3.2901 \(\int \frac{1}{(c e+d e x)^4 (a+b (c+d x)^3)^2} \, dx\)

Optimal. Leaf size=92 \[ -\frac{b}{3 a^2 d e^4 \left (a+b (c+d x)^3\right )}-\frac{2 b \log (c+d x)}{a^3 d e^4}+\frac{2 b \log \left (a+b (c+d x)^3\right )}{3 a^3 d e^4}-\frac{1}{3 a^2 d e^4 (c+d x)^3} \]

[Out]

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

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Rubi [A]  time = 0.0673652, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {372, 266, 44} \[ -\frac{b}{3 a^2 d e^4 \left (a+b (c+d x)^3\right )}-\frac{2 b \log (c+d x)}{a^3 d e^4}+\frac{2 b \log \left (a+b (c+d x)^3\right )}{3 a^3 d e^4}-\frac{1}{3 a^2 d e^4 (c+d x)^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((c*e + d*e*x)^4*(a + b*(c + d*x)^3)^2),x]

[Out]

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

Rule 372

Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m
*(a + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, m, n, p}, x] && LinearPairQ[u, v, x]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{(c e+d e x)^4 \left (a+b (c+d x)^3\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^4 \left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{d e^4}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^2} \, dx,x,(c+d x)^3\right )}{3 d e^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^2 x^2}-\frac{2 b}{a^3 x}+\frac{b^2}{a^2 (a+b x)^2}+\frac{2 b^2}{a^3 (a+b x)}\right ) \, dx,x,(c+d x)^3\right )}{3 d e^4}\\ &=-\frac{1}{3 a^2 d e^4 (c+d x)^3}-\frac{b}{3 a^2 d e^4 \left (a+b (c+d x)^3\right )}-\frac{2 b \log (c+d x)}{a^3 d e^4}+\frac{2 b \log \left (a+b (c+d x)^3\right )}{3 a^3 d e^4}\\ \end{align*}

Mathematica [A]  time = 0.0593806, size = 63, normalized size = 0.68 \[ -\frac{a \left (\frac{b}{a+b (c+d x)^3}+\frac{1}{(c+d x)^3}\right )-2 b \log \left (a+b (c+d x)^3\right )+6 b \log (c+d x)}{3 a^3 d e^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c*e + d*e*x)^4*(a + b*(c + d*x)^3)^2),x]

[Out]

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

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Maple [A]  time = 0.018, size = 131, normalized size = 1.4 \begin{align*} -{\frac{1}{3\,{a}^{2}d{e}^{4} \left ( dx+c \right ) ^{3}}}-2\,{\frac{b\ln \left ( dx+c \right ) }{{a}^{3}d{e}^{4}}}-{\frac{b}{3\,{a}^{2}d{e}^{4} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }}+{\frac{2\,b\ln \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }{3\,{a}^{3}d{e}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.01987, size = 336, normalized size = 3.65 \begin{align*} -\frac{2 \, b d^{3} x^{3} + 6 \, b c d^{2} x^{2} + 6 \, b c^{2} d x + 2 \, b c^{3} + a}{3 \,{\left (a^{2} b d^{7} e^{4} x^{6} + 6 \, a^{2} b c d^{6} e^{4} x^{5} + 15 \, a^{2} b c^{2} d^{5} e^{4} x^{4} +{\left (20 \, a^{2} b c^{3} + a^{3}\right )} d^{4} e^{4} x^{3} + 3 \,{\left (5 \, a^{2} b c^{4} + a^{3} c\right )} d^{3} e^{4} x^{2} + 3 \,{\left (2 \, a^{2} b c^{5} + a^{3} c^{2}\right )} d^{2} e^{4} x +{\left (a^{2} b c^{6} + a^{3} c^{3}\right )} d e^{4}\right )}} + \frac{2 \, b \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a^{3} d e^{4}} - \frac{2 \, b \log \left (d x + c\right )}{a^{3} d e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)^4/(a+b*(d*x+c)^3)^2,x, algorithm="maxima")

[Out]

-1/3*(2*b*d^3*x^3 + 6*b*c*d^2*x^2 + 6*b*c^2*d*x + 2*b*c^3 + a)/(a^2*b*d^7*e^4*x^6 + 6*a^2*b*c*d^6*e^4*x^5 + 15
*a^2*b*c^2*d^5*e^4*x^4 + (20*a^2*b*c^3 + a^3)*d^4*e^4*x^3 + 3*(5*a^2*b*c^4 + a^3*c)*d^3*e^4*x^2 + 3*(2*a^2*b*c
^5 + a^3*c^2)*d^2*e^4*x + (a^2*b*c^6 + a^3*c^3)*d*e^4) + 2/3*b*log(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b
*c^3 + a)/(a^3*d*e^4) - 2*b*log(d*x + c)/(a^3*d*e^4)

