\(\int \frac {c e+d e x}{(a+b (c+d x)^3)^3} \, dx\) [2905]

Optimal result

Integrand size = 22, antiderivative size = 207 \[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^3} \, dx=\frac {e (c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac {2 e (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}-\frac {2 e \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{7/3} b^{2/3} d}-\frac {2 e \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac {e \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{7/3} b^{2/3} d} \]

[Out]

1/6*e*(d*x+c)^2/a/d/(a+b*(d*x+c)^3)^2+2/9*e*(d*x+c)^2/a^2/d/(a+b*(d*x+c)^3)-2/27*e*ln(a^(1/3)+b^(1/3)*(d*x+c))
/a^(7/3)/b^(2/3)/d+1/27*e*ln(a^(2/3)-a^(1/3)*b^(1/3)*(d*x+c)+b^(2/3)*(d*x+c)^2)/a^(7/3)/b^(2/3)/d-2/27*e*arcta
n(1/3*(a^(1/3)-2*b^(1/3)*(d*x+c))/a^(1/3)*3^(1/2))/a^(7/3)/b^(2/3)/d*3^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {379, 296, 298, 31, 648, 631, 210, 642} \[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^3} \, dx=-\frac {2 e \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{7/3} b^{2/3} d}-\frac {2 e \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac {e \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{7/3} b^{2/3} d}+\frac {2 e (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}+\frac {e (c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2} \]

[In]

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

[Out]

(e*(c + d*x)^2)/(6*a*d*(a + b*(c + d*x)^3)^2) + (2*e*(c + d*x)^2)/(9*a^2*d*(a + b*(c + d*x)^3)) - (2*e*ArcTan[
(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(7/3)*b^(2/3)*d) - (2*e*Log[a^(1/3) + b^(1/3)
*(c + d*x)])/(27*a^(7/3)*b^(2/3)*d) + (e*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2])/(27*a
^(7/3)*b^(2/3)*d)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 296

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 298

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Dist[-(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 379

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 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = \frac {e \text {Subst}\left (\int \frac {x}{\left (a+b x^3\right )^3} \, dx,x,c+d x\right )}{d} \\ & = \frac {e (c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac {(2 e) \text {Subst}\left (\int \frac {x}{\left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{3 a d} \\ & = \frac {e (c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac {2 e (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}+\frac {(2 e) \text {Subst}\left (\int \frac {x}{a+b x^3} \, dx,x,c+d x\right )}{9 a^2 d} \\ & = \frac {e (c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac {2 e (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}-\frac {(2 e) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{27 a^{7/3} \sqrt [3]{b} d}+\frac {(2 e) \text {Subst}\left (\int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{27 a^{7/3} \sqrt [3]{b} d} \\ & = \frac {e (c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac {2 e (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}-\frac {2 e \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac {e \text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{27 a^{7/3} b^{2/3} d}+\frac {e \text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{9 a^2 \sqrt [3]{b} d} \\ & = \frac {e (c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac {2 e (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}-\frac {2 e \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac {e \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{7/3} b^{2/3} d}+\frac {(2 e) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{9 a^{7/3} b^{2/3} d} \\ & = \frac {e (c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac {2 e (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}-\frac {2 e \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{9 \sqrt {3} a^{7/3} b^{2/3} d}-\frac {2 e \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac {e \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{7/3} b^{2/3} d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.87 \[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^3} \, dx=\frac {e \left (\frac {9 a^{4/3} (c+d x)^2}{\left (a+b (c+d x)^3\right )^2}+\frac {12 \sqrt [3]{a} (c+d x)^2}{a+b (c+d x)^3}+\frac {4 \sqrt {3} \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{2/3}}-\frac {4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{b^{2/3}}+\frac {2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{b^{2/3}}\right )}{54 a^{7/3} d} \]

[In]

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

[Out]

