Integrand size = 22, antiderivative size = 176 \[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^2} \, dx=\frac {e (c+d x)^2}{3 a d \left (a+b (c+d x)^3\right )}-\frac {e \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{4/3} b^{2/3} d}-\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{4/3} b^{2/3} d}+\frac {e \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{4/3} b^{2/3} d} \] Output:
1/3*e*(d*x+c)^2/a/d/(a+b*(d*x+c)^3)-1/9*e*arctan(1/3*(a^(1/3)-2*b^(1/3)*(d *x+c))*3^(1/2)/a^(1/3))*3^(1/2)/a^(4/3)/b^(2/3)/d-1/9*e*ln(a^(1/3)+b^(1/3) *(d*x+c))/a^(4/3)/b^(2/3)/d+1/18*e*ln(a^(2/3)-a^(1/3)*b^(1/3)*(d*x+c)+b^(2 /3)*(d*x+c)^2)/a^(4/3)/b^(2/3)/d
Time = 0.03 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.87 \[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^2} \, dx=\frac {e \left (\frac {6 \sqrt [3]{a} (c+d x)^2}{a+b (c+d x)^3}+\frac {2 \sqrt {3} \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{2/3}}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{b^{2/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{b^{2/3}}\right )}{18 a^{4/3} d} \] Input:
Integrate[(c*e + d*e*x)/(a + b*(c + d*x)^3)^2,x]
Output:
(e*((6*a^(1/3)*(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))])/b^(2/3) - (2*Log[a^(1/3) + b^(1/3)*(c + d*x)])/b^(2/3) + Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^ (2/3)*(c + d*x)^2]/b^(2/3)))/(18*a^(4/3)*d)
Time = 0.53 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.98, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {895, 819, 821, 16, 1142, 25, 27, 1082, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 895 |
| \(\displaystyle \frac {e \int \frac {c+d x}{\left (b (c+d x)^3+a\right )^2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {e \left (\frac {\int \frac {c+d x}{b (c+d x)^3+a}d(c+d x)}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}\right )}{d}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {e \left (\frac {\frac {\int \frac {\sqrt [3]{b} (c+d x)+\sqrt [3]{a}}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{b} (c+d x)+\sqrt [3]{a}}d(c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}\right )}{d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {e \left (\frac {\frac {\int \frac {\sqrt [3]{b} (c+d x)+\sqrt [3]{a}}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}\right )}{d}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {e \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)\right )}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)\right )}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}\right )}{d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {e \left (\frac {\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}\right )}{d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {e \left (\frac {\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}\right )}{d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {e \left (\frac {\frac {\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}\right )}{d}\) |
Input:
Int[(c*e + d*e*x)/(a + b*(c + d*x)^3)^2,x]
Output:
(e*((c + d*x)^2/(3*a*(a + b*(c + d*x)^3)) + (-1/3*Log[a^(1/3) + b^(1/3)*(c + d*x)]/(a^(1/3)*b^(2/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*(c + d*x)) /a^(1/3))/Sqrt[3]])/b^(1/3)) + Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b ^(2/3)*(c + d*x)^2]/(2*b^(1/3)))/(3*a^(1/3)*b^(1/3)))/(3*a)))/d
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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] + Simp[(m + n*(p + 1) + 1)/(a*n*(p + 1)) Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a , b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[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]
Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Simp[u^m/(Coeff icient[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]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[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])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.26 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.83
| method | result | size |
| default | \(e \left (\frac {\frac {d \,x^{2}}{3 a}+\frac {2 c x}{3 a}+\frac {c^{2}}{3 a d}}{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\left (\textit {\_R} d +c \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}}{9 a b d}\right )\) | \(146\) |
| risch | \(\frac {\frac {d e \,x^{2}}{3 a}+\frac {2 c e x}{3 a}+\frac {c^{2} e}{3 d a}}{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a}+\frac {e \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\left (\textit {\_R} d +c \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}\right )}{9 a b d}\) | \(148\) |
Input:
int((d*e*x+c*e)/(a+b*(d*x+c)^3)^2,x,method=_RETURNVERBOSE)
Output:
e*((1/3*d/a*x^2+2/3/a*c*x+1/3*c^2/a/d)/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d* x+b*c^3+a)+1/9/a/b/d*sum((_R*d+c)/(_R^2*d^2+2*_R*c*d+c^2)*ln(x-_R),_R=Root Of(_Z^3*b*d^3+3*_Z^2*b*c*d^2+3*_Z*b*c^2*d+b*c^3+a)))
Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (137) = 274\).
