3.2909 \(\int \frac {1}{(c e+d e x)^4 (a+b (c+d x)^3)^3} \, dx\)

Optimal. Leaf size=116 \[ -\frac {3 b \log (c+d x)}{a^4 d e^4}+\frac {b \log \left (a+b (c+d x)^3\right )}{a^4 d e^4}-\frac {2 b}{3 a^3 d e^4 \left (a+b (c+d x)^3\right )}-\frac {1}{3 a^3 d e^4 (c+d x)^3}-\frac {b}{6 a^2 d e^4 \left (a+b (c+d x)^3\right )^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 116, 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} \[ -\frac {2 b}{3 a^3 d e^4 \left (a+b (c+d x)^3\right )}-\frac {b}{6 a^2 d e^4 \left (a+b (c+d x)^3\right )^2}-\frac {3 b \log (c+d x)}{a^4 d e^4}+\frac {b \log \left (a+b (c+d x)^3\right )}{a^4 d e^4}-\frac {1}{3 a^3 d e^4 (c+d x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(3*a^3*d*e^4*(c + d*x)^3) - b/(6*a^2*d*e^4*(a + b*(c + d*x)^3)^2) - (2*b)/(3*a^3*d*e^4*(a + b*(c + d*x)^3))
- (3*b*Log[c + d*x])/(a^4*d*e^4) + (b*Log[a + b*(c + d*x)^3])/(a^4*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 )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^4 \left (a+b x^3\right )^3} \, dx,x,c+d x\right )}{d e^4}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^3} \, dx,x,(c+d x)^3\right )}{3 d e^4}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a^3 x^2}-\frac {3 b}{a^4 x}+\frac {b^2}{a^2 (a+b x)^3}+\frac {2 b^2}{a^3 (a+b x)^2}+\frac {3 b^2}{a^4 (a+b x)}\right ) \, dx,x,(c+d x)^3\right )}{3 d e^4}\\ &=-\frac {1}{3 a^3 d e^4 (c+d x)^3}-\frac {b}{6 a^2 d e^4 \left (a+b (c+d x)^3\right )^2}-\frac {2 b}{3 a^3 d e^4 \left (a+b (c+d x)^3\right )}-\frac {3 b \log (c+d x)}{a^4 d e^4}+\frac {b \log \left (a+b (c+d x)^3\right )}{a^4 d e^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 83, normalized size = 0.72 \[ \frac {a \left (-\frac {4 b}{a+b (c+d x)^3}-\frac {a b}{\left (a+b (c+d x)^3\right )^2}-\frac {2}{(c+d x)^3}\right )+6 b \log \left (a+b (c+d x)^3\right )-18 b \log (c+d x)}{6 a^4 d e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(-2/(c + d*x)^3 - (a*b)/(a + b*(c + d*x)^3)^2 - (4*b)/(a + b*(c + d*x)^3)) - 18*b*Log[c + d*x] + 6*b*Log[a
+ b*(c + d*x)^3])/(6*a^4*d*e^4)