(e*((9*a^(4/3)*(c + d*x)^2)/(a + b*(c + d*x)^3)^2 + (12*a^(1/3)*(c + d*x)^2)/(a + b*(c + d*x)^3) + (4*Sqrt[3]*
ArcTan[(-a^(1/3) + 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))])/b^(2/3) - (4*Log[a^(1/3) + b^(1/3)*(c + d*x)])/b^(
2/3) + (2*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2])/b^(2/3)))/(54*a^(7/3)*d)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.10 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.04

method	result	size
default	\(e \left (\frac {\frac {2 b \,d^{4} x^{5}}{9 a^{2}}+\frac {10 b c \,d^{3} x^{4}}{9 a^{2}}+\frac {20 b \,c^{2} d^{2} x^{3}}{9 a^{2}}+\frac {d \left (40 c^{3} b +7 a \right ) x^{2}}{18 a^{2}}+\frac {c \left (10 c^{3} b +7 a \right ) x}{9 a^{2}}+\frac {c^{2} \left (4 c^{3} b +7 a \right )}{18 d \,a^{2}}}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )^{2}}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b \,d^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\left (\textit {\_R} d +c \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}\right )}{27 a^{2} b d}\right )\)	\(216\)
risch	\(\frac {\frac {2 b \,d^{4} e \,x^{5}}{9 a^{2}}+\frac {10 b c \,d^{3} e \,x^{4}}{9 a^{2}}+\frac {20 b \,c^{2} d^{2} e \,x^{3}}{9 a^{2}}+\frac {d e \left (40 c^{3} b +7 a \right ) x^{2}}{18 a^{2}}+\frac {c e \left (10 c^{3} b +7 a \right ) x}{9 a^{2}}+\frac {c^{2} e \left (4 c^{3} b +7 a \right )}{18 d \,a^{2}}}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )^{2}}+\frac {2 e \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b \,d^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\left (\textit {\_R} d +c \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}\right )}{27 a^{2} b d}\)	\(221\)

[In]

int((d*e*x+c*e)/(a+b*(d*x+c)^3)^3,x,method=_RETURNVERBOSE)

[Out]

e*((2/9*b*d^4/a^2*x^5+10/9*b*c*d^3/a^2*x^4+20/9*b*c^2*d^2/a^2*x^3+1/18*d*(40*b*c^3+7*a)/a^2*x^2+1/9*c*(10*b*c^
3+7*a)/a^2*x+1/18*c^2/d*(4*b*c^3+7*a)/a^2)/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2+2/27/a^2/b/d*sum((_
R*d+c)/(_R^2*d^2+2*_R*c*d+c^2)*ln(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)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 829 vs. \(2 (166) = 332\).