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Fricas [B]  time = 1.80417, size = 934, normalized size = 10.15 \begin{align*} -\frac{2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + a^{2} - 2 \,{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} +{\left (20 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + a b c^{3} + 3 \,{\left (5 \, b^{2} c^{4} + a b c\right )} d^{2} x^{2} + 3 \,{\left (2 \, b^{2} c^{5} + a b c^{2}\right )} d x\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) + 6 \,{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} +{\left (20 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + a b c^{3} + 3 \,{\left (5 \, b^{2} c^{4} + a b c\right )} d^{2} x^{2} + 3 \,{\left (2 \, b^{2} c^{5} + a b c^{2}\right )} d x\right )} \log \left (d x + c\right )}{3 \,{\left (a^{3} b d^{7} e^{4} x^{6} + 6 \, a^{3} b c d^{6} e^{4} x^{5} + 15 \, a^{3} b c^{2} d^{5} e^{4} x^{4} +{\left (20 \, a^{3} b c^{3} + a^{4}\right )} d^{4} e^{4} x^{3} + 3 \,{\left (5 \, a^{3} b c^{4} + a^{4} c\right )} d^{3} e^{4} x^{2} + 3 \,{\left (2 \, a^{3} b c^{5} + a^{4} c^{2}\right )} d^{2} e^{4} x +{\left (a^{3} b c^{6} + a^{4} c^{3}\right )} d e^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)^4/(a+b*(d*x+c)^3)^2,x, algorithm="fricas")

[Out]

-1/3*(2*a*b*d^3*x^3 + 6*a*b*c*d^2*x^2 + 6*a*b*c^2*d*x + 2*a*b*c^3 + a^2 - 2*(b^2*d^6*x^6 + 6*b^2*c*d^5*x^5 + 1
5*b^2*c^2*d^4*x^4 + b^2*c^6 + (20*b^2*c^3 + a*b)*d^3*x^3 + a*b*c^3 + 3*(5*b^2*c^4 + a*b*c)*d^2*x^2 + 3*(2*b^2*
c^5 + a*b*c^2)*d*x)*log(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a) + 6*(b^2*d^6*x^6 + 6*b^2*c*d^5*x^
5 + 15*b^2*c^2*d^4*x^4 + b^2*c^6 + (20*b^2*c^3 + a*b)*d^3*x^3 + a*b*c^3 + 3*(5*b^2*c^4 + a*b*c)*d^2*x^2 + 3*(2
*b^2*c^5 + a*b*c^2)*d*x)*log(d*x + c))/(a^3*b*d^7*e^4*x^6 + 6*a^3*b*c*d^6*e^4*x^5 + 15*a^3*b*c^2*d^5*e^4*x^4 +
(20*a^3*b*c^3 + a^4)*d^4*e^4*x^3 + 3*(5*a^3*b*c^4 + a^4*c)*d^3*e^4*x^2 + 3*(2*a^3*b*c^5 + a^4*c^2)*d^2*e^4*x
+ (a^3*b*c^6 + a^4*c^3)*d*e^4)

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Sympy [B]  time = 174.189, size = 292, normalized size = 3.17 \begin{align*} - \frac{a + 2 b c^{3} + 6 b c^{2} d x + 6 b c d^{2} x^{2} + 2 b d^{3} x^{3}}{3 a^{3} c^{3} d e^{4} + 3 a^{2} b c^{6} d e^{4} + 45 a^{2} b c^{2} d^{5} e^{4} x^{4} + 18 a^{2} b c d^{6} e^{4} x^{5} + 3 a^{2} b d^{7} e^{4} x^{6} + x^{3} \left (3 a^{3} d^{4} e^{4} + 60 a^{2} b c^{3} d^{4} e^{4}\right ) + x^{2} \left (9 a^{3} c d^{3} e^{4} + 45 a^{2} b c^{4} d^{3} e^{4}\right ) + x \left (9 a^{3} c^{2} d^{2} e^{4} + 18 a^{2} b c^{5} d^{2} e^{4}\right )} - \frac{2 b \log{\left (\frac{c}{d} + x \right )}}{a^{3} d e^{4}} + \frac{2 b \log{\left (\frac{3 c^{2} x}{d^{2}} + \frac{3 c x^{2}}{d} + x^{3} + \frac{a + b c^{3}}{b d^{3}} \right )}}{3 a^{3} d e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a + 2*b*c**3 + 6*b*c**2*d*x + 6*b*c*d**2*x**2 + 2*b*d**3*x**3)/(3*a**3*c**3*d*e**4 + 3*a**2*b*c**6*d*e**4 +
45*a**2*b*c**2*d**5*e**4*x**4 + 18*a**2*b*c*d**6*e**4*x**5 + 3*a**2*b*d**7*e**4*x**6 + x**3*(3*a**3*d**4*e**4
+ 60*a**2*b*c**3*d**4*e**4) + x**2*(9*a**3*c*d**3*e**4 + 45*a**2*b*c**4*d**3*e**4) + x*(9*a**3*c**2*d**2*e**4
+ 18*a**2*b*c**5*d**2*e**4)) - 2*b*log(c/d + x)/(a**3*d*e**4) + 2*b*log(3*c**2*x/d**2 + 3*c*x**2/d + x**3 + (a
+ b*c**3)/(b*d**3))/(3*a**3*d*e**4)

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Giac [A]  time = 1.11145, size = 213, normalized size = 2.32 \begin{align*} \frac{2 \, b e^{\left (-4\right )} \log \left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, a^{3} d} - \frac{2 \, b e^{\left (-4\right )} \log \left ({\left | d x + c \right |}\right )}{a^{3} d} - \frac{{\left (2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + a^{2}\right )} e^{\left (-4\right )}}{3 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{\left (d x + c\right )}^{3} a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)^4/(a+b*(d*x+c)^3)^2,x, algorithm="giac")

[Out]

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