________________________________________________________________________________________

fricas [B]  time = 1.16, size = 919, normalized size = 7.92 \[ -\frac {6 \, a b^{2} d^{6} x^{6} + 36 \, a b^{2} c d^{5} x^{5} + 90 \, a b^{2} c^{2} d^{4} x^{4} + 6 \, a b^{2} c^{6} + 3 \, {\left (40 \, a b^{2} c^{3} + 3 \, a^{2} b\right )} d^{3} x^{3} + 9 \, a^{2} b c^{3} + 9 \, {\left (10 \, a b^{2} c^{4} + 3 \, a^{2} b c\right )} d^{2} x^{2} + 2 \, a^{3} + 9 \, {\left (4 \, a b^{2} c^{5} + 3 \, a^{2} b c^{2}\right )} d x - 6 \, {\left (b^{3} d^{9} x^{9} + 9 \, b^{3} c d^{8} x^{8} + 36 \, b^{3} c^{2} d^{7} x^{7} + 2 \, {\left (42 \, b^{3} c^{3} + a b^{2}\right )} d^{6} x^{6} + b^{3} c^{9} + 6 \, {\left (21 \, b^{3} c^{4} + 2 \, a b^{2} c\right )} d^{5} x^{5} + 2 \, a b^{2} c^{6} + 6 \, {\left (21 \, b^{3} c^{5} + 5 \, a b^{2} c^{2}\right )} d^{4} x^{4} + {\left (84 \, b^{3} c^{6} + 40 \, a b^{2} c^{3} + a^{2} b\right )} d^{3} x^{3} + a^{2} b c^{3} + 3 \, {\left (12 \, b^{3} c^{7} + 10 \, a b^{2} c^{4} + a^{2} b c\right )} d^{2} x^{2} + 3 \, {\left (3 \, b^{3} c^{8} + 4 \, a b^{2} c^{5} + a^{2} 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 ) + 18 \, {\left (b^{3} d^{9} x^{9} + 9 \, b^{3} c d^{8} x^{8} + 36 \, b^{3} c^{2} d^{7} x^{7} + 2 \, {\left (42 \, b^{3} c^{3} + a b^{2}\right )} d^{6} x^{6} + b^{3} c^{9} + 6 \, {\left (21 \, b^{3} c^{4} + 2 \, a b^{2} c\right )} d^{5} x^{5} + 2 \, a b^{2} c^{6} + 6 \, {\left (21 \, b^{3} c^{5} + 5 \, a b^{2} c^{2}\right )} d^{4} x^{4} + {\left (84 \, b^{3} c^{6} + 40 \, a b^{2} c^{3} + a^{2} b\right )} d^{3} x^{3} + a^{2} b c^{3} + 3 \, {\left (12 \, b^{3} c^{7} + 10 \, a b^{2} c^{4} + a^{2} b c\right )} d^{2} x^{2} + 3 \, {\left (3 \, b^{3} c^{8} + 4 \, a b^{2} c^{5} + a^{2} b c^{2}\right )} d x\right )} \log \left (d x + c\right )}{6 \, {\left (a^{4} b^{2} d^{10} e^{4} x^{9} + 9 \, a^{4} b^{2} c d^{9} e^{4} x^{8} + 36 \, a^{4} b^{2} c^{2} d^{8} e^{4} x^{7} + 2 \, {\left (42 \, a^{4} b^{2} c^{3} + a^{5} b\right )} d^{7} e^{4} x^{6} + 6 \, {\left (21 \, a^{4} b^{2} c^{4} + 2 \, a^{5} b c\right )} d^{6} e^{4} x^{5} + 6 \, {\left (21 \, a^{4} b^{2} c^{5} + 5 \, a^{5} b c^{2}\right )} d^{5} e^{4} x^{4} + {\left (84 \, a^{4} b^{2} c^{6} + 40 \, a^{5} b c^{3} + a^{6}\right )} d^{4} e^{4} x^{3} + 3 \, {\left (12 \, a^{4} b^{2} c^{7} + 10 \, a^{5} b c^{4} + a^{6} c\right )} d^{3} e^{4} x^{2} + 3 \, {\left (3 \, a^{4} b^{2} c^{8} + 4 \, a^{5} b c^{5} + a^{6} c^{2}\right )} d^{2} e^{4} x + {\left (a^{4} b^{2} c^{9} + 2 \, a^{5} b c^{6} + a^{6} c^{3}\right )} 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)^3,x, algorithm="fricas")

[Out]