Time = 0.31 (sec) , antiderivative size = 1780, normalized size of antiderivative = 8.60 \[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/54*(12*a*b^3*d^5*e*x^5 + 60*a*b^3*c*d^4*e*x^4 + 120*a*b^3*c^2*d^3*e*x^3 + 3*(40*a*b^3*c^3 + 7*a^2*b^2)*d^2*
e*x^2 + 6*(10*a*b^3*c^4 + 7*a^2*b^2*c)*d*e*x + 6*sqrt(1/3)*(a*b^3*d^6*e*x^6 + 6*a*b^3*c*d^5*e*x^5 + 15*a*b^3*c
^2*d^4*e*x^4 + 2*(10*a*b^3*c^3 + a^2*b^2)*d^3*e*x^3 + 3*(5*a*b^3*c^4 + 2*a^2*b^2*c)*d^2*e*x^2 + 6*(a*b^3*c^5 +
a^2*b^2*c^2)*d*e*x + (a*b^3*c^6 + 2*a^2*b^2*c^3 + a^3*b)*e)*sqrt((-a*b^2)^(1/3)/a)*log((2*b^2*d^3*x^3 + 6*b^2
*c*d^2*x^2 + 6*b^2*c^2*d*x + 2*b^2*c^3 - a*b + 3*sqrt(1/3)*(a*b*d*x + a*b*c + 2*(d^2*x^2 + 2*c*d*x + c^2)*(-a*
b^2)^(2/3) + (-a*b^2)^(1/3)*a)*sqrt((-a*b^2)^(1/3)/a) - 3*(-a*b^2)^(2/3)*(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)) + 2*(b^2*d^6*e*x^6 + 6*b^2*c*d^5*e*x^5 + 15*b^2*c^2*d^4*e*x^4 + 2*(10*b^2*c^3 + a
*b)*d^3*e*x^3 + 3*(5*b^2*c^4 + 2*a*b*c)*d^2*e*x^2 + 6*(b^2*c^5 + a*b*c^2)*d*e*x + (b^2*c^6 + 2*a*b*c^3 + a^2)*
e)*(-a*b^2)^(2/3)*log(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + (-a*b^2)^(1/3)*(b*d*x + b*c) + (-a*b^2)^(2/3)) - 4
*(b^2*d^6*e*x^6 + 6*b^2*c*d^5*e*x^5 + 15*b^2*c^2*d^4*e*x^4 + 2*(10*b^2*c^3 + a*b)*d^3*e*x^3 + 3*(5*b^2*c^4 + 2
*a*b*c)*d^2*e*x^2 + 6*(b^2*c^5 + a*b*c^2)*d*e*x + (b^2*c^6 + 2*a*b*c^3 + a^2)*e)*(-a*b^2)^(2/3)*log(b*d*x + b*
c - (-a*b^2)^(1/3)) + 3*(4*a*b^3*c^5 + 7*a^2*b^2*c^2)*e)/(a^3*b^4*d^7*x^6 + 6*a^3*b^4*c*d^6*x^5 + 15*a^3*b^4*c
^2*d^5*x^4 + 2*(10*a^3*b^4*c^3 + a^4*b^3)*d^4*x^3 + 3*(5*a^3*b^4*c^4 + 2*a^4*b^3*c)*d^3*x^2 + 6*(a^3*b^4*c^5 +
a^4*b^3*c^2)*d^2*x + (a^3*b^4*c^6 + 2*a^4*b^3*c^3 + a^5*b^2)*d), 1/54*(12*a*b^3*d^5*e*x^5 + 60*a*b^3*c*d^4*e*
x^4 + 120*a*b^3*c^2*d^3*e*x^3 + 3*(40*a*b^3*c^3 + 7*a^2*b^2)*d^2*e*x^2 + 6*(10*a*b^3*c^4 + 7*a^2*b^2*c)*d*e*x
+ 12*sqrt(1/3)*(a*b^3*d^6*e*x^6 + 6*a*b^3*c*d^5*e*x^5 + 15*a*b^3*c^2*d^4*e*x^4 + 2*(10*a*b^3*c^3 + a^2*b^2)*d^
3*e*x^3 + 3*(5*a*b^3*c^4 + 2*a^2*b^2*c)*d^2*e*x^2 + 6*(a*b^3*c^5 + a^2*b^2*c^2)*d*e*x + (a*b^3*c^6 + 2*a^2*b^2
*c^3 + a^3*b)*e)*sqrt(-(-a*b^2)^(1/3)/a)*arctan(sqrt(1/3)*(2*b*d*x + 2*b*c + (-a*b^2)^(1/3))*sqrt(-(-a*b^2)^(1
/3)/a)/b) + 2*(b^2*d^6*e*x^6 + 6*b^2*c*d^5*e*x^5 + 15*b^2*c^2*d^4*e*x^4 + 2*(10*b^2*c^3 + a*b)*d^3*e*x^3 + 3*(
5*b^2*c^4 + 2*a*b*c)*d^2*e*x^2 + 6*(b^2*c^5 + a*b*c^2)*d*e*x + (b^2*c^6 + 2*a*b*c^3 + a^2)*e)*(-a*b^2)^(2/3)*l
og(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + (-a*b^2)^(1/3)*(b*d*x + b*c) + (-a*b^2)^(2/3)) - 4*(b^2*d^6*e*x^6 + 6
*b^2*c*d^5*e*x^5 + 15*b^2*c^2*d^4*e*x^4 + 2*(10*b^2*c^3 + a*b)*d^3*e*x^3 + 3*(5*b^2*c^4 + 2*a*b*c)*d^2*e*x^2 +
6*(b^2*c^5 + a*b*c^2)*d*e*x + (b^2*c^6 + 2*a*b*c^3 + a^2)*e)*(-a*b^2)^(2/3)*log(b*d*x + b*c - (-a*b^2)^(1/3))
+ 3*(4*a*b^3*c^5 + 7*a^2*b^2*c^2)*e)/(a^3*b^4*d^7*x^6 + 6*a^3*b^4*c*d^6*x^5 + 15*a^3*b^4*c^2*d^5*x^4 + 2*(10*
a^3*b^4*c^3 + a^4*b^3)*d^4*x^3 + 3*(5*a^3*b^4*c^4 + 2*a^4*b^3*c)*d^3*x^2 + 6*(a^3*b^4*c^5 + a^4*b^3*c^2)*d^2*x
+ (a^3*b^4*c^6 + 2*a^4*b^3*c^3 + a^5*b^2)*d)]

Sympy [A] (verification not implemented)

