\(\int \frac {(c+d x)^4}{(a+b (c+d x)^3)^2} \, dx\) [2868]

Optimal result

Integrand size = 21, antiderivative size = 172 \[ \int \frac {(c+d x)^4}{\left (a+b (c+d x)^3\right )^2} \, dx=-\frac {(c+d x)^2}{3 b d \left (a+b (c+d x)^3\right )}-\frac {2 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} \sqrt [3]{a} b^{5/3} d}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 \sqrt [3]{a} b^{5/3} d}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{9 \sqrt [3]{a} b^{5/3} d} \]

[Out]

-1/3*(d*x+c)^2/b/d/(a+b*(d*x+c)^3)-2/9*ln(a^(1/3)+b^(1/3)*(d*x+c))/a^(1/3)/b^(5/3)/d+1/9*ln(a^(2/3)-a^(1/3)*b^
(1/3)*(d*x+c)+b^(2/3)*(d*x+c)^2)/a^(1/3)/b^(5/3)/d-2/9*arctan(1/3*(a^(1/3)-2*b^(1/3)*(d*x+c))/a^(1/3)*3^(1/2))
/a^(1/3)/b^(5/3)/d*3^(1/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {379, 294, 298, 31, 648, 631, 210, 642} \[ \int \frac {(c+d x)^4}{\left (a+b (c+d x)^3\right )^2} \, dx=\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{9 \sqrt [3]{a} b^{5/3} d}-\frac {2 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} \sqrt [3]{a} b^{5/3} d}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 \sqrt [3]{a} b^{5/3} d}-\frac {(c+d x)^2}{3 b d \left (a+b (c+d x)^3\right )} \]

[In]

Int[(c + d*x)^4/(a + b*(c + d*x)^3)^2,x]

[Out]

-1/3*(c + d*x)^2/(b*d*(a + b*(c + d*x)^3)) - (2*ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))])/(3*
Sqrt[3]*a^(1/3)*b^(5/3)*d) - (2*Log[a^(1/3) + b^(1/3)*(c + d*x)])/(9*a^(1/3)*b^(5/3)*d) + Log[a^(2/3) - a^(1/3
)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2]/(9*a^(1/3)*b^(5/3)*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 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& 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 {\text {Subst}\left (\int \frac {x^4}{\left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{d} \\ & = -\frac {(c+d x)^2}{3 b d \left (a+b (c+d x)^3\right )}+\frac {2 \text {Subst}\left (\int \frac {x}{a+b x^3} \, dx,x,c+d x\right )}{3 b d} \\ & = -\frac {(c+d x)^2}{3 b d \left (a+b (c+d x)^3\right )}-\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{9 \sqrt [3]{a} b^{4/3} d}+\frac {2 \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 )}{9 \sqrt [3]{a} b^{4/3} d} \\ & = -\frac {(c+d x)^2}{3 b d \left (a+b (c+d x)^3\right )}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 \sqrt [3]{a} b^{5/3} d}+\frac {\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 )}{9 \sqrt [3]{a} b^{5/3} d}+\frac {\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 )}{3 b^{4/3} d} \\ & = -\frac {(c+d x)^2}{3 b d \left (a+b (c+d x)^3\right )}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 \sqrt [3]{a} b^{5/3} d}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{9 \sqrt [3]{a} b^{5/3} d}+\frac {2 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{3 \sqrt [3]{a} b^{5/3} d} \\ & = -\frac {(c+d x)^2}{3 b d \left (a+b (c+d x)^3\right )}-\frac {2 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{a} b^{5/3} d}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 \sqrt [3]{a} b^{5/3} d}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{9 \sqrt [3]{a} b^{5/3} d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.88 \[ \int \frac {(c+d x)^4}{\left (a+b (c+d x)^3\right )^2} \, dx=\frac {-\frac {3 b^{2/3} (c+d x)^2}{a+b (c+d x)^3}+\frac {2 \sqrt {3} \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a}}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{\sqrt [3]{a}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{\sqrt [3]{a}}}{9 b^{5/3} d} \]

[In]

Integrate[(c + d*x)^4/(a + b*(c + d*x)^3)^2,x]

[Out]

((-3*b^(2/3)*(c + d*x)^2)/(a + b*(c + d*x)^3) + (2*Sqrt[3]*ArcTan[(-a^(1/3) + 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^
(1/3))])/a^(1/3) - (2*Log[a^(1/3) + b^(1/3)*(c + d*x)])/a^(1/3) + Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^
(2/3)*(c + d*x)^2]/a^(1/3))/(9*b^(5/3)*d)

Maple [C] (verified)

