3.2898 \(\int \frac{c e+d e x}{\left (a+b (c+d x)^3\right )^3} \, dx\)

Optimal. Leaf size=207 \[ -\frac{2 e \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac{e \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{7/3} b^{2/3} d}-\frac{2 e \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} b^{2/3} d}+\frac{2 e (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}+\frac{e (c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2} \]

[Out]

(e*(c + d*x)^2)/(6*a*d*(a + b*(c + d*x)^3)^2) + (2*e*(c + d*x)^2)/(9*a^2*d*(a +
b*(c + d*x)^3)) - (2*e*ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))]
)/(9*Sqrt[3]*a^(7/3)*b^(2/3)*d) - (2*e*Log[a^(1/3) + b^(1/3)*(c + d*x)])/(27*a^(
7/3)*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])/(27*a^(7/3)*b^(2/3)*d)

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Rubi [A]  time = 0.422248, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{2 e \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac{e \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{7/3} b^{2/3} d}-\frac{2 e \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} b^{2/3} d}+\frac{2 e (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}+\frac{e (c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(e*(c + d*x)^2)/(6*a*d*(a + b*(c + d*x)^3)^2) + (2*e*(c + d*x)^2)/(9*a^2*d*(a +
b*(c + d*x)^3)) - (2*e*ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))]
)/(9*Sqrt[3]*a^(7/3)*b^(2/3)*d) - (2*e*Log[a^(1/3) + b^(1/3)*(c + d*x)])/(27*a^(
7/3)*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])/(27*a^(7/3)*b^(2/3)*d)

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Rubi in Sympy [A]  time = 47.1994, size = 196, normalized size = 0.95 \[ \frac{e \left (c + d x\right )^{2}}{6 a d \left (a + b \left (c + d x\right )^{3}\right )^{2}} + \frac{2 e \left (c + d x\right )^{2}}{9 a^{2} d \left (a + b \left (c + d x\right )^{3}\right )} - \frac{2 e \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{27 a^{\frac{7}{3}} b^{\frac{2}{3}} d} + \frac{e \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{27 a^{\frac{7}{3}} b^{\frac{2}{3}} d} - \frac{2 \sqrt{3} e \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{7}{3}} b^{\frac{2}{3}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*(c + d*x)**2/(6*a*d*(a + b*(c + d*x)**3)**2) + 2*e*(c + d*x)**2/(9*a**2*d*(a +
b*(c + d*x)**3)) - 2*e*log(a**(1/3) + b**(1/3)*(c + d*x))/(27*a**(7/3)*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)/(27
*a**(7/3)*b**(2/3)*d) - 2*sqrt(3)*e*atan(sqrt(3)*(a**(1/3)/3 + b**(1/3)*(-2*c/3
- 2*d*x/3))/a**(1/3))/(27*a**(7/3)*b**(2/3)*d)

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Mathematica [A]  time = 0.219268, size = 181, normalized size = 0.87 \[ \frac{e \left (\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{b^{2/3}}+\frac{9 a^{4/3} (c+d x)^2}{\left (a+b (c+d x)^3\right )^2}-\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{b^{2/3}}+\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{b^{2/3}}+\frac{12 \sqrt [3]{a} (c+d x)^2}{a+b (c+d x)^3}\right )}{54 a^{7/3} d} \]

Antiderivative was successfully verified.

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

[Out]

(e*((9*a^(4/3)*(c + d*x)^2)/(a + b*(c + d*x)^3)^2 + (12*a^(1/3)*(c + d*x)^2)/(a
+ b*(c + d*x)^3) + (4*Sqrt[3]*ArcTan[(-a^(1/3) + 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a
^(1/3))])/b^(2/3) - (4*Log[a^(1/3) + b^(1/3)*(c + d*x)])/b^(2/3) + (2*Log[a^(2/3
) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2])/b^(2/3)))/(54*a^(7/3)*d)

