\(\int \frac {1}{(c+d x)^2 (a+b (c+d x)^3)^3} \, dx\) [2883]

Optimal result

Integrand size = 21, antiderivative size = 219 \[ \int \frac {1}{(c+d x)^2 \left (a+b (c+d x)^3\right )^3} \, dx=-\frac {14}{9 a^3 d (c+d x)}+\frac {1}{6 a d (c+d x) \left (a+b (c+d x)^3\right )^2}+\frac {7}{18 a^2 d (c+d x) \left (a+b (c+d x)^3\right )}+\frac {14 \sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{10/3} d}+\frac {14 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{10/3} d}-\frac {7 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{10/3} d} \]

[Out]

-14/9/a^3/d/(d*x+c)+1/6/a/d/(d*x+c)/(a+b*(d*x+c)^3)^2+7/18/a^2/d/(d*x+c)/(a+b*(d*x+c)^3)+14/27*b^(1/3)*ln(a^(1
/3)+b^(1/3)*(d*x+c))/a^(10/3)/d-7/27*b^(1/3)*ln(a^(2/3)-a^(1/3)*b^(1/3)*(d*x+c)+b^(2/3)*(d*x+c)^2)/a^(10/3)/d+
14/27*b^(1/3)*arctan(1/3*(a^(1/3)-2*b^(1/3)*(d*x+c))/a^(1/3)*3^(1/2))/a^(10/3)/d*3^(1/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {379, 296, 331, 298, 31, 648, 631, 210, 642} \[ \int \frac {1}{(c+d x)^2 \left (a+b (c+d x)^3\right )^3} \, dx=\frac {14 \sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{10/3} d}-\frac {7 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{10/3} d}+\frac {14 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{10/3} d}-\frac {14}{9 a^3 d (c+d x)}+\frac {7}{18 a^2 d (c+d x) \left (a+b (c+d x)^3\right )}+\frac {1}{6 a d (c+d x) \left (a+b (c+d x)^3\right )^2} \]

[In]

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

[Out]

-14/(9*a^3*d*(c + d*x)) + 1/(6*a*d*(c + d*x)*(a + b*(c + d*x)^3)^2) + 7/(18*a^2*d*(c + d*x)*(a + b*(c + d*x)^3
)) + (14*b^(1/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(10/3)*d) + (14*b^(1/
3)*Log[a^(1/3) + b^(1/3)*(c + d*x)])/(27*a^(10/3)*d) - (7*b^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^
(2/3)*(c + d*x)^2])/(27*a^(10/3)*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 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, 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 {\text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^3\right )^3} \, dx,x,c+d x\right )}{d} \\ & = \frac {1}{6 a d (c+d x) \left (a+b (c+d x)^3\right )^2}+\frac {7 \text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{6 a d} \\ & = \frac {1}{6 a d (c+d x) \left (a+b (c+d x)^3\right )^2}+\frac {7}{18 a^2 d (c+d x) \left (a+b (c+d x)^3\right )}+\frac {14 \text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^3\right )} \, dx,x,c+d x\right )}{9 a^2 d} \\ & = -\frac {14}{9 a^3 d (c+d x)}+\frac {1}{6 a d (c+d x) \left (a+b (c+d x)^3\right )^2}+\frac {7}{18 a^2 d (c+d x) \left (a+b (c+d x)^3\right )}-\frac {(14 b) \text {Subst}\left (\int \frac {x}{a+b x^3} \, dx,x,c+d x\right )}{9 a^3 d} \\ & = -\frac {14}{9 a^3 d (c+d x)}+\frac {1}{6 a d (c+d x) \left (a+b (c+d x)^3\right )^2}+\frac {7}{18 a^2 d (c+d x) \left (a+b (c+d x)^3\right )}+\frac {\left (14 b^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{27 a^{10/3} d}-\frac {\left (14 b^{2/3}\right ) \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^{10/3} d} \\ & = -\frac {14}{9 a^3 d (c+d x)}+\frac {1}{6 a d (c+d x) \left (a+b (c+d x)^3\right )^2}+\frac {7}{18 a^2 d (c+d x) \left (a+b (c+d x)^3\right )}+\frac {14 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{10/3} d}-\frac {\left (7 \sqrt [3]{b}\right ) \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^{10/3} d}-\frac {\left (7 b^{2/3}\right ) \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^3 d} \\ & = -\frac {14}{9 a^3 d (c+d x)}+\frac {1}{6 a d (c+d x) \left (a+b (c+d x)^3\right )^2}+\frac {7}{18 a^2 d (c+d x) \left (a+b (c+d x)^3\right )}+\frac {14 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{10/3} d}-\frac {7 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{10/3} d}-\frac {\left (14 \sqrt [3]{b}\right ) \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^{10/3} d} \\ & = -\frac {14}{9 a^3 d (c+d x)}+\frac {1}{6 a d (c+d x) \left (a+b (c+d x)^3\right )^2}+\frac {7}{18 a^2 d (c+d x) \left (a+b (c+d x)^3\right )}+\frac {14 \sqrt [3]{b} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{9 \sqrt {3} a^{10/3} d}+\frac {14 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{10/3} d}-\frac {7 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{10/3} d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.89 \[ \int \frac {1}{(c+d x)^2 \left (a+b (c+d x)^3\right )^3} \, dx=\frac {-\frac {54 \sqrt [3]{a}}{c+d x}-\frac {9 a^{4/3} b (c+d x)^2}{\left (a+b (c+d x)^3\right )^2}-\frac {30 \sqrt [3]{a} b (c+d x)^2}{a+b (c+d x)^3}-28 \sqrt {3} \sqrt [3]{b} \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )+28 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )-14 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{10/3} d} \]