Time = 0.10 (sec) , antiderivative size = 894, normalized size of antiderivative = 5.08 \[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^2} \, dx =\text {Too large to display} \] Input:
integrate((d*e*x+c*e)/(a+b*(d*x+c)^3)^2,x, algorithm="fricas")
Output:
[1/18*(6*a*b^2*d^2*e*x^2 + 12*a*b^2*c*d*e*x + 6*a*b^2*c^2*e + 3*sqrt(1/3)* (a*b^2*d^3*e*x^3 + 3*a*b^2*c*d^2*e*x^2 + 3*a*b^2*c^2*d*e*x + (a*b^2*c^3 + a^2*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)) + (b*d^3*e*x^3 + 3*b*c*d^2*e*x^2 + 3*b*c^2*d*e*x + (b*c^3 + a) *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)) - 2*(b*d^3*e*x^3 + 3*b*c*d^2*e*x^2 + 3*b *c^2*d*e*x + (b*c^3 + a)*e)*(-a*b^2)^(2/3)*log(b*d*x + b*c - (-a*b^2)^(1/3 )))/(a^2*b^3*d^4*x^3 + 3*a^2*b^3*c*d^3*x^2 + 3*a^2*b^3*c^2*d^2*x + (a^2*b^ 3*c^3 + a^3*b^2)*d), 1/18*(6*a*b^2*d^2*e*x^2 + 12*a*b^2*c*d*e*x + 6*a*b^2* c^2*e + 6*sqrt(1/3)*(a*b^2*d^3*e*x^3 + 3*a*b^2*c*d^2*e*x^2 + 3*a*b^2*c^2*d *e*x + (a*b^2*c^3 + a^2*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) + (b*d^3*e*x^3 + 3*b*c*d^2*e*x^2 + 3*b*c^2*d*e*x + (b*c^3 + a)*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)) - 2*(b*d^3*e*x^3 + 3*b*c*d^2*e*x^2 + 3*b*c^2*d*e*x + (b*c^3 + a)*e) *(-a*b^2)^(2/3)*log(b*d*x + b*c - (-a*b^2)^(1/3)))/(a^2*b^3*d^4*x^3 + 3*a^ 2*b^3*c*d^3*x^2 + 3*a^2*b^3*c^2*d^2*x + (a^2*b^3*c^3 + a^3*b^2)*d)]
Time = 0.67 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.69 \[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^2} \, dx=\frac {c^{2} e + 2 c d e x + d^{2} e x^{2}}{3 a^{2} d + 3 a b c^{3} d + 9 a b c^{2} d^{2} x + 9 a b c d^{3} x^{2} + 3 a b d^{4} x^{3}} + \frac {e \operatorname {RootSum} {\left (729 t^{3} a^{4} b^{2} + 1, \left ( t \mapsto t \log {\left (x + \frac {81 t^{2} a^{3} b e^{2} + c e^{2}}{d e^{2}} \right )} \right )\right )}}{d} \] Input:
integrate((d*e*x+c*e)/(a+b*(d*x+c)**3)**2,x)
Output:
(c**2*e + 2*c*d*e*x + d**2*e*x**2)/(3*a**2*d + 3*a*b*c**3*d + 9*a*b*c**2*d **2*x + 9*a*b*c*d**3*x**2 + 3*a*b*d**4*x**3) + e*RootSum(729*_t**3*a**4*b* *2 + 1, Lambda(_t, _t*log(x + (81*_t**2*a**3*b*e**2 + c*e**2)/(d*e**2))))/ d
\[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^2} \, dx=\int { \frac {d e x + c e}{{\left ({\left (d x + c\right )}^{3} b + a\right )}^{2}} \,d x } \] Input:
integrate((d*e*x+c*e)/(a+b*(d*x+c)^3)^2,x, algorithm="maxima")
Output:
1/3*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 + 1/3*(d^2*e*x^2 + 2*c*d*e*x + c^2*e)/(a*b*d^4*x^3 + 3*a*b*c*d ^3*x^2 + 3*a*b*c^2*d^2*x + (a*b*c^3 + a^2)*d)
Time = 0.14 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.24 \[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^2} \, dx=-\frac {2 \, \sqrt {3} \left (-\frac {e^{3}}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}}{3 \, \left (-a^{2} b\right )^{\frac {2}{3}}}\right ) + \left (-\frac {e^{3}}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}^{2} + 3 \, \left (-a^{2} b\right )^{\frac {4}{3}}\right ) - 2 \, \left (-\frac {e^{3}}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | a b d x + a b c + \left (-a^{2} b\right )^{\frac {2}{3}} \right |}\right )}{18 \, a} + \frac {d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{3 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} a d} \] Input:
integrate((d*e*x+c*e)/(a+b*(d*x+c)^3)^2,x, algorithm="giac")
Output:
-1/18*(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 + 1/3*(d^2*e* x^2 + 2*c*d*e*x + 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)*a*d)
Time = 0.72 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.09 \[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^2} \, dx=\frac {\frac {c^2\,e}{3\,a\,d}+\frac {2\,c\,e\,x}{3\,a}+\frac {d\,e\,x^2}{3\,a}}{b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a}-\frac {e\,\ln \left (b^{1/3}\,c+a^{1/3}+b^{1/3}\,d\,x\right )}{9\,a^{4/3}\,b^{2/3}\,d}+\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,c-2\,b^{1/3}\,d\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (e-\sqrt {3}\,e\,1{}\mathrm {i}\right )}{18\,a^{4/3}\,b^{2/3}\,d}+\frac {\ln \left (2\,b^{1/3}\,c-a^{1/3}+2\,b^{1/3}\,d\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (e+\sqrt {3}\,e\,1{}\mathrm {i}\right )}{18\,a^{4/3}\,b^{2/3}\,d} \] Input:
int((c*e + d*e*x)/(a + b*(c + d*x)^3)^2,x)
Output:
((c^2*e)/(3*a*d) + (2*c*e*x)/(3*a) + (d*e*x^2)/(3*a))/(a + b*c^3 + b*d^3*x ^3 + 3*b*c^2*d*x + 3*b*c*d^2*x^2) - (e*log(b^(1/3)*c + a^(1/3) + b^(1/3)*d *x))/(9*a^(4/3)*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))/(18*a^(4/3)*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) )/(18*a^(4/3)*b^(2/3)*d)
Time = 0.24 (sec) , antiderivative size = 706, normalized size of antiderivative = 4.01 \[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^2} \, dx =\text {Too large to display} \] Input:
int((d*e*x+c*e)/(a+b*(d*x+c)^3)^2,x)
Output:
(e*( - 2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*c - 2*b**(1/3)*d*x)/(a**(1/3) *sqrt(3)))*a*b*c - 2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*c - 2*b**(1/3)*d* x)/(a**(1/3)*sqrt(3)))*b**2*c**4 - 6*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*c - 2*b**(1/3)*d*x)/(a**(1/3)*sqrt(3)))*b**2*c**3*d*x - 6*sqrt(3)*atan((a** (1/3) - 2*b**(1/3)*c - 2*b**(1/3)*d*x)/(a**(1/3)*sqrt(3)))*b**2*c**2*d**2* x**2 - 2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*c - 2*b**(1/3)*d*x)/(a**(1/3) *sqrt(3)))*b**2*c*d**3*x**3 - 2*b**(2/3)*a**(1/3)*a + 4*b**(2/3)*a**(1/3)* b*c**3 + 6*b**(2/3)*a**(1/3)*b*c**2*d*x - 2*b**(2/3)*a**(1/3)*b*d**3*x**3 + log(a**(2/3) - b**(1/3)*a**(1/3)*c - b**(1/3)*a**(1/3)*d*x + b**(2/3)*c* *2 + 2*b**(2/3)*c*d*x + b**(2/3)*d**2*x**2)*a*b*c + log(a**(2/3) - b**(1/3 )*a**(1/3)*c - b**(1/3)*a**(1/3)*d*x + b**(2/3)*c**2 + 2*b**(2/3)*c*d*x + b**(2/3)*d**2*x**2)*b**2*c**4 + 3*log(a**(2/3) - b**(1/3)*a**(1/3)*c - b** (1/3)*a**(1/3)*d*x + b**(2/3)*c**2 + 2*b**(2/3)*c*d*x + b**(2/3)*d**2*x**2 )*b**2*c**3*d*x + 3*log(a**(2/3) - b**(1/3)*a**(1/3)*c - b**(1/3)*a**(1/3) *d*x + b**(2/3)*c**2 + 2*b**(2/3)*c*d*x + b**(2/3)*d**2*x**2)*b**2*c**2*d* *2*x**2 + log(a**(2/3) - b**(1/3)*a**(1/3)*c - b**(1/3)*a**(1/3)*d*x + b** (2/3)*c**2 + 2*b**(2/3)*c*d*x + b**(2/3)*d**2*x**2)*b**2*c*d**3*x**3 - 2*l og(a**(1/3) + b**(1/3)*c + b**(1/3)*d*x)*a*b*c - 2*log(a**(1/3) + b**(1/3) *c + b**(1/3)*d*x)*b**2*c**4 - 6*log(a**(1/3) + b**(1/3)*c + b**(1/3)*d*x) *b**2*c**3*d*x - 6*log(a**(1/3) + b**(1/3)*c + b**(1/3)*d*x)*b**2*c**2*...