3.2879 \(\int \frac{(c e+d e x)^4}{a+b (c+d x)^3} \, dx\)

Optimal. Leaf size=168 \[ \frac{a^{2/3} e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{5/3} d}-\frac{a^{2/3} e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{5/3} d}+\frac{a^{2/3} e^4 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3} d}+\frac{e^4 (c+d x)^2}{2 b d} \]

[Out]

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

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Rubi [A]  time = 0.350035, antiderivative size = 168, 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 \[ \frac{a^{2/3} e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{5/3} d}-\frac{a^{2/3} e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{5/3} d}+\frac{a^{2/3} e^4 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3} d}+\frac{e^4 (c+d x)^2}{2 b d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 41.8468, size = 160, normalized size = 0.95 \[ \frac{a^{\frac{2}{3}} e^{4} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{3 b^{\frac{5}{3}} d} - \frac{a^{\frac{2}{3}} e^{4} \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{6 b^{\frac{5}{3}} d} + \frac{\sqrt{3} a^{\frac{2}{3}} e^{4} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{5}{3}} d} + \frac{e^{4} \left (c + d x\right )^{2}}{2 b d} \]

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),x)

[Out]

a**(2/3)*e**4*log(a**(1/3) + b**(1/3)*(c + d*x))/(3*b**(5/3)*d) - a**(2/3)*e**4*
log(a**(2/3) + a**(1/3)*b**(1/3)*(-c - d*x) + b**(2/3)*(c + d*x)**2)/(6*b**(5/3)
*d) + sqrt(3)*a**(2/3)*e**4*atan(sqrt(3)*(a**(1/3)/3 + b**(1/3)*(-2*c/3 - 2*d*x/
3))/a**(1/3))/(3*b**(5/3)*d) + e**4*(c + d*x)**2/(2*b*d)

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Mathematica [A]  time = 0.0704846, size = 163, normalized size = 0.97 \[ e^4 \left (\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{5/3} d}-\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{5/3} d}-\frac{a^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3} d}+\frac{(c+d x)^2}{2 b d}\right ) \]

Antiderivative was successfully verified.

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

[Out]

e^4*((c + d*x)^2/(2*b*d) - (a^(2/3)*ArcTan[(-a^(1/3) + 2*b^(1/3)*(c + d*x))/(Sqr
t[3]*a^(1/3))])/(Sqrt[3]*b^(5/3)*d) + (a^(2/3)*Log[a^(1/3) + b^(1/3)*(c + d*x)])
/(3*b^(5/3)*d) - (a^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c +
d*x)^2])/(6*b^(5/3)*d))

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Maple [C]  time = 0.004, size = 102, normalized size = 0.6 \[{\frac{{e}^{4}d{x}^{2}}{2\,b}}+{\frac{{e}^{4}cx}{b}}-{\frac{{e}^{4}a}{3\,{b}^{2}d}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{ \left ({\it \_R}\,d+c \right ) \ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*e^4/b*d*x^2+e^4/b*c*x-1/3*e^4*a/b^2/d*sum((_R*d+c)/(_R^2*d^2+2*_R*c*d+c^2)*l
n(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. \[ -\frac{a e^{4} \int \frac{d x + c}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}\,{d x}}{b} + \frac{d e^{4} x^{2} + 2 \, c e^{4} x}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*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 + 1/2*(d*e^4*x^2 + 2*c*e^4*x)/b

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Fricas [A]  time = 0.213111, size = 266, normalized size = 1.58 \[ -\frac{\sqrt{3}{\left (\sqrt{3} e^{4} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a d^{2} x^{2} + 2 \, a c d x + a c^{2} -{\left (b d x + b c\right )} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} + a \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3} e^{4} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a d x + a c + b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right ) - 6 \, e^{4} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3} b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} - 2 \, \sqrt{3}{\left (a d x + a c\right )}}{3 \, b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}\right ) - 3 \, \sqrt{3}{\left (d^{2} e^{4} x^{2} + 2 \, c d e^{4} x\right )}\right )}}{18 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18*sqrt(3)*(sqrt(3)*e^4*(a^2/b^2)^(1/3)*log(a*d^2*x^2 + 2*a*c*d*x + a*c^2 - (
b*d*x + b*c)*(a^2/b^2)^(2/3) + a*(a^2/b^2)^(1/3)) - 2*sqrt(3)*e^4*(a^2/b^2)^(1/3
)*log(a*d*x + a*c + b*(a^2/b^2)^(2/3)) - 6*e^4*(a^2/b^2)^(1/3)*arctan(1/3*(sqrt(
3)*b*(a^2/b^2)^(2/3) - 2*sqrt(3)*(a*d*x + a*c))/(b*(a^2/b^2)^(2/3))) - 3*sqrt(3)
*(d^2*e^4*x^2 + 2*c*d*e^4*x))/(b*d)

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Sympy [A]  time = 2.248, size = 66, normalized size = 0.39 \[ \frac{e^{4} \operatorname{RootSum}{\left (27 t^{3} b^{5} - a^{2}, \left ( t \mapsto t \log{\left (x + \frac{9 t^{2} b^{3} e^{8} + a c e^{8}}{a d e^{8}} \right )} \right )\right )}}{d} + \frac{c e^{4} x}{b} + \frac{d e^{4} x^{2}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e**4*RootSum(27*_t**3*b**5 - a**2, Lambda(_t, _t*log(x + (9*_t**2*b**3*e**8 + a*
c*e**8)/(a*d*e**8))))/d + c*e**4*x/b + d*e**4*x**2/(2*b)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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