-1/6*(6*a*b^2*d^6*x^6 + 36*a*b^2*c*d^5*x^5 + 90*a*b^2*c^2*d^4*x^4 + 6*a*b^2*c^6 + 3*(40*a*b^2*c^3 + 3*a^2*b)*d
^3*x^3 + 9*a^2*b*c^3 + 9*(10*a*b^2*c^4 + 3*a^2*b*c)*d^2*x^2 + 2*a^3 + 9*(4*a*b^2*c^5 + 3*a^2*b*c^2)*d*x - 6*(b
^3*d^9*x^9 + 9*b^3*c*d^8*x^8 + 36*b^3*c^2*d^7*x^7 + 2*(42*b^3*c^3 + a*b^2)*d^6*x^6 + b^3*c^9 + 6*(21*b^3*c^4 +
2*a*b^2*c)*d^5*x^5 + 2*a*b^2*c^6 + 6*(21*b^3*c^5 + 5*a*b^2*c^2)*d^4*x^4 + (84*b^3*c^6 + 40*a*b^2*c^3 + a^2*b)
*d^3*x^3 + a^2*b*c^3 + 3*(12*b^3*c^7 + 10*a*b^2*c^4 + a^2*b*c)*d^2*x^2 + 3*(3*b^3*c^8 + 4*a*b^2*c^5 + a^2*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) + 18*(b^3*d^9*x^9 + 9*b^3*c*d^8*x^8 + 36*b^3*
c^2*d^7*x^7 + 2*(42*b^3*c^3 + a*b^2)*d^6*x^6 + b^3*c^9 + 6*(21*b^3*c^4 + 2*a*b^2*c)*d^5*x^5 + 2*a*b^2*c^6 + 6*
(21*b^3*c^5 + 5*a*b^2*c^2)*d^4*x^4 + (84*b^3*c^6 + 40*a*b^2*c^3 + a^2*b)*d^3*x^3 + a^2*b*c^3 + 3*(12*b^3*c^7 +
10*a*b^2*c^4 + a^2*b*c)*d^2*x^2 + 3*(3*b^3*c^8 + 4*a*b^2*c^5 + a^2*b*c^2)*d*x)*log(d*x + c))/(a^4*b^2*d^10*e^
4*x^9 + 9*a^4*b^2*c*d^9*e^4*x^8 + 36*a^4*b^2*c^2*d^8*e^4*x^7 + 2*(42*a^4*b^2*c^3 + a^5*b)*d^7*e^4*x^6 + 6*(21*
a^4*b^2*c^4 + 2*a^5*b*c)*d^6*e^4*x^5 + 6*(21*a^4*b^2*c^5 + 5*a^5*b*c^2)*d^5*e^4*x^4 + (84*a^4*b^2*c^6 + 40*a^5
*b*c^3 + a^6)*d^4*e^4*x^3 + 3*(12*a^4*b^2*c^7 + 10*a^5*b*c^4 + a^6*c)*d^3*e^4*x^2 + 3*(3*a^4*b^2*c^8 + 4*a^5*b
*c^5 + a^6*c^2)*d^2*e^4*x + (a^4*b^2*c^9 + 2*a^5*b*c^6 + a^6*c^3)*d*e^4)

________________________________________________________________________________________

giac [B]  time = 0.25, size = 259, normalized size = 2.23 \[ \frac {b e^{\left (-4\right )} \log \left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{a^{4} d} - \frac {3 \, b e^{\left (-4\right )} \log \left ({\left | d x + c \right |}\right )}{a^{4} d} - \frac {{\left (6 \, a b^{2} d^{6} x^{6} + 36 \, a b^{2} c d^{5} x^{5} + 90 \, a b^{2} c^{2} d^{4} x^{4} + 6 \, a b^{2} c^{6} + 9 \, a^{2} b c^{3} + 3 \, {\left (40 \, a b^{2} c^{3} d^{3} + 3 \, a^{2} b d^{3}\right )} x^{3} + 2 \, a^{3} + 9 \, {\left (10 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c d^{2}\right )} x^{2} + 9 \, {\left (4 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{2} d\right )} x\right )} e^{\left (-4\right )}}{6 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}^{2} {\left (d x + c\right )}^{3} a^{4} d} \]

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)^3,x, algorithm="giac")

[Out]

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^4*d) - 3*b*e^(-4)*log(abs(d*x + c))/
(a^4*d) - 1/6*(6*a*b^2*d^6*x^6 + 36*a*b^2*c*d^5*x^5 + 90*a*b^2*c^2*d^4*x^4 + 6*a*b^2*c^6 + 9*a^2*b*c^3 + 3*(40
*a*b^2*c^3*d^3 + 3*a^2*b*d^3)*x^3 + 2*a^3 + 9*(10*a*b^2*c^4*d^2 + 3*a^2*b*c*d^2)*x^2 + 9*(4*a*b^2*c^5*d + 3*a^
2*b*c^2*d)*x)*e^(-4)/((b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)^2*(d*x + c)^3*a^4*d)

________________________________________________________________________________________

maple [B]  time = 0.03, size = 335, normalized size = 2.89 \[ -\frac {2 b^{2} d^{2} x^{3}}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{3} e^{4}}-\frac {2 b^{2} c d \,x^{2}}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{3} e^{4}}-\frac {2 b^{2} c^{2} x}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{3} e^{4}}-\frac {2 b^{2} c^{3}}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{3} d \,e^{4}}-\frac {5 b}{6 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{2} d \,e^{4}}-\frac {3 b \ln \left (d x +c \right )}{a^{4} d \,e^{4}}+\frac {b \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )}{a^{4} d \,e^{4}}-\frac {1}{3 \left (d x +c \right )^{3} a^{3} d \,e^{4}} \]