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

Time = 3.84 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.82

method	result	size
default	\(\frac {-\frac {d \,x^{2}}{3 b}-\frac {2 c x}{3 b}-\frac {c^{2}}{3 d b}}{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\left (\textit {\_R} d +c \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}\right )}{9 b^{2} d}\)	\(141\)
risch	\(\frac {-\frac {d \,x^{2}}{3 b}-\frac {2 c x}{3 b}-\frac {c^{2}}{3 d b}}{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\left (\textit {\_R} d +c \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}\right )}{9 b^{2} d}\)	\(141\)

[In]

int((d*x+c)^4/(a+b*(d*x+c)^3)^2,x,method=_RETURNVERBOSE)

[Out]

(-1/3*d*x^2/b-2/3*c*x/b-1/3*c^2/d/b)/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)+2/9/b^2/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. 358 vs. \(2 (133) = 266\).

Time = 0.30 (sec) , antiderivative size = 838, normalized size of antiderivative = 4.87 \[ \int \frac {(c+d x)^4}{\left (a+b (c+d x)^3\right )^2} \, dx=\left [-\frac {3 \, a b^{2} d^{2} x^{2} + 6 \, a b^{2} c d x + 3 \, a b^{2} c^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (a b^{2} d^{3} x^{3} + 3 \, a b^{2} c d^{2} x^{2} + 3 \, a b^{2} c^{2} d x + a b^{2} c^{3} + a^{2} b\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {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 {\frac {1}{3}} {\left (a b d x + a b c + 2 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} + \left (-a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} {\left (d x + c\right )}}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}\right ) - {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} {\left (b d x + b c\right )} + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) + 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 )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b d x + b c - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{9 \, {\left (a b^{4} d^{4} x^{3} + 3 \, a b^{4} c d^{3} x^{2} + 3 \, a b^{4} c^{2} d^{2} x + {\left (a b^{4} c^{3} + a^{2} b^{3}\right )} d\right )}}, -\frac {3 \, a b^{2} d^{2} x^{2} + 6 \, a b^{2} c d x + 3 \, a b^{2} c^{2} - 6 \, \sqrt {\frac {1}{3}} {\left (a b^{2} d^{3} x^{3} + 3 \, a b^{2} c d^{2} x^{2} + 3 \, a b^{2} c^{2} d x + a b^{2} c^{3} + a^{2} b\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b d x + 2 \, b c + \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) - {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} {\left (b d x + b c\right )} + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) + 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 )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b d x + b c - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{9 \, {\left (a b^{4} d^{4} x^{3} + 3 \, a b^{4} c d^{3} x^{2} + 3 \, a b^{4} c^{2} d^{2} x + {\left (a b^{4} c^{3} + a^{2} b^{3}\right )} d\right )}}\right ] \]

[In]

integrate((d*x+c)^4/(a+b*(d*x+c)^3)^2,x, algorithm="fricas")

[Out]

[-1/9*(3*a*b^2*d^2*x^2 + 6*a*b^2*c*d*x + 3*a*b^2*c^2 - 3*sqrt(1/3)*(a*b^2*d^3*x^3 + 3*a*b^2*c*d^2*x^2 + 3*a*b^
2*c^2*d*x + a*b^2*c^3 + a^2*b)*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)) -
(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)*(-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)) + 2*(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)*(-a
*b^2)^(2/3)*log(b*d*x + b*c - (-a*b^2)^(1/3)))/(a*b^4*d^4*x^3 + 3*a*b^4*c*d^3*x^2 + 3*a*b^4*c^2*d^2*x + (a*b^4
*c^3 + a^2*b^3)*d), -1/9*(3*a*b^2*d^2*x^2 + 6*a*b^2*c*d*x + 3*a*b^2*c^2 - 6*sqrt(1/3)*(a*b^2*d^3*x^3 + 3*a*b^2
*c*d^2*x^2 + 3*a*b^2*c^2*d*x + a*b^2*c^3 + a^2*b)*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) - (b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)*(-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)) + 2*(b*d^3*x^3 +
3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)*(-a*b^2)^(2/3)*log(b*d*x + b*c - (-a*b^2)^(1/3)))/(a*b^4*d^4*x^3 + 3*
a*b^4*c*d^3*x^2 + 3*a*b^4*c^2*d^2*x + (a*b^4*c^3 + a^2*b^3)*d)]

Sympy [A] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.63 \[ \int \frac {(c+d x)^4}{\left (a+b (c+d x)^3\right )^2} \, dx=\frac {- c^{2} - 2 c d x - d^{2} x^{2}}{3 a b d + 3 b^{2} c^{3} d + 9 b^{2} c^{2} d^{2} x + 9 b^{2} c d^{3} x^{2} + 3 b^{2} d^{4} x^{3}} + \frac {\operatorname {RootSum} {\left (729 t^{3} a b^{5} + 8, \left ( t \mapsto t \log {\left (x + \frac {81 t^{2} a b^{3} + 4 c}{4 d} \right )} \right )\right )}}{d} \]