[In]

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

[Out]

((-54*a^(1/3))/(c + d*x) - (9*a^(4/3)*b*(c + d*x)^2)/(a + b*(c + d*x)^3)^2 - (30*a^(1/3)*b*(c + d*x)^2)/(a + b
*(c + d*x)^3) - 28*Sqrt[3]*b^(1/3)*ArcTan[(-a^(1/3) + 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))] + 28*b^(1/3)*Log
[a^(1/3) + b^(1/3)*(c + d*x)] - 14*b^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2])/(54
*a^(10/3)*d)

Maple [C] (verified)

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

Time = 3.92 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.98

method	result	size
default	\(-\frac {1}{a^{3} d \left (d x +c \right )}-\frac {b \left (\frac {\frac {5 b \,d^{4} x^{5}}{9}+\frac {25 b c \,d^{3} x^{4}}{9}+\frac {50 d^{2} b \,c^{2} x^{3}}{9}+\left (\frac {50}{9} b \,c^{3} d +\frac {13}{18} a d \right ) x^{2}+\frac {c \left (25 c^{3} b +13 a \right ) x}{9}+\frac {c^{2} \left (10 c^{3} b +13 a \right )}{18 d}}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )^{2}}+\frac {14 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\left (\textit {\_R} d +c \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}\right )}{27 b d}\right )}{a^{3}}\)	\(215\)
risch	\(\frac {-\frac {14 b^{2} d^{5} x^{6}}{9 a^{3}}-\frac {28 b^{2} c \,d^{4} x^{5}}{3 a^{3}}-\frac {70 b^{2} c^{2} d^{3} x^{4}}{3 a^{3}}-\frac {7 b \,d^{2} \left (80 c^{3} b +7 a \right ) x^{3}}{18 a^{3}}-\frac {7 b c d \left (20 c^{3} b +7 a \right ) x^{2}}{6 a^{3}}-\frac {7 \left (8 c^{3} b +7 a \right ) b \,c^{2} x}{6 a^{3}}-\frac {28 b^{2} c^{6}+49 a b \,c^{3}+18 a^{2}}{18 d \,a^{3}}}{\left (d x +c \right ) \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )^{2}}+\frac {14 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{10} d^{3} \textit {\_Z}^{3}-b \right )}{\sum }\textit {\_R} \ln \left (\left (-4 a^{10} d^{4} \textit {\_R}^{3}+3 b d \right ) x -4 a^{10} c \,d^{3} \textit {\_R}^{3}-a^{7} d^{2} \textit {\_R}^{2}+3 b c \right )\right )}{27}\)	\(253\)

[In]

int(1/(d*x+c)^2/(a+b*(d*x+c)^3)^3,x,method=_RETURNVERBOSE)

[Out]

-1/a^3/d/(d*x+c)-1/a^3*b*((5/9*b*d^4*x^5+25/9*b*c*d^3*x^4+50/9*d^2*b*c^2*x^3+(50/9*b*c^3*d+13/18*a*d)*x^2+1/9*
c*(25*b*c^3+13*a)*x+1/18*c^2/d*(10*b*c^3+13*a))/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2+14/27/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. 855 vs. \(2 (176) = 352\).