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)^3,x)

[Out]

-1/3/a^3/d/e^4/(d*x+c)^3-3*b*ln(d*x+c)/a^4/d/e^4-2/3/e^4/a^3*b^2/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)
^2*d^2*x^3-2/e^4/a^3*b^2/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*c*d*x^2-2/e^4/a^3*b^2/(b*d^3*x^3+3*b*
c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*c^2*x-2/3/e^4/a^3*b^2/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2/d*c^3-5
/6/e^4/a^2*b/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2/d+1/e^4/a^4*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 [B]  time = 0.71, size = 474, normalized size = 4.09 \[ -\frac {6 \, b^{2} d^{6} x^{6} + 36 \, b^{2} c d^{5} x^{5} + 90 \, b^{2} c^{2} d^{4} x^{4} + 6 \, b^{2} c^{6} + 3 \, {\left (40 \, b^{2} c^{3} + 3 \, a b\right )} d^{3} x^{3} + 9 \, a b c^{3} + 9 \, {\left (10 \, b^{2} c^{4} + 3 \, a b c\right )} d^{2} x^{2} + 9 \, {\left (4 \, b^{2} c^{5} + 3 \, a b c^{2}\right )} d x + 2 \, a^{2}}{6 \, {\left (a^{3} b^{2} d^{10} e^{4} x^{9} + 9 \, a^{3} b^{2} c d^{9} e^{4} x^{8} + 36 \, a^{3} b^{2} c^{2} d^{8} e^{4} x^{7} + 2 \, {\left (42 \, a^{3} b^{2} c^{3} + a^{4} b\right )} d^{7} e^{4} x^{6} + 6 \, {\left (21 \, a^{3} b^{2} c^{4} + 2 \, a^{4} b c\right )} d^{6} e^{4} x^{5} + 6 \, {\left (21 \, a^{3} b^{2} c^{5} + 5 \, a^{4} b c^{2}\right )} d^{5} e^{4} x^{4} + {\left (84 \, a^{3} b^{2} c^{6} + 40 \, a^{4} b c^{3} + a^{5}\right )} d^{4} e^{4} x^{3} + 3 \, {\left (12 \, a^{3} b^{2} c^{7} + 10 \, a^{4} b c^{4} + a^{5} c\right )} d^{3} e^{4} x^{2} + 3 \, {\left (3 \, a^{3} b^{2} c^{8} + 4 \, a^{4} b c^{5} + a^{5} c^{2}\right )} d^{2} e^{4} x + {\left (a^{3} b^{2} c^{9} + 2 \, a^{4} b c^{6} + a^{5} c^{3}\right )} d e^{4}\right )}} + \frac {b \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{a^{4} d e^{4}} - \frac {3 \, b \log \left (d x + c\right )}{a^{4} d e^{4}} \]

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)^3,x, algorithm="maxima")

[Out]

-1/6*(6*b^2*d^6*x^6 + 36*b^2*c*d^5*x^5 + 90*b^2*c^2*d^4*x^4 + 6*b^2*c^6 + 3*(40*b^2*c^3 + 3*a*b)*d^3*x^3 + 9*a
*b*c^3 + 9*(10*b^2*c^4 + 3*a*b*c)*d^2*x^2 + 9*(4*b^2*c^5 + 3*a*b*c^2)*d*x + 2*a^2)/(a^3*b^2*d^10*e^4*x^9 + 9*a
^3*b^2*c*d^9*e^4*x^8 + 36*a^3*b^2*c^2*d^8*e^4*x^7 + 2*(42*a^3*b^2*c^3 + a^4*b)*d^7*e^4*x^6 + 6*(21*a^3*b^2*c^4
+ 2*a^4*b*c)*d^6*e^4*x^5 + 6*(21*a^3*b^2*c^5 + 5*a^4*b*c^2)*d^5*e^4*x^4 + (84*a^3*b^2*c^6 + 40*a^4*b*c^3 + a^
5)*d^4*e^4*x^3 + 3*(12*a^3*b^2*c^7 + 10*a^4*b*c^4 + a^5*c)*d^3*e^4*x^2 + 3*(3*a^3*b^2*c^8 + 4*a^4*b*c^5 + a^5*
c^2)*d^2*e^4*x + (a^3*b^2*c^9 + 2*a^4*b*c^6 + a^5*c^3)*d*e^4) + 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^4*d*e^4) - 3*b*log(d*x + c)/(a^4*d*e^4)