[In]

integrate((d*x+c)**4/(a+b*(d*x+c)**3)**2,x)

[Out]

(-c**2 - 2*c*d*x - d**2*x**2)/(3*a*b*d + 3*b**2*c**3*d + 9*b**2*c**2*d**2*x + 9*b**2*c*d**3*x**2 + 3*b**2*d**4
*x**3) + RootSum(729*_t**3*a*b**5 + 8, Lambda(_t, _t*log(x + (81*_t**2*a*b**3 + 4*c)/(4*d))))/d

Maxima [F]

\[ \int \frac {(c+d x)^4}{\left (a+b (c+d x)^3\right )^2} \, dx=\int { \frac {{\left (d x + c\right )}^{4}}{{\left ({\left (d x + c\right )}^{3} b + a\right )}^{2}} \,d x } \]

[In]

integrate((d*x+c)^4/(a+b*(d*x+c)^3)^2,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.19 \[ \int \frac {(c+d x)^4}{\left (a+b (c+d x)^3\right )^2} \, dx=-\frac {2 \, \sqrt {3} \left (-\frac {1}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}}{3 \, \left (-a^{2} b\right )^{\frac {2}{3}}}\right ) + \left (-\frac {1}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}^{2} + 3 \, \left (-a^{2} b\right )^{\frac {4}{3}}\right ) - 2 \, \left (-\frac {1}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | a b d x + a b c + \left (-a^{2} b\right )^{\frac {2}{3}} \right |}\right )}{9 \, b} - \frac {d^{2} x^{2} + 2 \, c d x + c^{2}}{3 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} b d} \]

[In]

integrate((d*x+c)^4/(a+b*(d*x+c)^3)^2,x, algorithm="giac")

[Out]

-1/9*(2*sqrt(3)*(-1/(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
)) + (-1/(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*(-1/(a*b^2*d^
3))^(1/3)*log(abs(a*b*d*x + a*b*c + (-a^2*b)^(2/3))))/b - 1/3*(d^2*x^2 + 2*c*d*x + c^2)/((b*d^3*x^3 + 3*b*c*d^
2*x^2 + 3*b*c^2*d*x + b*c^3 + a)*b*d)

Mupad [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.40 \[ \int \frac {(c+d x)^4}{\left (a+b (c+d x)^3\right )^2} \, dx=\frac {2\,\ln \left (\frac {4\,c\,d^4}{9\,b}-\frac {4\,{\left (-a\right )}^{1/3}\,d^4}{9\,b^{4/3}}+\frac {4\,d^5\,x}{9\,b}\right )}{9\,{\left (-a\right )}^{1/3}\,b^{5/3}\,d}-\frac {\frac {d\,x^2}{3\,b}+\frac {c^2}{3\,b\,d}+\frac {2\,c\,x}{3\,b}}{b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a}+\frac {\ln \left (\frac {4\,c\,d^4}{9\,b}+\frac {4\,d^5\,x}{9\,b}-\frac {{\left (-a\right )}^{1/3}\,d^4\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{9\,b^{4/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,{\left (-a\right )}^{1/3}\,b^{5/3}\,d}-\frac {\ln \left (\frac {4\,c\,d^4}{9\,b}+\frac {4\,d^5\,x}{9\,b}-\frac {{\left (-a\right )}^{1/3}\,d^4\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{9\,b^{4/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,{\left (-a\right )}^{1/3}\,b^{5/3}\,d} \]

[In]

int((c + d*x)^4/(a + b*(c + d*x)^3)^2,x)

[Out]

(2*log((4*c*d^4)/(9*b) - (4*(-a)^(1/3)*d^4)/(9*b^(4/3)) + (4*d^5*x)/(9*b)))/(9*(-a)^(1/3)*b^(5/3)*d) - ((d*x^2
)/(3*b) + c^2/(3*b*d) + (2*c*x)/(3*b))/(a + b*c^3 + b*d^3*x^3 + 3*b*c^2*d*x + 3*b*c*d^2*x^2) + (log((4*c*d^4)/
(9*b) + (4*d^5*x)/(9*b) - ((-a)^(1/3)*d^4*(3^(1/2)*1i - 1)^2)/(9*b^(4/3)))*(3^(1/2)*1i - 1))/(9*(-a)^(1/3)*b^(
5/3)*d) - (log((4*c*d^4)/(9*b) + (4*d^5*x)/(9*b) - ((-a)^(1/3)*d^4*(3^(1/2)*1i + 1)^2)/(9*b^(4/3)))*(3^(1/2)*1
i + 1))/(9*(-a)^(1/3)*b^(5/3)*d)