Time = 0.28 (sec) , antiderivative size = 855, normalized size of antiderivative = 3.90 \[ \int \frac {1}{(c+d x)^2 \left (a+b (c+d x)^3\right )^3} \, dx=-\frac {84 \, b^{2} d^{6} x^{6} + 504 \, b^{2} c d^{5} x^{5} + 1260 \, b^{2} c^{2} d^{4} x^{4} + 84 \, b^{2} c^{6} + 21 \, {\left (80 \, b^{2} c^{3} + 7 \, a b\right )} d^{3} x^{3} + 147 \, a b c^{3} + 63 \, {\left (20 \, b^{2} c^{4} + 7 \, a b c\right )} d^{2} x^{2} + 63 \, {\left (8 \, b^{2} c^{5} + 7 \, a b c^{2}\right )} d x + 28 \, \sqrt {3} {\left (b^{2} d^{7} x^{7} + 7 \, b^{2} c d^{6} x^{6} + 21 \, b^{2} c^{2} d^{5} x^{5} + b^{2} c^{7} + {\left (35 \, b^{2} c^{3} + 2 \, a b\right )} d^{4} x^{4} + {\left (35 \, b^{2} c^{4} + 8 \, a b c\right )} d^{3} x^{3} + 2 \, a b c^{4} + 3 \, {\left (7 \, b^{2} c^{5} + 4 \, a b c^{2}\right )} d^{2} x^{2} + a^{2} c + {\left (7 \, b^{2} c^{6} + 8 \, a b c^{3} + a^{2}\right )} d x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (d x + c\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 14 \, {\left (b^{2} d^{7} x^{7} + 7 \, b^{2} c d^{6} x^{6} + 21 \, b^{2} c^{2} d^{5} x^{5} + b^{2} c^{7} + {\left (35 \, b^{2} c^{3} + 2 \, a b\right )} d^{4} x^{4} + {\left (35 \, b^{2} c^{4} + 8 \, a b c\right )} d^{3} x^{3} + 2 \, a b c^{4} + 3 \, {\left (7 \, b^{2} c^{5} + 4 \, a b c^{2}\right )} d^{2} x^{2} + a^{2} c + {\left (7 \, b^{2} c^{6} + 8 \, a b c^{3} + a^{2}\right )} d x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} - {\left (a d x + a c\right )} \left (\frac {b}{a}\right )^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 28 \, {\left (b^{2} d^{7} x^{7} + 7 \, b^{2} c d^{6} x^{6} + 21 \, b^{2} c^{2} d^{5} x^{5} + b^{2} c^{7} + {\left (35 \, b^{2} c^{3} + 2 \, a b\right )} d^{4} x^{4} + {\left (35 \, b^{2} c^{4} + 8 \, a b c\right )} d^{3} x^{3} + 2 \, a b c^{4} + 3 \, {\left (7 \, b^{2} c^{5} + 4 \, a b c^{2}\right )} d^{2} x^{2} + a^{2} c + {\left (7 \, b^{2} c^{6} + 8 \, a b c^{3} + a^{2}\right )} d x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b d x + b c + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 54 \, a^{2}}{54 \, {\left (a^{3} b^{2} d^{8} x^{7} + 7 \, a^{3} b^{2} c d^{7} x^{6} + 21 \, a^{3} b^{2} c^{2} d^{6} x^{5} + {\left (35 \, a^{3} b^{2} c^{3} + 2 \, a^{4} b\right )} d^{5} x^{4} + {\left (35 \, a^{3} b^{2} c^{4} + 8 \, a^{4} b c\right )} d^{4} x^{3} + 3 \, {\left (7 \, a^{3} b^{2} c^{5} + 4 \, a^{4} b c^{2}\right )} d^{3} x^{2} + {\left (7 \, a^{3} b^{2} c^{6} + 8 \, a^{4} b c^{3} + a^{5}\right )} d^{2} x + {\left (a^{3} b^{2} c^{7} + 2 \, a^{4} b c^{4} + a^{5} c\right )} d\right )}} \]

[In]

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

[Out]