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Maple [C]  time = 0.009, size = 507, normalized size = 2.5 \[{\frac{2\,be{d}^{4}{x}^{5}}{9\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}{a}^{2}}}+{\frac{10\,bce{d}^{3}{x}^{4}}{9\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}{a}^{2}}}+{\frac{20\,e{c}^{2}{d}^{2}b{x}^{3}}{9\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}{a}^{2}}}+{\frac{20\,de{x}^{2}b{c}^{3}}{9\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}{a}^{2}}}+{\frac{7\,de{x}^{2}}{18\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}a}}+{\frac{10\,e{c}^{4}xb}{9\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}{a}^{2}}}+{\frac{7\,cex}{9\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}a}}+{\frac{2\,e{c}^{5}b}{9\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}d{a}^{2}}}+{\frac{7\,e{c}^{2}}{18\, \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}da}}+{\frac{2\,e}{27\,{a}^{2}bd}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{ \left ({\it \_R}\,d+c \right ) \ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*e/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*b*d^4/a^2*x^5+10/9*e/(b*d^
3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*b*c*d^3/a^2*x^4+20/9*e/(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^2*b/a^2*x^3+20/9*e/(b*d^3*x^3+3*b*c*d^2*
x^2+3*b*c^2*d*x+b*c^3+a)^2*d/a^2*x^2*b*c^3+7/18*e/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c
^2*d*x+b*c^3+a)^2*d/a*x^2+10/9*e/(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/a^2*x*b+7/9*e/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2/a*c*x+2/9*e/(
b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*c^5/d/a^2*b+7/18*e/(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+2/27*e/a^2/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. \[ \frac{4 \, b d^{5} e x^{5} + 20 \, b c d^{4} e x^{4} + 40 \, b c^{2} d^{3} e x^{3} +{\left (40 \, b c^{3} + 7 \, a\right )} d^{2} e x^{2} + 2 \,{\left (10 \, b c^{4} + 7 \, a c\right )} d e x +{\left (4 \, b c^{5} + 7 \, a c^{2}\right )} e}{18 \,{\left (a^{2} b^{2} d^{7} x^{6} + 6 \, a^{2} b^{2} c d^{6} x^{5} + 15 \, a^{2} b^{2} c^{2} d^{5} x^{4} + 2 \,{\left (10 \, a^{2} b^{2} c^{3} + a^{3} b\right )} d^{4} x^{3} + 3 \,{\left (5 \, a^{2} b^{2} c^{4} + 2 \, a^{3} b c\right )} d^{3} x^{2} + 6 \,{\left (a^{2} b^{2} c^{5} + a^{3} b c^{2}\right )} d^{2} x +{\left (a^{2} b^{2} c^{6} + 2 \, a^{3} b c^{3} + a^{4}\right )} d\right )}} + \frac{2 \, e \int \frac{d x + c}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}\,{d x}}{9 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*(4*b*d^5*e*x^5 + 20*b*c*d^4*e*x^4 + 40*b*c^2*d^3*e*x^3 + (40*b*c^3 + 7*a)*d
^2*e*x^2 + 2*(10*b*c^4 + 7*a*c)*d*e*x + (4*b*c^5 + 7*a*c^2)*e)/(a^2*b^2*d^7*x^6
+ 6*a^2*b^2*c*d^6*x^5 + 15*a^2*b^2*c^2*d^5*x^4 + 2*(10*a^2*b^2*c^3 + a^3*b)*d^4*
x^3 + 3*(5*a^2*b^2*c^4 + 2*a^3*b*c)*d^3*x^2 + 6*(a^2*b^2*c^5 + a^3*b*c^2)*d^2*x
+ (a^2*b^2*c^6 + 2*a^3*b*c^3 + a^4)*d) + 2/9*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^2