________________________________________________________________________________________

mupad [B]  time = 3.57, size = 507, normalized size = 4.37 \[ \frac {b\,\ln \left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )}{a^4\,d\,e^4}-\frac {\frac {2\,a^2+9\,a\,b\,c^3+6\,b^2\,c^6}{6\,a^3\,d}+\frac {3\,x^2\,\left (10\,d\,b^2\,c^4+3\,a\,d\,b\,c\right )}{2\,a^3}+\frac {3\,x\,\left (4\,b^2\,c^5+3\,a\,b\,c^2\right )}{2\,a^3}+\frac {x^3\,\left (40\,b^2\,c^3\,d^2+3\,a\,b\,d^2\right )}{2\,a^3}+\frac {b^2\,d^5\,x^6}{a^3}+\frac {15\,b^2\,c^2\,d^3\,x^4}{a^3}+\frac {6\,b^2\,c\,d^4\,x^5}{a^3}}{x^5\,\left (126\,b^2\,c^4\,d^5\,e^4+12\,a\,b\,c\,d^5\,e^4\right )+x^3\,\left (a^2\,d^3\,e^4+40\,a\,b\,c^3\,d^3\,e^4+84\,b^2\,c^6\,d^3\,e^4\right )+x\,\left (3\,d\,a^2\,c^2\,e^4+12\,d\,a\,b\,c^5\,e^4+9\,d\,b^2\,c^8\,e^4\right )+x^4\,\left (126\,b^2\,c^5\,d^4\,e^4+30\,a\,b\,c^2\,d^4\,e^4\right )+x^2\,\left (3\,a^2\,c\,d^2\,e^4+30\,a\,b\,c^4\,d^2\,e^4+36\,b^2\,c^7\,d^2\,e^4\right )+x^6\,\left (84\,b^2\,c^3\,d^6\,e^4+2\,a\,b\,d^6\,e^4\right )+a^2\,c^3\,e^4+b^2\,c^9\,e^4+b^2\,d^9\,e^4\,x^9+2\,a\,b\,c^6\,e^4+36\,b^2\,c^2\,d^7\,e^4\,x^7+9\,b^2\,c\,d^8\,e^4\,x^8}-\frac {3\,b\,\ln \left (c+d\,x\right )}{a^4\,d\,e^4} \]

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)^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))/(a^4*d*e^4) - ((2*a^2 + 6*b^2*c^6 + 9*a*b*c^3)/(6
*a^3*d) + (3*x^2*(10*b^2*c^4*d + 3*a*b*c*d))/(2*a^3) + (3*x*(4*b^2*c^5 + 3*a*b*c^2))/(2*a^3) + (x^3*(40*b^2*c^
3*d^2 + 3*a*b*d^2))/(2*a^3) + (b^2*d^5*x^6)/a^3 + (15*b^2*c^2*d^3*x^4)/a^3 + (6*b^2*c*d^4*x^5)/a^3)/(x^5*(126*
b^2*c^4*d^5*e^4 + 12*a*b*c*d^5*e^4) + x^3*(a^2*d^3*e^4 + 84*b^2*c^6*d^3*e^4 + 40*a*b*c^3*d^3*e^4) + x*(3*a^2*c
^2*d*e^4 + 9*b^2*c^8*d*e^4 + 12*a*b*c^5*d*e^4) + x^4*(126*b^2*c^5*d^4*e^4 + 30*a*b*c^2*d^4*e^4) + x^2*(3*a^2*c
*d^2*e^4 + 36*b^2*c^7*d^2*e^4 + 30*a*b*c^4*d^2*e^4) + x^6*(84*b^2*c^3*d^6*e^4 + 2*a*b*d^6*e^4) + a^2*c^3*e^4 +
b^2*c^9*e^4 + b^2*d^9*e^4*x^9 + 2*a*b*c^6*e^4 + 36*b^2*c^2*d^7*e^4*x^7 + 9*b^2*c*d^8*e^4*x^8) - (3*b*log(c +
d*x))/(a^4*d*e^4)

