Integrand size = 21, antiderivative size = 59 \[ \int \frac {1}{(c+d x) \left (a+b (c+d x)^3\right )^2} \, dx=\frac {1}{3 a d \left (a+b (c+d x)^3\right )}+\frac {\log (c+d x)}{a^2 d}-\frac {\log \left (a+b (c+d x)^3\right )}{3 a^2 d} \] Output:
1/3/a/d/(a+b*(d*x+c)^3)+ln(d*x+c)/a^2/d-1/3*ln(a+b*(d*x+c)^3)/a^2/d
Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(c+d x) \left (a+b (c+d x)^3\right )^2} \, dx=\frac {\frac {a}{a+b (c+d x)^3}+3 \log (c+d x)-\log \left (a+b (c+d x)^3\right )}{3 a^2 d} \] Input:
Integrate[1/((c + d*x)*(a + b*(c + d*x)^3)^2),x]
Output:
(a/(a + b*(c + d*x)^3) + 3*Log[c + d*x] - Log[a + b*(c + d*x)^3])/(3*a^2*d )
Time = 0.33 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {895, 798, 54, 2009}
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.
\(\displaystyle \int \frac {1}{(c+d x) \left (a+b (c+d x)^3\right )^2} \, dx\) |
\(\Big \downarrow \) 895 |
\(\displaystyle \frac {\int \frac {1}{(c+d x) \left (b (c+d x)^3+a\right )^2}d(c+d x)}{d}\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\int \frac {1}{(c+d x)^3 \left (b (c+d x)^3+a\right )^2}d(c+d x)^3}{3 d}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {\int \left (-\frac {b}{a^2 \left (b (c+d x)^3+a\right )}-\frac {b}{a \left (b (c+d x)^3+a\right )^2}+\frac {1}{a^2 (c+d x)^3}\right )d(c+d x)^3}{3 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {\log \left (a+b (c+d x)^3\right )}{a^2}+\frac {\log \left ((c+d x)^3\right )}{a^2}+\frac {1}{a \left (a+b (c+d x)^3\right )}}{3 d}\) |
Input:
Int[1/((c + d*x)*(a + b*(c + d*x)^3)^2),x]
Output:
(1/(a*(a + b*(c + d*x)^3)) + Log[(c + d*x)^3]/a^2 - Log[a + b*(c + d*x)^3] /a^2)/(3*d)
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
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]
Time = 0.79 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.69
method | result | size |
norman | \(\frac {1}{3 a d \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )}+\frac {\ln \left (x d +c \right )}{a^{2} d}-\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )}{3 a^{2} d}\) | \(100\) |
risch | \(\frac {1}{3 a d \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )}+\frac {\ln \left (x d +c \right )}{a^{2} d}-\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )}{3 a^{2} d}\) | \(100\) |
default | \(\frac {\ln \left (x d +c \right )}{a^{2} d}-\frac {b \left (-\frac {a}{3 d b \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )}+\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )}{3 b d}\right )}{a^{2}}\) | \(108\) |
parallelrisch | \(\frac {3 \ln \left (x d +c \right ) x^{3} b^{2} d^{5}-\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right ) x^{3} b^{2} d^{5}+9 \ln \left (x d +c \right ) x^{2} b^{2} c \,d^{4}-3 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right ) x^{2} b^{2} c \,d^{4}+9 \ln \left (x d +c \right ) x \,b^{2} c^{2} d^{3}-3 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right ) x \,b^{2} c^{2} d^{3}+3 \ln \left (x d +c \right ) b^{2} c^{3} d^{2}-\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right ) b^{2} c^{3} d^{2}+3 \ln \left (x d +c \right ) a b \,d^{2}-\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right ) a b \,d^{2}+a b \,d^{2}}{3 a^{2} d^{3} b \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )}\) | \(360\) |
Input:
int(1/(d*x+c)/(a+b*(d*x+c)^3)^2,x,method=_RETURNVERBOSE)
Output:
1/3/a/d/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)+ln(d*x+c)/a^2/d-1/3/ a^2/d*ln(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)
Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (55) = 110\).