Time = 1.28 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.56 \[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^3} \, dx=\frac {7 a c^{2} e + 4 b c^{5} e + 40 b c^{2} d^{3} e x^{3} + 20 b c d^{4} e x^{4} + 4 b d^{5} e x^{5} + x^{2} \cdot \left (7 a d^{2} e + 40 b c^{3} d^{2} e\right ) + x \left (14 a c d e + 20 b c^{4} d e\right )}{18 a^{4} d + 36 a^{3} b c^{3} d + 18 a^{2} b^{2} c^{6} d + 270 a^{2} b^{2} c^{2} d^{5} x^{4} + 108 a^{2} b^{2} c d^{6} x^{5} + 18 a^{2} b^{2} d^{7} x^{6} + x^{3} \cdot \left (36 a^{3} b d^{4} + 360 a^{2} b^{2} c^{3} d^{4}\right ) + x^{2} \cdot \left (108 a^{3} b c d^{3} + 270 a^{2} b^{2} c^{4} d^{3}\right ) + x \left (108 a^{3} b c^{2} d^{2} + 108 a^{2} b^{2} c^{5} d^{2}\right )} + \frac {e \operatorname {RootSum} {\left (19683 t^{3} a^{7} b^{2} + 8, \left ( t \mapsto t \log {\left (x + \frac {729 t^{2} a^{5} b e^{2} + 4 c e^{2}}{4 d e^{2}} \right )} \right )\right )}}{d} \]

[In]

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

[Out]

(7*a*c**2*e + 4*b*c**5*e + 40*b*c**2*d**3*e*x**3 + 20*b*c*d**4*e*x**4 + 4*b*d**5*e*x**5 + x**2*(7*a*d**2*e + 4
0*b*c**3*d**2*e) + x*(14*a*c*d*e + 20*b*c**4*d*e))/(18*a**4*d + 36*a**3*b*c**3*d + 18*a**2*b**2*c**6*d + 270*a
**2*b**2*c**2*d**5*x**4 + 108*a**2*b**2*c*d**6*x**5 + 18*a**2*b**2*d**7*x**6 + x**3*(36*a**3*b*d**4 + 360*a**2
*b**2*c**3*d**4) + x**2*(108*a**3*b*c*d**3 + 270*a**2*b**2*c**4*d**3) + x*(108*a**3*b*c**2*d**2 + 108*a**2*b**
2*c**5*d**2)) + e*RootSum(19683*_t**3*a**7*b**2 + 8, Lambda(_t, _t*log(x + (729*_t**2*a**5*b*e**2 + 4*c*e**2)/
(4*d*e**2))))/d

Maxima [F]

\[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^3} \, dx=\int { \frac {d e x + c e}{{\left ({\left (d x + c\right )}^{3} b + a\right )}^{3}} \,d x } \]

[In]

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

[Out]

1/18*(4*b*d^5*e*x^5 + 20*b*c*d^4*e*x^4 + 40*b*c^2*d^3*e*x^3 + (40*b*c^3 + 7*a)*d^2*e*x^2 + 2*(10*b*c^4 + 7*a*c
)*d*e*x + (4*b*c^5 + 7*a*c^2)*e)/(a^2*b^2*d^7*x^6 + 6*a^2*b^2*c*d^6*x^5 + 15*a^2*b^2*c^2*d^5*x^4 + 2*(10*a^2*b
^2*c^3 + a^3*b)*d^4*x^3 + 3*(5*a^2*b^2*c^4 + 2*a^3*b*c)*d^3*x^2 + 6*(a^2*b^2*c^5 + a^3*b*c^2)*d^2*x + (a^2*b^2
*c^6 + 2*a^3*b*c^3 + a^4)*d) + 2/9*e*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)/a^2

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.38 \[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^3} \, dx=-\frac {2 \, \sqrt {3} \left (-\frac {e^{3}}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}}{3 \, \left (-a^{2} b\right )^{\frac {2}{3}}}\right ) + \left (-\frac {e^{3}}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}^{2} + 3 \, \left (-a^{2} b\right )^{\frac {4}{3}}\right ) - 2 \, \left (-\frac {e^{3}}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | a b d x + a b c + \left (-a^{2} b\right )^{\frac {2}{3}} \right |}\right )}{27 \, a^{2}} + \frac {4 \, b d^{5} e x^{5} + 20 \, b c d^{4} e x^{4} + 40 \, b c^{2} d^{3} e x^{3} + 40 \, b c^{3} d^{2} e x^{2} + 20 \, b c^{4} d e x + 4 \, b c^{5} e + 7 \, a d^{2} e x^{2} + 14 \, a c d e x + 7 \, a c^{2} e}{18 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}^{2} a^{2} d} \]

