3) + b^(1/3)*(c + d*x)])/(27*a^(
5/3)*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])/(54*a^(5/3)*b^(4/3)*d)

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Rubi [A]  time = 0.420451, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{e^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{5/3} b^{4/3} d}-\frac{e^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{5/3} b^{4/3} d}-\frac{e^3 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{5/3} b^{4/3} d}+\frac{e^3 (c+d x)}{18 a b d \left (a+b (c+d x)^3\right )}-\frac{e^3 (c+d x)}{6 b d \left (a+b (c+d x)^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(e^3*(c + d*x))/(6*b*d*(a + b*(c + d*x)^3)^2) + (e^3*(c + d*x))/(18*a*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))]
)/(9*Sqrt[3]*a^(5/3)*b^(4/3)*d) + (e^3*Log[a^(1/3) + b^(1/3)*(c + d*x)])/(27*a^(
5/3)*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])/(54*a^(5/3)*b^(4/3)*d)

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Rubi in Sympy [A]  time = 48.57, size = 196, normalized size = 0.91 \[ - \frac{e^{3} \left (c + d x\right )}{6 b d \left (a + b \left (c + d x\right )^{3}\right )^{2}} + \frac{e^{3} \left (c + d x\right )}{18 a b d \left (a + b \left (c + d x\right )^{3}\right )} + \frac{e^{3} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{27 a^{\frac{5}{3}} b^{\frac{4}{3}} d} - \frac{e^{3} \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{54 a^{\frac{5}{3}} b^{\frac{4}{3}} d} - \frac{\sqrt{3} e^{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{5}{3}} b^{\frac{4}{3}} d} \]

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)**3,x)

[Out]

-e**3*(c + d*x)/(6*b*d*(a + b*(c + d*x)**3)**2) + e**3*(c + d*x)/(18*a*b*d*(a +
b*(c + d*x)**3)) + e**3*log(a**(1/3) + b**(1/3)*(c + d*x))/(27*a**(5/3)*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)/
(54*a**(5/3)*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))/(27*a**(5/3)*b**(4/3)*d)

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Mathematica [A]  time = 0.230917, size = 182, normalized size = 0.84 \[ \frac{e^3 \left (-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{a^{5/3}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{a^{5/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{5/3}}+\frac{3 \sqrt [3]{b} (c+d x)}{a \left (a+b (c+d x)^3\right )}-\frac{9 \sqrt [3]{b} (c+d x)}{\left (a+b (c+d x)^3\right )^2}\right )}{54 b^{4/3} d} \]

Antiderivative was successfully verified.

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

[Out]

(e^3*((-9*b^(1/3)*(c + d*x))/(a + b*(c + d*x)^3)^2 + (3*b^(1/3)*(c + d*x))/(a*(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^(5/3) + (2*Log[a^(1/3) + b^(1/3)*(c + d*x)])/a^(5/3) - Log[a^(2/3)
- a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2]/a^(5/3)))/(54*b^(4/3)*d)

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Maple [C]  time = 0.008, size = 414, normalized size = 1.9 \[{\frac{{e}^{3}{d}^{3}{x}^{4}}{18\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}a}}+{\frac{2\,{e}^{3}c{d}^{2}{x}^{3}}{9\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}a}}+{\frac{{e}^{3}{c}^{2}d{x}^{2}}{3\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}a}}+{\frac{2\,{e}^{3}x{c}^{3}}{9\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}a}}-{\frac{{e}^{3}x}{9\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}b}}+{\frac{{e}^{3}{c}^{4}}{18\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}da}}-{\frac{{e}^{3}c}{9\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}db}}+{\frac{{e}^{3}}{27\,a{b}^{2}d}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*e^3/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*d^3/a*x^4+2/9*e^3/(b*d^
3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*c*d^2/a*x^3+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)^2*c^2*d/a*x^2+2/9*e^3/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*
c^2*d*x+b*c^3+a)^2/a*x*c^3-1/9*e^3/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)
^2/b*x+1/18*e^3/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*c^4/d/a-1/9*e^3/
(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*c/d/b+1/27*e^3/b^2/a/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. \[ \frac{e^{3} \int \frac{1}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}\,{d x}}{9 \, a b} + \frac{b d^{4} e^{3} x^{4} + 4 \, b c d^{3} e^{3} x^{3} + 6 \, b c^{2} d^{2} e^{3} x^{2} + 2 \,{\left (2 \, b c^{3} - a\right )} d e^{3} x +{\left (b c^{4} - 2 \, a c\right )} e^{3}}{18 \,{\left (a b^{3} d^{7} x^{6} + 6 \, a b^{3} c d^{6} x^{5} + 15 \, a b^{3} c^{2} d^{5} x^{4} + 2 \,{\left (10 \, a b^{3} c^{3} + a^{2} b^{2}\right )} d^{4} x^{3} + 3 \,{\left (5 \, a b^{3} c^{4} + 2 \, a^{2} b^{2} c\right )} d^{3} x^{2} + 6 \,{\left (a b^{3} c^{5} + a^{2} b^{2} c^{2}\right )} d^{2} x +{\left (a b^{3} c^{6} + 2 \, a^{2} b^{2} c^{3} + a^{3} b\right )} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*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)/(a
*b) + 1/18*(b*d^4*e^3*x^4 + 4*b*c*d^3*e^3*x^3 + 6*b*c^2*d^2*e^3*x^2 + 2*(2*b*c^3
- a)*d*e^3*x + (b*c^4 - 2*a*c)*e^3)/(a*b^3*d^7*x^6 + 6*a*b^3*c*d^6*x^5 + 15*a*b
^3*c^2*d^5*x^4 + 2*(10*a*b^3*c^3 + a^2*b^2)*d^4*x^3 + 3*(5*a*b^3*c^4 + 2*a^2*b^2
*c)*d^3*x^2 + 6*(a*b^3*c^5 + a^2*b^2*c^2)*d^2*x + (a*b^3*c^6 + 2*a^2*b^2*c^3 + a
^3*b)*d)

