3.2893 \(\int \frac{1}{(c e+d e x)^4 (a+b (c+d x)^3)} \, dx\)

Optimal. Leaf size=65 \[ -\frac{b \log (c+d x)}{a^2 d e^4}+\frac{b \log \left (a+b (c+d x)^3\right )}{3 a^2 d e^4}-\frac{1}{3 a d e^4 (c+d x)^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0452679, antiderivative size = 65, 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} \[ -\frac{b \log (c+d x)}{a^2 d e^4}+\frac{b \log \left (a+b (c+d x)^3\right )}{3 a^2 d e^4}-\frac{1}{3 a d e^4 (c+d x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0159609, size = 47, normalized size = 0.72 \[ \frac{b \log \left (a+b (c+d x)^3\right )-\frac{a}{(c+d x)^3}-3 b \log (c+d x)}{3 a^2 d e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 84, normalized size = 1.3 \begin{align*} -{\frac{1}{3\,ad{e}^{4} \left ( dx+c \right ) ^{3}}}-{\frac{b\ln \left ( dx+c \right ) }{{a}^{2}d{e}^{4}}}+{\frac{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}^{2}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),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.982409, size = 157, normalized size = 2.42 \begin{align*} -\frac{1}{3 \,{\left (a d^{4} e^{4} x^{3} + 3 \, a c d^{3} e^{4} x^{2} + 3 \, a c^{2} d^{2} e^{4} x + a c^{3} d e^{4}\right )}} + \frac{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^{2} d e^{4}} - \frac{b \log \left (d x + c\right )}{a^{2} 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),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 1.56151, size = 355, normalized size = 5.46 \begin{align*} \frac{{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\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 ) - 3 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (d x + c\right ) - a}{3 \,{\left (a^{2} d^{4} e^{4} x^{3} + 3 \, a^{2} c d^{3} e^{4} x^{2} + 3 \, a^{2} c^{2} d^{2} e^{4} x + a^{2} c^{3} 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),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 2.19192, size = 121, normalized size = 1.86 \begin{align*} - \frac{1}{3 a c^{3} d e^{4} + 9 a c^{2} d^{2} e^{4} x + 9 a c d^{3} e^{4} x^{2} + 3 a d^{4} e^{4} x^{3}} - \frac{b \log{\left (\frac{c}{d} + x \right )}}{a^{2} d e^{4}} + \frac{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^{2} 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),x)

[Out]

-1/(3*a*c**3*d*e**4 + 9*a*c**2*d**2*e**4*x + 9*a*c*d**3*e**4*x**2 + 3*a*d**4*e**4*x**3) - b*log(c/d + x)/(a**2
*d*e**4) + b*log(3*c**2*x/d**2 + 3*c*x**2/d + x**3 + (a + b*c**3)/(b*d**3))/(3*a**2*d*e**4)

________________________________________________________________________________________

Giac [A]  time = 1.15164, size = 111, normalized size = 1.71 \begin{align*} \frac{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^{2} d} - \frac{b e^{\left (-4\right )} \log \left ({\left | d x + c \right |}\right )}{a^{2} d} - \frac{e^{\left (-4\right )}}{3 \,{\left (d x + c\right )}^{3} a 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),x, algorithm="giac")

[Out]

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