[In]

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

[Out]

-1/27*(2*sqrt(3)*(-e^3/(a*b^2*d^3))^(1/3)*arctan(1/3*sqrt(3)*(2*a*b*d*x + 2*a*b*c - (-a^2*b)^(2/3))/(-a^2*b)^(
2/3)) + (-e^3/(a*b^2*d^3))^(1/3)*log((2*a*b*d*x + 2*a*b*c - (-a^2*b)^(2/3))^2 + 3*(-a^2*b)^(4/3)) - 2*(-e^3/(a
*b^2*d^3))^(1/3)*log(abs(a*b*d*x + a*b*c + (-a^2*b)^(2/3))))/a^2 + 1/18*(4*b*d^5*e*x^5 + 20*b*c*d^4*e*x^4 + 40
*b*c^2*d^3*e*x^3 + 40*b*c^3*d^2*e*x^2 + 20*b*c^4*d*e*x + 4*b*c^5*e + 7*a*d^2*e*x^2 + 14*a*c*d*e*x + 7*a*c^2*e)
/((b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)^2*a^2*d)

Mupad [B] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.72 \[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^3} \, dx=\frac {\frac {4\,b\,e\,c^5+7\,a\,e\,c^2}{18\,a^2\,d}+\frac {x^2\,\left (40\,b\,d\,e\,c^3+7\,a\,d\,e\right )}{18\,a^2}+\frac {c\,x\,\left (10\,b\,e\,c^3+7\,a\,e\right )}{9\,a^2}+\frac {2\,b\,d^4\,e\,x^5}{9\,a^2}+\frac {20\,b\,c^2\,d^2\,e\,x^3}{9\,a^2}+\frac {10\,b\,c\,d^3\,e\,x^4}{9\,a^2}}{x^3\,\left (20\,b^2\,c^3\,d^3+2\,a\,b\,d^3\right )+x^2\,\left (15\,b^2\,c^4\,d^2+6\,a\,b\,c\,d^2\right )+a^2+x\,\left (6\,d\,b^2\,c^5+6\,a\,d\,b\,c^2\right )+b^2\,c^6+b^2\,d^6\,x^6+2\,a\,b\,c^3+6\,b^2\,c\,d^5\,x^5+15\,b^2\,c^2\,d^4\,x^4}-\frac {2\,e\,\ln \left (b^{1/3}\,c+a^{1/3}+b^{1/3}\,d\,x\right )}{27\,a^{7/3}\,b^{2/3}\,d}+\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,c-2\,b^{1/3}\,d\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (e-\sqrt {3}\,e\,1{}\mathrm {i}\right )}{27\,a^{7/3}\,b^{2/3}\,d}+\frac {\ln \left (2\,b^{1/3}\,c-a^{1/3}+2\,b^{1/3}\,d\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (e+\sqrt {3}\,e\,1{}\mathrm {i}\right )}{27\,a^{7/3}\,b^{2/3}\,d} \]

[In]

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

[Out]

((7*a*c^2*e + 4*b*c^5*e)/(18*a^2*d) + (x^2*(7*a*d*e + 40*b*c^3*d*e))/(18*a^2) + (c*x*(7*a*e + 10*b*c^3*e))/(9*
a^2) + (2*b*d^4*e*x^5)/(9*a^2) + (20*b*c^2*d^2*e*x^3)/(9*a^2) + (10*b*c*d^3*e*x^4)/(9*a^2))/(x^3*(20*b^2*c^3*d
^3 + 2*a*b*d^3) + x^2*(15*b^2*c^4*d^2 + 6*a*b*c*d^2) + a^2 + x*(6*b^2*c^5*d + 6*a*b*c^2*d) + b^2*c^6 + b^2*d^6
*x^6 + 2*a*b*c^3 + 6*b^2*c*d^5*x^5 + 15*b^2*c^2*d^4*x^4) - (2*e*log(b^(1/3)*c + a^(1/3) + b^(1/3)*d*x))/(27*a^
(7/3)*b^(2/3)*d) + (log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*c + a^(1/3) - 2*b^(1/3)*d*x)*(e - 3^(1/2)*e*1i))/(27*a^
(7/3)*b^(2/3)*d) + (log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*c - a^(1/3) + 2*b^(1/3)*d*x)*(e + 3^(1/2)*e*1i))/(27*a^
(7/3)*b^(2/3)*d)