-1/54*(84*b^2*d^6*x^6 + 504*b^2*c*d^5*x^5 + 1260*b^2*c^2*d^4*x^4 + 84*b^2*c^6 + 21*(80*b^2*c^3 + 7*a*b)*d^3*x^
3 + 147*a*b*c^3 + 63*(20*b^2*c^4 + 7*a*b*c)*d^2*x^2 + 63*(8*b^2*c^5 + 7*a*b*c^2)*d*x + 28*sqrt(3)*(b^2*d^7*x^7
+ 7*b^2*c*d^6*x^6 + 21*b^2*c^2*d^5*x^5 + b^2*c^7 + (35*b^2*c^3 + 2*a*b)*d^4*x^4 + (35*b^2*c^4 + 8*a*b*c)*d^3*
x^3 + 2*a*b*c^4 + 3*(7*b^2*c^5 + 4*a*b*c^2)*d^2*x^2 + a^2*c + (7*b^2*c^6 + 8*a*b*c^3 + a^2)*d*x)*(b/a)^(1/3)*a
rctan(2/3*sqrt(3)*(d*x + c)*(b/a)^(1/3) - 1/3*sqrt(3)) + 14*(b^2*d^7*x^7 + 7*b^2*c*d^6*x^6 + 21*b^2*c^2*d^5*x^
5 + b^2*c^7 + (35*b^2*c^3 + 2*a*b)*d^4*x^4 + (35*b^2*c^4 + 8*a*b*c)*d^3*x^3 + 2*a*b*c^4 + 3*(7*b^2*c^5 + 4*a*b
*c^2)*d^2*x^2 + a^2*c + (7*b^2*c^6 + 8*a*b*c^3 + a^2)*d*x)*(b/a)^(1/3)*log(b*d^2*x^2 + 2*b*c*d*x + b*c^2 - (a*
d*x + a*c)*(b/a)^(2/3) + a*(b/a)^(1/3)) - 28*(b^2*d^7*x^7 + 7*b^2*c*d^6*x^6 + 21*b^2*c^2*d^5*x^5 + b^2*c^7 + (
35*b^2*c^3 + 2*a*b)*d^4*x^4 + (35*b^2*c^4 + 8*a*b*c)*d^3*x^3 + 2*a*b*c^4 + 3*(7*b^2*c^5 + 4*a*b*c^2)*d^2*x^2 +
a^2*c + (7*b^2*c^6 + 8*a*b*c^3 + a^2)*d*x)*(b/a)^(1/3)*log(b*d*x + b*c + a*(b/a)^(2/3)) + 54*a^2)/(a^3*b^2*d^
8*x^7 + 7*a^3*b^2*c*d^7*x^6 + 21*a^3*b^2*c^2*d^6*x^5 + (35*a^3*b^2*c^3 + 2*a^4*b)*d^5*x^4 + (35*a^3*b^2*c^4 +
8*a^4*b*c)*d^4*x^3 + 3*(7*a^3*b^2*c^5 + 4*a^4*b*c^2)*d^3*x^2 + (7*a^3*b^2*c^6 + 8*a^4*b*c^3 + a^5)*d^2*x + (a^
3*b^2*c^7 + 2*a^4*b*c^4 + a^5*c)*d)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 391 vs. \(2 (194) = 388\).

Time = 2.04 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.79 \[ \int \frac {1}{(c+d x)^2 \left (a+b (c+d x)^3\right )^3} \, dx=\frac {- 18 a^{2} - 49 a b c^{3} - 28 b^{2} c^{6} - 420 b^{2} c^{2} d^{4} x^{4} - 168 b^{2} c d^{5} x^{5} - 28 b^{2} d^{6} x^{6} + x^{3} \left (- 49 a b d^{3} - 560 b^{2} c^{3} d^{3}\right ) + x^{2} \left (- 147 a b c d^{2} - 420 b^{2} c^{4} d^{2}\right ) + x \left (- 147 a b c^{2} d - 168 b^{2} c^{5} d\right )}{18 a^{5} c d + 36 a^{4} b c^{4} d + 18 a^{3} b^{2} c^{7} d + 378 a^{3} b^{2} c^{2} d^{6} x^{5} + 126 a^{3} b^{2} c d^{7} x^{6} + 18 a^{3} b^{2} d^{8} x^{7} + x^{4} \cdot \left (36 a^{4} b d^{5} + 630 a^{3} b^{2} c^{3} d^{5}\right ) + x^{3} \cdot \left (144 a^{4} b c d^{4} + 630 a^{3} b^{2} c^{4} d^{4}\right ) + x^{2} \cdot \left (216 a^{4} b c^{2} d^{3} + 378 a^{3} b^{2} c^{5} d^{3}\right ) + x \left (18 a^{5} d^{2} + 144 a^{4} b c^{3} d^{2} + 126 a^{3} b^{2} c^{6} d^{2}\right )} + \frac {\operatorname {RootSum} {\left (19683 t^{3} a^{10} - 2744 b, \left ( t \mapsto t \log {\left (x + \frac {729 t^{2} a^{7} + 196 b c}{196 b d} \right )} \right )\right )}}{d} \]

