Integrand size = 22, antiderivative size = 207 \[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^3} \, dx=\frac {e (c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac {2 e (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}-\frac {2 e \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{7/3} b^{2/3} d}-\frac {2 e \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac {e \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{7/3} b^{2/3} d} \] Output:
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*arctan(1/3*(a^(1/3)-2*b^(1/3)*(d*x+c))*3^(1/2)/a^(1/3))*3^(1/2)/a ^(7/3)/b^(2/3)/d-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
Time = 0.01 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.87 \[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^3} \, dx=\frac {e \left (\frac {9 a^{4/3} (c+d x)^2}{\left (a+b (c+d x)^3\right )^2}+\frac {12 \sqrt [3]{a} (c+d x)^2}{a+b (c+d x)^3}+\frac {4 \sqrt {3} \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{2/3}}-\frac {4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{b^{2/3}}+\frac {2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{b^{2/3}}\right )}{54 a^{7/3} d} \] Input:
Integrate[(c*e + d*e*x)/(a + b*(c + d*x)^3)^3,x]
Output:
(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)
Time = 0.55 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {895, 819, 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.
| \(\displaystyle \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 895 |
| \(\displaystyle \frac {e \int \frac {c+d x}{\left (b (c+d x)^3+a\right )^3}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {e \left (\frac {2 \int \frac {c+d x}{\left (b (c+d x)^3+a\right )^2}d(c+d x)}{3 a}+\frac {(c+d x)^2}{6 a \left (a+b (c+d x)^3\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {e \left (\frac {2 \left (\frac {\int \frac {c+d x}{b (c+d x)^3+a}d(c+d x)}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}\right )}{3 a}+\frac {(c+d x)^2}{6 a \left (a+b (c+d x)^3\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {e \left (\frac {2 \left (\frac {\frac {\int \frac {\sqrt [3]{b} (c+d x)+\sqrt [3]{a}}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{b} (c+d x)+\sqrt [3]{a}}d(c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}\right )}{3 a}+\frac {(c+d x)^2}{6 a \left (a+b (c+d x)^3\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {e \left (\frac {2 \left (\frac {\frac {\int \frac {\sqrt [3]{b} (c+d x)+\sqrt [3]{a}}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}\right )}{3 a}+\frac {(c+d x)^2}{6 a \left (a+b (c+d x)^3\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {e \left (\frac {2 \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)\right )}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}\right )}{3 a}+\frac {(c+d x)^2}{6 a \left (a+b (c+d x)^3\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e \left (\frac {2 \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)\right )}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}\right )}{3 a}+\frac {(c+d x)^2}{6 a \left (a+b (c+d x)^3\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \left (\frac {2 \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}\right )}{3 a}+\frac {(c+d x)^2}{6 a \left (a+b (c+d x)^3\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {e \left (\frac {2 \left (\frac {\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}\right )}{3 a}+\frac {(c+d x)^2}{6 a \left (a+b (c+d x)^3\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {e \left (\frac {2 \left (\frac {\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}\right )}{3 a}+\frac {(c+d x)^2}{6 a \left (a+b (c+d x)^3\right )^2}\right )}{d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {e \left (\frac {2 \left (\frac {\frac {\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}\right )}{3 a}+\frac {(c+d x)^2}{6 a \left (a+b (c+d x)^3\right )^2}\right )}{d}\) |
Input:
Int[(c*e + d*e*x)/(a + b*(c + d*x)^3)^3,x]
Output:
(e*((c + d*x)^2/(6*a*(a + b*(c + d*x)^3)^2) + (2*((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)))/(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 = 0.34 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.04
| method | result | size |
| default | \(e \left (\frac {\frac {2 b \,d^{4} x^{5}}{9 a^{2}}+\frac {10 b c \,d^{3} x^{4}}{9 a^{2}}+\frac {20 b \,c^{2} d^{2} x^{3}}{9 a^{2}}+\frac {d \left (40 c^{3} b +7 a \right ) x^{2}}{18 a^{2}}+\frac {c \left (10 c^{3} b +7 a \right ) x}{9 a^{2}}+\frac {c^{2} \left (4 c^{3} b +7 a \right )}{18 d \,a^{2}}}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )^{2}}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\left (\textit {\_R} d +c \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}\right )}{27 a^{2} b d}\right )\) | \(216\) |
| risch | \(\frac {\frac {2 b \,d^{4} e \,x^{5}}{9 a^{2}}+\frac {10 b c \,d^{3} e \,x^{4}}{9 a^{2}}+\frac {20 b \,c^{2} d^{2} e \,x^{3}}{9 a^{2}}+\frac {d e \left (40 c^{3} b +7 a \right ) x^{2}}{18 a^{2}}+\frac {c e \left (10 c^{3} b +7 a \right ) x}{9 a^{2}}+\frac {c^{2} e \left (4 c^{3} b +7 a \right )}{18 d \,a^{2}}}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )^{2}}+\frac {2 e \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\left (\textit {\_R} d +c \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}\right )}{27 a^{2} b d}\) | \(221\) |
Input:
int((d*e*x+c*e)/(a+b*(d*x+c)^3)^3,x,method=_RETURNVERBOSE)
Output:
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)))
Leaf count of result is larger than twice the leaf count of optimal. 829 vs. \(2 (166) = 332\).