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Fricas [A]  time = 0.244029, size = 1004, normalized size = 4.85 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/162*sqrt(3)*(4*sqrt(3)*(b^2*d^6*e*x^6 + 6*b^2*c*d^5*e*x^5 + 15*b^2*c^2*d^4*e*x
^4 + 2*(10*b^2*c^3 + a*b)*d^3*e*x^3 + 3*(5*b^2*c^4 + 2*a*b*c)*d^2*e*x^2 + 6*(b^2
*c^5 + a*b*c^2)*d*e*x + (b^2*c^6 + 2*a*b*c^3 + a^2)*e)*log(a*b + (-a*b^2)^(2/3)*
(d*x + c)) - 2*sqrt(3)*(b^2*d^6*e*x^6 + 6*b^2*c*d^5*e*x^5 + 15*b^2*c^2*d^4*e*x^4
+ 2*(10*b^2*c^3 + a*b)*d^3*e*x^3 + 3*(5*b^2*c^4 + 2*a*b*c)*d^2*e*x^2 + 6*(b^2*c
^5 + a*b*c^2)*d*e*x + (b^2*c^6 + 2*a*b*c^3 + a^2)*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)) - 12*(b^2*d^6*e*x^6 +
6*b^2*c*d^5*e*x^5 + 15*b^2*c^2*d^4*e*x^4 + 2*(10*b^2*c^3 + a*b)*d^3*e*x^3 + 3*(
5*b^2*c^4 + 2*a*b*c)*d^2*e*x^2 + 6*(b^2*c^5 + a*b*c^2)*d*e*x + (b^2*c^6 + 2*a*b*
c^3 + a^2)*e)*arctan(-1/3*(sqrt(3)*a*b - 2*sqrt(3)*(-a*b^2)^(2/3)*(d*x + c))/(a*
b)) + 3*sqrt(3)*(4*b*d^5*e*x^5 + 20*b*c*d^4*e*x^4 + 40*b*c^2*d^3*e*x^3 + (40*b*c
^3 + 7*a)*d^2*e*x^2 + 2*(10*b*c^4 + 7*a*c)*d*e*x + (4*b*c^5 + 7*a*c^2)*e)*(-a*b^
2)^(1/3))/((a^2*b^2*d^7*x^6 + 6*a^2*b^2*c*d^6*x^5 + 15*a^2*b^2*c^2*d^5*x^4 + 2*(
10*a^2*b^2*c^3 + a^3*b)*d^4*x^3 + 3*(5*a^2*b^2*c^4 + 2*a^3*b*c)*d^3*x^2 + 6*(a^2
*b^2*c^5 + a^3*b*c^2)*d^2*x + (a^2*b^2*c^6 + 2*a^3*b*c^3 + a^4)*d)*(-a*b^2)^(1/3
))

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Sympy [A]  time = 69.453, size = 323, normalized size = 1.56 \[ \frac{7 a c^{2} e + 4 b c^{5} e + 40 b c^{2} d^{3} e x^{3} + 20 b c d^{4} e x^{4} + 4 b d^{5} e x^{5} + x^{2} \left (7 a d^{2} e + 40 b c^{3} d^{2} e\right ) + x \left (14 a c d e + 20 b c^{4} d e\right )}{18 a^{4} d + 36 a^{3} b c^{3} d + 18 a^{2} b^{2} c^{6} d + 270 a^{2} b^{2} c^{2} d^{5} x^{4} + 108 a^{2} b^{2} c d^{6} x^{5} + 18 a^{2} b^{2} d^{7} x^{6} + x^{3} \left (36 a^{3} b d^{4} + 360 a^{2} b^{2} c^{3} d^{4}\right ) + x^{2} \left (108 a^{3} b c d^{3} + 270 a^{2} b^{2} c^{4} d^{3}\right ) + x \left (108 a^{3} b c^{2} d^{2} + 108 a^{2} b^{2} c^{5} d^{2}\right )} + \frac{e \operatorname{RootSum}{\left (19683 t^{3} a^{7} b^{2} + 8, \left ( t \mapsto t \log{\left (x + \frac{729 t^{2} a^{5} b e^{2} + 4 c e^{2}}{4 d e^{2}} \right )} \right )\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(7*a*c**2*e + 4*b*c**5*e + 40*b*c**2*d**3*e*x**3 + 20*b*c*d**4*e*x**4 + 4*b*d**5
*e*x**5 + x**2*(7*a*d**2*e + 40*b*c**3*d**2*e) + x*(14*a*c*d*e + 20*b*c**4*d*e))
/(18*a**4*d + 36*a**3*b*c**3*d + 18*a**2*b**2*c**6*d + 270*a**2*b**2*c**2*d**5*x
**4 + 108*a**2*b**2*c*d**6*x**5 + 18*a**2*b**2*d**7*x**6 + x**3*(36*a**3*b*d**4
+ 360*a**2*b**2*c**3*d**4) + x**2*(108*a**3*b*c*d**3 + 270*a**2*b**2*c**4*d**3)
+ x*(108*a**3*b*c**2*d**2 + 108*a**2*b**2*c**5*d**2)) + e*RootSum(19683*_t**3*a*
*7*b**2 + 8, Lambda(_t, _t*log(x + (729*_t**2*a**5*b*e**2 + 4*c*e**2)/(4*d*e**2)
)))/d

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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