________________________________________________________________________________________

sympy [B]  time = 8.76, size = 578, normalized size = 4.98 \[ \frac {- 2 a^{2} - 9 a b c^{3} - 6 b^{2} c^{6} - 90 b^{2} c^{2} d^{4} x^{4} - 36 b^{2} c d^{5} x^{5} - 6 b^{2} d^{6} x^{6} + x^{3} \left (- 9 a b d^{3} - 120 b^{2} c^{3} d^{3}\right ) + x^{2} \left (- 27 a b c d^{2} - 90 b^{2} c^{4} d^{2}\right ) + x \left (- 27 a b c^{2} d - 36 b^{2} c^{5} d\right )}{6 a^{5} c^{3} d e^{4} + 12 a^{4} b c^{6} d e^{4} + 6 a^{3} b^{2} c^{9} d e^{4} + 216 a^{3} b^{2} c^{2} d^{8} e^{4} x^{7} + 54 a^{3} b^{2} c d^{9} e^{4} x^{8} + 6 a^{3} b^{2} d^{10} e^{4} x^{9} + x^{6} \left (12 a^{4} b d^{7} e^{4} + 504 a^{3} b^{2} c^{3} d^{7} e^{4}\right ) + x^{5} \left (72 a^{4} b c d^{6} e^{4} + 756 a^{3} b^{2} c^{4} d^{6} e^{4}\right ) + x^{4} \left (180 a^{4} b c^{2} d^{5} e^{4} + 756 a^{3} b^{2} c^{5} d^{5} e^{4}\right ) + x^{3} \left (6 a^{5} d^{4} e^{4} + 240 a^{4} b c^{3} d^{4} e^{4} + 504 a^{3} b^{2} c^{6} d^{4} e^{4}\right ) + x^{2} \left (18 a^{5} c d^{3} e^{4} + 180 a^{4} b c^{4} d^{3} e^{4} + 216 a^{3} b^{2} c^{7} d^{3} e^{4}\right ) + x \left (18 a^{5} c^{2} d^{2} e^{4} + 72 a^{4} b c^{5} d^{2} e^{4} + 54 a^{3} b^{2} c^{8} d^{2} e^{4}\right )} - \frac {3 b \log {\left (\frac {c}{d} + x \right )}}{a^{4} d e^{4}} + \frac {b \log {\left (\frac {3 c^{2} x}{d^{2}} + \frac {3 c x^{2}}{d} + x^{3} + \frac {a + b c^{3}}{b d^{3}} \right )}}{a^{4} d e^{4}} \]

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)**3,x)

[Out]

(-2*a**2 - 9*a*b*c**3 - 6*b**2*c**6 - 90*b**2*c**2*d**4*x**4 - 36*b**2*c*d**5*x**5 - 6*b**2*d**6*x**6 + x**3*(
-9*a*b*d**3 - 120*b**2*c**3*d**3) + x**2*(-27*a*b*c*d**2 - 90*b**2*c**4*d**2) + x*(-27*a*b*c**2*d - 36*b**2*c*
*5*d))/(6*a**5*c**3*d*e**4 + 12*a**4*b*c**6*d*e**4 + 6*a**3*b**2*c**9*d*e**4 + 216*a**3*b**2*c**2*d**8*e**4*x*
*7 + 54*a**3*b**2*c*d**9*e**4*x**8 + 6*a**3*b**2*d**10*e**4*x**9 + x**6*(12*a**4*b*d**7*e**4 + 504*a**3*b**2*c
**3*d**7*e**4) + x**5*(72*a**4*b*c*d**6*e**4 + 756*a**3*b**2*c**4*d**6*e**4) + x**4*(180*a**4*b*c**2*d**5*e**4
+ 756*a**3*b**2*c**5*d**5*e**4) + x**3*(6*a**5*d**4*e**4 + 240*a**4*b*c**3*d**4*e**4 + 504*a**3*b**2*c**6*d**
4*e**4) + x**2*(18*a**5*c*d**3*e**4 + 180*a**4*b*c**4*d**3*e**4 + 216*a**3*b**2*c**7*d**3*e**4) + x*(18*a**5*c
**2*d**2*e**4 + 72*a**4*b*c**5*d**2*e**4 + 54*a**3*b**2*c**8*d**2*e**4)) - 3*b*log(c/d + x)/(a**4*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))/(a**4*d*e**4)

________________________________________________________________________________________