Time = 0.18 (sec) , antiderivative size = 1780, normalized size of antiderivative = 8.60 \[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((d*e*x+c*e)/(a+b*(d*x+c)^3)^3,x, algorithm="fricas")
Output:
[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*...
Time = 1.65 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.56 \[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^3} \, dx=\frac {7 a c^{2} e + 4 b c^{5} e + 40 b c^{2} d^{3} e x^{3} + 20 b c d^{4} e x^{4} + 4 b d^{5} e x^{5} + x^{2} \cdot \left (7 a d^{2} e + 40 b c^{3} d^{2} e\right ) + x \left (14 a c d e + 20 b c^{4} d e\right )}{18 a^{4} d + 36 a^{3} b c^{3} d + 18 a^{2} b^{2} c^{6} d + 270 a^{2} b^{2} c^{2} d^{5} x^{4} + 108 a^{2} b^{2} c d^{6} x^{5} + 18 a^{2} b^{2} d^{7} x^{6} + x^{3} \cdot \left (36 a^{3} b d^{4} + 360 a^{2} b^{2} c^{3} d^{4}\right ) + x^{2} \cdot \left (108 a^{3} b c d^{3} + 270 a^{2} b^{2} c^{4} d^{3}\right ) + x \left (108 a^{3} b c^{2} d^{2} + 108 a^{2} b^{2} c^{5} d^{2}\right )} + \frac {e \operatorname {RootSum} {\left (19683 t^{3} a^{7} b^{2} + 8, \left ( t \mapsto t \log {\left (x + \frac {729 t^{2} a^{5} b e^{2} + 4 c e^{2}}{4 d e^{2}} \right )} \right )\right )}}{d} \] Input:
integrate((d*e*x+c*e)/(a+b*(d*x+c)**3)**3,x)
Output:
(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 + 40*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 + (72 9*_t**2*a**5*b*e**2 + 4*c*e**2)/(4*d*e**2))))/d
\[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^3} \, dx=\int { \frac {d e x + c e}{{\left ({\left (d x + c\right )}^{3} b + a\right )}^{3}} \,d x } \] Input:
integrate((d*e*x+c*e)/(a+b*(d*x+c)^3)^3,x, algorithm="maxima")
Output:
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*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) /a^2
Time = 0.16 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.38 \[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^3} \, dx=-\frac {2 \, \sqrt {3} \left (-\frac {e^{3}}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}}{3 \, \left (-a^{2} b\right )^{\frac {2}{3}}}\right ) + \left (-\frac {e^{3}}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}^{2} + 3 \, \left (-a^{2} b\right )^{\frac {4}{3}}\right ) - 2 \, \left (-\frac {e^{3}}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | a b d x + a b c + \left (-a^{2} b\right )^{\frac {2}{3}} \right |}\right )}{27 \, a^{2}} + \frac {4 \, b d^{5} e x^{5} + 20 \, b c d^{4} e x^{4} + 40 \, b c^{2} d^{3} e x^{3} + 40 \, b c^{3} d^{2} e x^{2} + 20 \, b c^{4} d e x + 4 \, b c^{5} e + 7 \, a d^{2} e x^{2} + 14 \, a c d e x + 7 \, a c^{2} e}{18 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}^{2} a^{2} d} \] Input:
integrate((d*e*x+c*e)/(a+b*(d*x+c)^3)^3,x, algorithm="giac")
Output:
-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)
