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

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Rubi [A]  time = 0.05, antiderivative size = 65, normalized size of antiderivative = 1.00, 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 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])

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 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]

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*}

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Mathematica [A]  time = 0.02, 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)

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fricas [B]  time = 1.06, size = 170, normalized size = 2.62 \[ \frac {{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) - 3 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (d x + c\right ) - a}{3 \, {\left (a^{2} d^{4} e^{4} x^{3} + 3 \, a^{2} c d^{3} e^{4} x^{2} + 3 \, a^{2} c^{2} d^{2} e^{4} x + a^{2} c^{3} d e^{4}\right )}} \]

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)

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giac [A]  time = 0.19, size = 82, normalized size = 1.26 \[ \frac {b e^{\left (-4\right )} \log \left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, a^{2} d} - \frac {b e^{\left (-4\right )} \log \left ({\left | d x + c \right |}\right )}{a^{2} d} - \frac {e^{\left (-4\right )}}{3 \, {\left (d x + c\right )}^{3} a d} \]

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)

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maple [A]  time = 0.01, size = 84, normalized size = 1.29 \[ -\frac {b \ln \left (d x +c \right )}{a^{2} d \,e^{4}}+\frac {b \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )}{3 a^{2} d \,e^{4}}-\frac {1}{3 \left (d x +c \right )^{3} a d \,e^{4}} \]

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)

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maxima [A]  time = 0.69, size = 116, normalized size = 1.78 \[ -\frac {1}{3 \, {\left (a d^{4} e^{4} x^{3} + 3 \, a c d^{3} e^{4} x^{2} + 3 \, a c^{2} d^{2} e^{4} x + a c^{3} d e^{4}\right )}} + \frac {b \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a^{2} d e^{4}} - \frac {b \log \left (d x + c\right )}{a^{2} d e^{4}} \]

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)

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mupad [B]  time = 1.44, size = 118, normalized size = 1.82 \[ \frac {b\,\ln \left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )}{3\,a^2\,d\,e^4}-\frac {1}{3\,\left (a\,c^3\,d\,e^4+3\,a\,c^2\,d^2\,e^4\,x+3\,a\,c\,d^3\,e^4\,x^2+a\,d^4\,e^4\,x^3\right )}-\frac {b\,\ln \left (c+d\,x\right )}{a^2\,d\,e^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 2.10, size = 121, normalized size = 1.86 \[ - \frac {1}{3 a c^{3} d e^{4} + 9 a c^{2} d^{2} e^{4} x + 9 a c d^{3} e^{4} x^{2} + 3 a d^{4} e^{4} x^{3}} - \frac {b \log {\left (\frac {c}{d} + x \right )}}{a^{2} d e^{4}} + \frac {b \log {\left (\frac {3 c^{2} x}{d^{2}} + \frac {3 c x^{2}}{d} + x^{3} + \frac {a + b c^{3}}{b d^{3}} \right )}}{3 a^{2} d e^{4}} \]

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)

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