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Fricas [A]  time = 0.245218, size = 1021, normalized size = 4.73 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/162*sqrt(3)*(sqrt(3)*(b^2*d^6*e^3*x^6 + 6*b^2*c*d^5*e^3*x^5 + 15*b^2*c^2*d^4*
e^3*x^4 + 2*(10*b^2*c^3 + a*b)*d^3*e^3*x^3 + 3*(5*b^2*c^4 + 2*a*b*c)*d^2*e^3*x^2
+ 6*(b^2*c^5 + a*b*c^2)*d*e^3*x + (b^2*c^6 + 2*a*b*c^3 + a^2)*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^2*d^6*e^3*x^6 + 6*b^2*c*d^5*e^3*x^5 + 15*b^2*c^2*d^4*e^3*x^4 + 2*(10*b^2*c^3
+ a*b)*d^3*e^3*x^3 + 3*(5*b^2*c^4 + 2*a*b*c)*d^2*e^3*x^2 + 6*(b^2*c^5 + a*b*c^2
)*d*e^3*x + (b^2*c^6 + 2*a*b*c^3 + a^2)*e^3)*log((a^2*b)^(1/3)*(d*x + c) + a) -
6*(b^2*d^6*e^3*x^6 + 6*b^2*c*d^5*e^3*x^5 + 15*b^2*c^2*d^4*e^3*x^4 + 2*(10*b^2*c^
3 + a*b)*d^3*e^3*x^3 + 3*(5*b^2*c^4 + 2*a*b*c)*d^2*e^3*x^2 + 6*(b^2*c^5 + a*b*c^
2)*d*e^3*x + (b^2*c^6 + 2*a*b*c^3 + a^2)*e^3)*arctan(1/3*(2*sqrt(3)*(a^2*b)^(1/3
)*(d*x + c) - sqrt(3)*a)/a) - 3*sqrt(3)*(b*d^4*e^3*x^4 + 4*b*c*d^3*e^3*x^3 + 6*b
*c^2*d^2*e^3*x^2 + 2*(2*b*c^3 - a)*d*e^3*x + (b*c^4 - 2*a*c)*e^3)*(a^2*b)^(1/3))
/((a*b^3*d^7*x^6 + 6*a*b^3*c*d^6*x^5 + 15*a*b^3*c^2*d^5*x^4 + 2*(10*a*b^3*c^3 +
a^2*b^2)*d^4*x^3 + 3*(5*a*b^3*c^4 + 2*a^2*b^2*c)*d^3*x^2 + 6*(a*b^3*c^5 + a^2*b^
2*c^2)*d^2*x + (a*b^3*c^6 + 2*a^2*b^2*c^3 + a^3*b)*d)*(a^2*b)^(1/3))

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Sympy [A]  time = 99.3193, size = 298, normalized size = 1.38 \[ \frac{- 2 a c e^{3} + b c^{4} e^{3} + 6 b c^{2} d^{2} e^{3} x^{2} + 4 b c d^{3} e^{3} x^{3} + b d^{4} e^{3} x^{4} + x \left (- 2 a d e^{3} + 4 b c^{3} d e^{3}\right )}{18 a^{3} b d + 36 a^{2} b^{2} c^{3} d + 18 a b^{3} c^{6} d + 270 a b^{3} c^{2} d^{5} x^{4} + 108 a b^{3} c d^{6} x^{5} + 18 a b^{3} d^{7} x^{6} + x^{3} \left (36 a^{2} b^{2} d^{4} + 360 a b^{3} c^{3} d^{4}\right ) + x^{2} \left (108 a^{2} b^{2} c d^{3} + 270 a b^{3} c^{4} d^{3}\right ) + x \left (108 a^{2} b^{2} c^{2} d^{2} + 108 a b^{3} c^{5} d^{2}\right )} + \frac{e^{3} \operatorname{RootSum}{\left (19683 t^{3} a^{5} b^{4} - 1, \left ( t \mapsto t \log{\left (x + \frac{27 t a^{2} b e^{3} + c e^{3}}{d e^{3}} \right )} \right )\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*a*c*e**3 + b*c**4*e**3 + 6*b*c**2*d**2*e**3*x**2 + 4*b*c*d**3*e**3*x**3 + b*
d**4*e**3*x**4 + x*(-2*a*d*e**3 + 4*b*c**3*d*e**3))/(18*a**3*b*d + 36*a**2*b**2*
c**3*d + 18*a*b**3*c**6*d + 270*a*b**3*c**2*d**5*x**4 + 108*a*b**3*c*d**6*x**5 +
18*a*b**3*d**7*x**6 + x**3*(36*a**2*b**2*d**4 + 360*a*b**3*c**3*d**4) + x**2*(1
08*a**2*b**2*c*d**3 + 270*a*b**3*c**4*d**3) + x*(108*a**2*b**2*c**2*d**2 + 108*a
*b**3*c**5*d**2)) + e**3*RootSum(19683*_t**3*a**5*b**4 - 1, Lambda(_t, _t*log(x
+ (27*_t*a**2*b*e**3 + c*e**3)/(d*e**3))))/d

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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