Time = 0.08 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.86 \[ \int \frac {1}{(c+d x) \left (a+b (c+d x)^3\right )^2} \, dx=-\frac {{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) - 3 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} \log \left (d x + c\right ) - a}{3 \, {\left (a^{2} b d^{4} x^{3} + 3 \, a^{2} b c d^{3} x^{2} + 3 \, a^{2} b c^{2} d^{2} x + {\left (a^{2} b c^{3} + a^{3}\right )} d\right )}} \] Input:
integrate(1/(d*x+c)/(a+b*(d*x+c)^3)^2,x, algorithm="fricas")
Output:
-1/3*((b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)*log(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a) - 3*(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)*log(d*x + c) - a)/(a^2*b*d^4*x^3 + 3*a^2*b*c*d^ 3*x^2 + 3*a^2*b*c^2*d^2*x + (a^2*b*c^3 + a^3)*d)
Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (46) = 92\).
Time = 0.97 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.86 \[ \int \frac {1}{(c+d x) \left (a+b (c+d x)^3\right )^2} \, dx=\frac {1}{3 a^{2} d + 3 a b c^{3} d + 9 a b c^{2} d^{2} x + 9 a b c d^{3} x^{2} + 3 a b d^{4} x^{3}} + \frac {\log {\left (\frac {c}{d} + x \right )}}{a^{2} d} - \frac {\log {\left (\frac {3 c^{2} x}{d^{2}} + \frac {3 c x^{2}}{d} + x^{3} + \frac {a + b c^{3}}{b d^{3}} \right )}}{3 a^{2} d} \] Input:
integrate(1/(d*x+c)/(a+b*(d*x+c)**3)**2,x)
Output:
1/(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) + log(c/d + x)/(a**2*d) - log(3*c**2*x/d**2 + 3*c*x**2/d + x** 3 + (a + b*c**3)/(b*d**3))/(3*a**2*d)
Time = 0.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.76 \[ \int \frac {1}{(c+d x) \left (a+b (c+d x)^3\right )^2} \, dx=\frac {1}{3 \, {\left (a b d^{4} x^{3} + 3 \, a b c d^{3} x^{2} + 3 \, a b c^{2} d^{2} x + {\left (a b c^{3} + a^{2}\right )} d\right )}} - \frac {\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a^{2} d} + \frac {\log \left (d x + c\right )}{a^{2} d} \] Input:
integrate(1/(d*x+c)/(a+b*(d*x+c)^3)^2,x, algorithm="maxima")
Output:
1/3/(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) - 1/3*log(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)/(a^2*d) + l og(d*x + c)/(a^2*d)
Time = 0.13 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.71 \[ \int \frac {1}{(c+d x) \left (a+b (c+d x)^3\right )^2} \, dx=-\frac {\log \left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, a^{2} d} + \frac {\log \left ({\left | d x + c \right |}\right )}{a^{2} d} + \frac {1}{3 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} a d} \] Input:
integrate(1/(d*x+c)/(a+b*(d*x+c)^3)^2,x, algorithm="giac")
Output:
-1/3*log(abs(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a))/(a^2*d) + log(abs(d*x + c))/(a^2*d) + 1/3/((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 = 0.58 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.78 \[ \int \frac {1}{(c+d x) \left (a+b (c+d x)^3\right )^2} \, dx=\frac {1}{3\,\left (a^2\,d+b\,a\,c^3\,d+3\,b\,a\,c^2\,d^2\,x+3\,b\,a\,c\,d^3\,x^2+b\,a\,d^4\,x^3\right )}-\frac {\ln \left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )}{3\,a^2\,d}+\frac {\ln \left (c+d\,x\right )}{a^2\,d} \] Input:
int(1/((a + b*(c + d*x)^3)^2*(c + d*x)),x)
Output:
1/(3*(a^2*d + a*b*c^3*d + a*b*d^4*x^3 + 3*a*b*c^2*d^2*x + 3*a*b*c*d^3*x^2) ) - log(a + b*c^3 + b*d^3*x^3 + 3*b*c^2*d*x + 3*b*c*d^2*x^2)/(3*a^2*d) + l og(c + d*x)/(a^2*d)
Time = 0.24 (sec) , antiderivative size = 508, normalized size of antiderivative = 8.61 \[ \int \frac {1}{(c+d x) \left (a+b (c+d x)^3\right )^2} \, dx=\frac {-\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} c -b^{\frac {1}{3}} a^{\frac {1}{3}} d x +b^{\frac {2}{3}} c^{2}+2 b^{\frac {2}{3}} c d x +b^{\frac {2}{3}} d^{2} x^{2}\right ) a -\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} c -b^{\frac {1}{3}} a^{\frac {1}{3}} d x +b^{\frac {2}{3}} c^{2}+2 b^{\frac {2}{3}} c d x +b^{\frac {2}{3}} d^{2} x^{2}\right ) b \,c^{3}-3 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} c -b^{\frac {1}{3}} a^{\frac {1}{3}} d x +b^{\frac {2}{3}} c^{2}+2 b^{\frac {2}{3}} c d x +b^{\frac {2}{3}} d^{2} x^{2}\right ) b \,c^{2} d x -3 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} c -b^{\frac {1}{3}} a^{\frac {1}{3}} d x +b^{\frac {2}{3}} c^{2}+2 b^{\frac {2}{3}} c d x +b^{\frac {2}{3}} d^{2} x^{2}\right ) b c \,d^{2} x^{2}-\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} c -b^{\frac {1}{3}} a^{\frac {1}{3}} d x +b^{\frac {2}{3}} c^{2}+2 b^{\frac {2}{3}} c d x +b^{\frac {2}{3}} d^{2} x^{2}\right ) b \,d^{3} x^{3}-\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} c +b^{\frac {1}{3}} d x \right ) a -\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} c +b^{\frac {1}{3}} d x \right ) b \,c^{3}-3 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} c +b^{\frac {1}{3}} d x \right ) b \,c^{2} d x -3 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} c +b^{\frac {1}{3}} d x \right ) b c \,d^{2} x^{2}-\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} c +b^{\frac {1}{3}} d x \right ) b \,d^{3} x^{3}+3 \,\mathrm {log}\left (d x +c \right ) a +3 \,\mathrm {log}\left (d x +c \right ) b \,c^{3}+9 \,\mathrm {log}\left (d x +c \right ) b \,c^{2} d x +9 \,\mathrm {log}\left (d x +c \right ) b c \,d^{2} x^{2}+3 \,\mathrm {log}\left (d x +c \right ) b \,d^{3} x^{3}+a}{3 a^{2} d \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )} \] Input:
int(1/(d*x+c)/(a+b*(d*x+c)^3)^2,x)
Output:
( - 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 - 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*c**3 - 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*c **2*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*c*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*d**3*x**3 - log(a**(1/3) + b**(1/3 )*c + b**(1/3)*d*x)*a - log(a**(1/3) + b**(1/3)*c + b**(1/3)*d*x)*b*c**3 - 3*log(a**(1/3) + b**(1/3)*c + b**(1/3)*d*x)*b*c**2*d*x - 3*log(a**(1/3) + b**(1/3)*c + b**(1/3)*d*x)*b*c*d**2*x**2 - log(a**(1/3) + b**(1/3)*c + b* *(1/3)*d*x)*b*d**3*x**3 + 3*log(c + d*x)*a + 3*log(c + d*x)*b*c**3 + 9*log (c + d*x)*b*c**2*d*x + 9*log(c + d*x)*b*c*d**2*x**2 + 3*log(c + d*x)*b*d** 3*x**3 + a)/(3*a**2*d*(a + b*c**3 + 3*b*c**2*d*x + 3*b*c*d**2*x**2 + b*d** 3*x**3))