3.2887 \(\int \frac{(c e+d e x)^4}{\left (a+b (c+d x)^3\right )^2} \, dx\)

Optimal. Leaf size=184 \[ \frac{e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{9 \sqrt [3]{a} b^{5/3} d}-\frac{2 e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 \sqrt [3]{a} b^{5/3} d}-\frac{2 e^4 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} \sqrt [3]{a} b^{5/3} d}-\frac{e^4 (c+d x)^2}{3 b d \left (a+b (c+d x)^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.351981, antiderivative size = 184, 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 \[ \frac{e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{9 \sqrt [3]{a} b^{5/3} d}-\frac{2 e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 \sqrt [3]{a} b^{5/3} d}-\frac{2 e^4 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} \sqrt [3]{a} b^{5/3} d}-\frac{e^4 (c+d x)^2}{3 b d \left (a+b (c+d x)^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(c*e + d*e*x)^4/(a + b*(c + d*x)^3)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 41.4329, size = 173, normalized size = 0.94 \[ - \frac{e^{4} \left (c + d x\right )^{2}}{3 b d \left (a + b \left (c + d x\right )^{3}\right )} - \frac{2 e^{4} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{9 \sqrt [3]{a} b^{\frac{5}{3}} d} + \frac{e^{4} \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{9 \sqrt [3]{a} b^{\frac{5}{3}} d} - \frac{2 \sqrt{3} e^{4} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{9 \sqrt [3]{a} b^{\frac{5}{3}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*e*x+c*e)**4/(a+b*(d*x+c)**3)**2,x)

[Out]

-e**4*(c + d*x)**2/(3*b*d*(a + b*(c + d*x)**3)) - 2*e**4*log(a**(1/3) + b**(1/3)
*(c + d*x))/(9*a**(1/3)*b**(5/3)*d) + e**4*log(a**(2/3) + a**(1/3)*b**(1/3)*(-c
- d*x) + b**(2/3)*(c + d*x)**2)/(9*a**(1/3)*b**(5/3)*d) - 2*sqrt(3)*e**4*atan(sq
rt(3)*(a**(1/3)/3 + b**(1/3)*(-2*c/3 - 2*d*x/3))/a**(1/3))/(9*a**(1/3)*b**(5/3)*
d)

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Mathematica [A]  time = 0.181068, size = 155, normalized size = 0.84 \[ \frac{e^4 \left (\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{\sqrt [3]{a}}-\frac{3 b^{2/3} (c+d x)^2}{a+b (c+d x)^3}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{\sqrt [3]{a}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{a}}\right )}{9 b^{5/3} d} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*e + d*e*x)^4/(a + b*(c + d*x)^3)^2,x]

[Out]

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

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Maple [C]  time = 0.007, size = 221, normalized size = 1.2 \[ -{\frac{{e}^{4}d{x}^{2}}{ \left ( 3\,b{d}^{3}{x}^{3}+9\,bc{d}^{2}{x}^{2}+9\,b{c}^{2}dx+3\,b{c}^{3}+3\,a \right ) b}}-{\frac{2\,{e}^{4}cx}{ \left ( 3\,b{d}^{3}{x}^{3}+9\,bc{d}^{2}{x}^{2}+9\,b{c}^{2}dx+3\,b{c}^{3}+3\,a \right ) b}}-{\frac{{e}^{4}{c}^{2}}{ \left ( 3\,b{d}^{3}{x}^{3}+9\,bc{d}^{2}{x}^{2}+9\,b{c}^{2}dx+3\,b{c}^{3}+3\,a \right ) bd}}+{\frac{2\,{e}^{4}}{9\,{b}^{2}d}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{ \left ({\it \_R}\,d+c \right ) \ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*e*x+c*e)^4/(a+b*(d*x+c)^3)^2,x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*e*x + c*e)^4/((d*x + c)^3*b + a)^2,x, algorithm="maxima")

[Out]

2/3*e^4*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)/b - 1/3*(d^2*e^4*x^2 + 2*c*d*e^4*x + c^2*e^4)/(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 = 0.219216, size = 483, normalized size = 2.62 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (b d^{3} e^{4} x^{3} + 3 \, b c d^{2} e^{4} x^{2} + 3 \, b c^{2} d e^{4} x +{\left (b c^{3} + a\right )} e^{4}\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}}{\left (d x + c\right )}\right ) - \sqrt{3}{\left (b d^{3} e^{4} x^{3} + 3 \, b c d^{2} e^{4} x^{2} + 3 \, b c^{2} d e^{4} x +{\left (b c^{3} + a\right )} e^{4}\right )} \log \left (-a b + \left (-a b^{2}\right )^{\frac{2}{3}}{\left (d x + c\right )} +{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right ) - 6 \,{\left (b d^{3} e^{4} x^{3} + 3 \, b c d^{2} e^{4} x^{2} + 3 \, b c^{2} d e^{4} x +{\left (b c^{3} + a\right )} e^{4}\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}}{\left (d x + c\right )}}{3 \, a b}\right ) - 3 \, \sqrt{3}{\left (d^{2} e^{4} x^{2} + 2 \, c d e^{4} x + c^{2} e^{4}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}}{27 \,{\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x +{\left (b^{2} c^{3} + a b\right )} d\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*e*x + c*e)^4/((d*x + c)^3*b + a)^2,x, algorithm="fricas")

[Out]

1/27*sqrt(3)*(2*sqrt(3)*(b*d^3*e^4*x^3 + 3*b*c*d^2*e^4*x^2 + 3*b*c^2*d*e^4*x + (
b*c^3 + a)*e^4)*log(a*b + (-a*b^2)^(2/3)*(d*x + c)) - sqrt(3)*(b*d^3*e^4*x^3 + 3
*b*c*d^2*e^4*x^2 + 3*b*c^2*d*e^4*x + (b*c^3 + a)*e^4)*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^4*x^3 +
3*b*c*d^2*e^4*x^2 + 3*b*c^2*d*e^4*x + (b*c^3 + a)*e^4)*arctan(-1/3*(sqrt(3)*a*b
- 2*sqrt(3)*(-a*b^2)^(2/3)*(d*x + c))/(a*b)) - 3*sqrt(3)*(d^2*e^4*x^2 + 2*c*d*e
^4*x + c^2*e^4)*(-a*b^2)^(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*b^2)^(1/3))

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Sympy [A]  time = 10.8324, size = 131, normalized size = 0.71 \[ - \frac{c^{2} e^{4} + 2 c d e^{4} x + d^{2} e^{4} x^{2}}{3 a b d + 3 b^{2} c^{3} d + 9 b^{2} c^{2} d^{2} x + 9 b^{2} c d^{3} x^{2} + 3 b^{2} d^{4} x^{3}} + \frac{e^{4} \operatorname{RootSum}{\left (729 t^{3} a b^{5} + 8, \left ( t \mapsto t \log{\left (x + \frac{81 t^{2} a b^{3} e^{8} + 4 c e^{8}}{4 d e^{8}} \right )} \right )\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*e*x+c*e)**4/(a+b*(d*x+c)**3)**2,x)

[Out]

-(c**2*e**4 + 2*c*d*e**4*x + d**2*e**4*x**2)/(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**4*RootSum(729*_t**3*a*b
**5 + 8, Lambda(_t, _t*log(x + (81*_t**2*a*b**3*e**8 + 4*c*e**8)/(4*d*e**8))))/d

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d e x + c e\right )}^{4}}{{\left ({\left (d x + c\right )}^{3} b + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*e*x + c*e)^4/((d*x + c)^3*b + a)^2,x, algorithm="giac")

[Out]

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