[In]

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

[Out]

(-18*a**2 - 49*a*b*c**3 - 28*b**2*c**6 - 420*b**2*c**2*d**4*x**4 - 168*b**2*c*d**5*x**5 - 28*b**2*d**6*x**6 +
x**3*(-49*a*b*d**3 - 560*b**2*c**3*d**3) + x**2*(-147*a*b*c*d**2 - 420*b**2*c**4*d**2) + x*(-147*a*b*c**2*d -
168*b**2*c**5*d))/(18*a**5*c*d + 36*a**4*b*c**4*d + 18*a**3*b**2*c**7*d + 378*a**3*b**2*c**2*d**6*x**5 + 126*a
**3*b**2*c*d**7*x**6 + 18*a**3*b**2*d**8*x**7 + x**4*(36*a**4*b*d**5 + 630*a**3*b**2*c**3*d**5) + x**3*(144*a*
*4*b*c*d**4 + 630*a**3*b**2*c**4*d**4) + x**2*(216*a**4*b*c**2*d**3 + 378*a**3*b**2*c**5*d**3) + x*(18*a**5*d*
*2 + 144*a**4*b*c**3*d**2 + 126*a**3*b**2*c**6*d**2)) + RootSum(19683*_t**3*a**10 - 2744*b, Lambda(_t, _t*log(
x + (729*_t**2*a**7 + 196*b*c)/(196*b*d))))/d

Maxima [F]

\[ \int \frac {1}{(c+d x)^2 \left (a+b (c+d x)^3\right )^3} \, dx=\int { \frac {1}{{\left ({\left (d x + c\right )}^{3} b + a\right )}^{3} {\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

-1/18*(28*b^2*d^6*x^6 + 168*b^2*c*d^5*x^5 + 420*b^2*c^2*d^4*x^4 + 28*b^2*c^6 + 7*(80*b^2*c^3 + 7*a*b)*d^3*x^3
+ 49*a*b*c^3 + 21*(20*b^2*c^4 + 7*a*b*c)*d^2*x^2 + 21*(8*b^2*c^5 + 7*a*b*c^2)*d*x + 18*a^2)/(a^3*b^2*d^8*x^7 +
7*a^3*b^2*c*d^7*x^6 + 21*a^3*b^2*c^2*d^6*x^5 + (35*a^3*b^2*c^3 + 2*a^4*b)*d^5*x^4 + (35*a^3*b^2*c^4 + 8*a^4*b
*c)*d^4*x^3 + 3*(7*a^3*b^2*c^5 + 4*a^4*b*c^2)*d^3*x^2 + (7*a^3*b^2*c^6 + 8*a^4*b*c^3 + a^5)*d^2*x + (a^3*b^2*c
^7 + 2*a^4*b*c^4 + a^5*c)*d) - 14/9*b*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^3

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(c+d x)^2 \left (a+b (c+d x)^3\right )^3} \, dx=\frac {14 \, \left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | -\left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}} - \frac {1}{{\left (d x + c\right )} d} \right |}\right )}{27 \, a^{3}} - \frac {14 \, \sqrt {3} \left (a^{2} b\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (\left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}} - \frac {2}{{\left (d x + c\right )} d}\right )}}{3 \, \left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}}}\right )}{27 \, a^{4} d} - \frac {7 \, \left (a^{2} b\right )^{\frac {1}{3}} \log \left (\left (\frac {b}{a d^{3}}\right )^{\frac {2}{3}} - \frac {\left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}}}{{\left (d x + c\right )} d} + \frac {1}{{\left (d x + c\right )}^{2} d^{2}}\right )}{27 \, a^{4} d} - \frac {\frac {10 \, b^{2}}{{\left (d x + c\right )} d} + \frac {13 \, a b}{{\left (d x + c\right )}^{4} d}}{18 \, a^{3} {\left (b + \frac {a}{{\left (d x + c\right )}^{3}}\right )}^{2}} - \frac {1}{{\left (d x + c\right )} a^{3} d} \]

[In]

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

[Out]