Time = 0.61 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.72 \[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^3} \, dx=\frac {\frac {4\,b\,e\,c^5+7\,a\,e\,c^2}{18\,a^2\,d}+\frac {x^2\,\left (40\,b\,d\,e\,c^3+7\,a\,d\,e\right )}{18\,a^2}+\frac {c\,x\,\left (10\,b\,e\,c^3+7\,a\,e\right )}{9\,a^2}+\frac {2\,b\,d^4\,e\,x^5}{9\,a^2}+\frac {20\,b\,c^2\,d^2\,e\,x^3}{9\,a^2}+\frac {10\,b\,c\,d^3\,e\,x^4}{9\,a^2}}{x^3\,\left (20\,b^2\,c^3\,d^3+2\,a\,b\,d^3\right )+x^2\,\left (15\,b^2\,c^4\,d^2+6\,a\,b\,c\,d^2\right )+a^2+x\,\left (6\,d\,b^2\,c^5+6\,a\,d\,b\,c^2\right )+b^2\,c^6+b^2\,d^6\,x^6+2\,a\,b\,c^3+6\,b^2\,c\,d^5\,x^5+15\,b^2\,c^2\,d^4\,x^4}-\frac {2\,e\,\ln \left (b^{1/3}\,c+a^{1/3}+b^{1/3}\,d\,x\right )}{27\,a^{7/3}\,b^{2/3}\,d}+\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,c-2\,b^{1/3}\,d\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (e-\sqrt {3}\,e\,1{}\mathrm {i}\right )}{27\,a^{7/3}\,b^{2/3}\,d}+\frac {\ln \left (2\,b^{1/3}\,c-a^{1/3}+2\,b^{1/3}\,d\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (e+\sqrt {3}\,e\,1{}\mathrm {i}\right )}{27\,a^{7/3}\,b^{2/3}\,d} \] Input:
int((c*e + d*e*x)/(a + b*(c + d*x)^3)^3,x)
Output:
((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)
Time = 0.24 (sec) , antiderivative size = 1829, normalized size of antiderivative = 8.84 \[ \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^3} \, dx =\text {Too large to display} \] Input:
int((d*e*x+c*e)/(a+b*(d*x+c)^3)^3,x)
Output:
(e*( - 4*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*c - 2*b**(1/3)*d*x)/(a**(1/3) *sqrt(3)))*a**2*b*c - 8*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*c - 2*b**(1/3) *d*x)/(a**(1/3)*sqrt(3)))*a*b**2*c**4 - 24*sqrt(3)*atan((a**(1/3) - 2*b**( 1/3)*c - 2*b**(1/3)*d*x)/(a**(1/3)*sqrt(3)))*a*b**2*c**3*d*x - 24*sqrt(3)* atan((a**(1/3) - 2*b**(1/3)*c - 2*b**(1/3)*d*x)/(a**(1/3)*sqrt(3)))*a*b**2 *c**2*d**2*x**2 - 8*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*c - 2*b**(1/3)*d*x )/(a**(1/3)*sqrt(3)))*a*b**2*c*d**3*x**3 - 4*sqrt(3)*atan((a**(1/3) - 2*b* *(1/3)*c - 2*b**(1/3)*d*x)/(a**(1/3)*sqrt(3)))*b**3*c**7 - 24*sqrt(3)*atan ((a**(1/3) - 2*b**(1/3)*c - 2*b**(1/3)*d*x)/(a**(1/3)*sqrt(3)))*b**3*c**6* d*x - 60*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*c - 2*b**(1/3)*d*x)/(a**(1/3) *sqrt(3)))*b**3*c**5*d**2*x**2 - 80*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*c - 2*b**(1/3)*d*x)/(a**(1/3)*sqrt(3)))*b**3*c**4*d**3*x**3 - 60*sqrt(3)*ata n((a**(1/3) - 2*b**(1/3)*c - 2*b**(1/3)*d*x)/(a**(1/3)*sqrt(3)))*b**3*c**3 *d**4*x**4 - 24*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*c - 2*b**(1/3)*d*x)/(a **(1/3)*sqrt(3)))*b**3*c**2*d**5*x**5 - 4*sqrt(3)*atan((a**(1/3) - 2*b**(1 /3)*c - 2*b**(1/3)*d*x)/(a**(1/3)*sqrt(3)))*b**3*c*d**6*x**6 - 2*b**(2/3)* a**(1/3)*a**2 + 17*b**(2/3)*a**(1/3)*a*b*c**3 + 30*b**(2/3)*a**(1/3)*a*b*c **2*d*x + 9*b**(2/3)*a**(1/3)*a*b*c*d**2*x**2 - 4*b**(2/3)*a**(1/3)*a*b*d* *3*x**3 + 10*b**(2/3)*a**(1/3)*b**2*c**6 + 48*b**(2/3)*a**(1/3)*b**2*c**5* d*x + 90*b**(2/3)*a**(1/3)*b**2*c**4*d**2*x**2 + 80*b**(2/3)*a**(1/3)*b...