14/27*(b/(a*d^3))^(1/3)*log(abs(-(b/(a*d^3))^(1/3) - 1/((d*x + c)*d)))/a^3 - 14/27*sqrt(3)*(a^2*b)^(1/3)*arcta
n(1/3*sqrt(3)*((b/(a*d^3))^(1/3) - 2/((d*x + c)*d))/(b/(a*d^3))^(1/3))/(a^4*d) - 7/27*(a^2*b)^(1/3)*log((b/(a*
d^3))^(2/3) - (b/(a*d^3))^(1/3)/((d*x + c)*d) + 1/((d*x + c)^2*d^2))/(a^4*d) - 1/18*(10*b^2/((d*x + c)*d) + 13
*a*b/((d*x + c)^4*d))/(a^3*(b + a/(d*x + c)^3)^2) - 1/((d*x + c)*a^3*d)

Mupad [B] (verification not implemented)

Time = 7.00 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.97 \[ \int \frac {1}{(c+d x)^2 \left (a+b (c+d x)^3\right )^3} \, dx=\frac {14\,b^{1/3}\,\ln \left (b^{1/3}\,c+a^{1/3}+b^{1/3}\,d\,x\right )}{27\,a^{10/3}\,d}-\frac {\frac {18\,a^2+49\,a\,b\,c^3+28\,b^2\,c^6}{18\,a^3\,d}+\frac {7\,x^2\,\left (20\,d\,b^2\,c^4+7\,a\,d\,b\,c\right )}{6\,a^3}+\frac {7\,x\,\left (8\,b^2\,c^5+7\,a\,b\,c^2\right )}{6\,a^3}+\frac {7\,x^3\,\left (80\,b^2\,c^3\,d^2+7\,a\,b\,d^2\right )}{18\,a^3}+\frac {14\,b^2\,d^5\,x^6}{9\,a^3}+\frac {70\,b^2\,c^2\,d^3\,x^4}{3\,a^3}+\frac {28\,b^2\,c\,d^4\,x^5}{3\,a^3}}{x^4\,\left (35\,b^2\,c^3\,d^4+2\,a\,b\,d^4\right )+x\,\left (d\,a^2+8\,d\,a\,b\,c^3+7\,d\,b^2\,c^6\right )+a^2\,c+x^3\,\left (35\,b^2\,c^4\,d^3+8\,a\,b\,c\,d^3\right )+b^2\,c^7+x^2\,\left (21\,b^2\,c^5\,d^2+12\,a\,b\,c^2\,d^2\right )+b^2\,d^7\,x^7+2\,a\,b\,c^4+7\,b^2\,c\,d^6\,x^6+21\,b^2\,c^2\,d^5\,x^5}+\frac {14\,b^{1/3}\,\ln \left (2\,b^{1/3}\,c-a^{1/3}+2\,b^{1/3}\,d\,x-\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{27\,a^{10/3}\,d}-\frac {14\,b^{1/3}\,\ln \left (2\,b^{1/3}\,c-a^{1/3}+2\,b^{1/3}\,d\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{27\,a^{10/3}\,d} \]

[In]

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

[Out]

(14*b^(1/3)*log(b^(1/3)*c + a^(1/3) + b^(1/3)*d*x))/(27*a^(10/3)*d) - ((18*a^2 + 28*b^2*c^6 + 49*a*b*c^3)/(18*
a^3*d) + (7*x^2*(20*b^2*c^4*d + 7*a*b*c*d))/(6*a^3) + (7*x*(8*b^2*c^5 + 7*a*b*c^2))/(6*a^3) + (7*x^3*(80*b^2*c
^3*d^2 + 7*a*b*d^2))/(18*a^3) + (14*b^2*d^5*x^6)/(9*a^3) + (70*b^2*c^2*d^3*x^4)/(3*a^3) + (28*b^2*c*d^4*x^5)/(
3*a^3))/(x^4*(35*b^2*c^3*d^4 + 2*a*b*d^4) + x*(a^2*d + 7*b^2*c^6*d + 8*a*b*c^3*d) + a^2*c + x^3*(35*b^2*c^4*d^
3 + 8*a*b*c*d^3) + b^2*c^7 + x^2*(21*b^2*c^5*d^2 + 12*a*b*c^2*d^2) + b^2*d^7*x^7 + 2*a*b*c^4 + 7*b^2*c*d^6*x^6
+ 21*b^2*c^2*d^5*x^5) + (14*b^(1/3)*log(2*b^(1/3)*c - 3^(1/2)*a^(1/3)*1i - a^(1/3) + 2*b^(1/3)*d*x)*((3^(1/2)
*1i)/2 - 1/2))/(27*a^(10/3)*d) - (14*b^(1/3)*log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*c - a^(1/3) + 2*b^(1/3)*d*x)*(
(3^(1/2)*1i)/2 + 1/2))/(27*a^(10/3)*d)