Integrand size = 13, antiderivative size = 66 \[ \int \frac {1}{x^4 \left (a+b x^3\right )^3} \, dx=-\frac {1}{3 a^3 x^3}-\frac {b}{6 a^2 \left (a+b x^3\right )^2}-\frac {2 b}{3 a^3 \left (a+b x^3\right )}-\frac {3 b \log (x)}{a^4}+\frac {b \log \left (a+b x^3\right )}{a^4} \] Output:
-1/3/a^3/x^3-1/6*b/a^2/(b*x^3+a)^2-2/3*b/a^3/(b*x^3+a)-3*b*ln(x)/a^4+b*ln( b*x^3+a)/a^4
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^4 \left (a+b x^3\right )^3} \, dx=-\frac {\frac {a \left (2 a^2+9 a b x^3+6 b^2 x^6\right )}{x^3 \left (a+b x^3\right )^2}+18 b \log (x)-6 b \log \left (a+b x^3\right )}{6 a^4} \] Input:
Integrate[1/(x^4*(a + b*x^3)^3),x]
Output:
-1/6*((a*(2*a^2 + 9*a*b*x^3 + 6*b^2*x^6))/(x^3*(a + b*x^3)^2) + 18*b*Log[x ] - 6*b*Log[a + b*x^3])/a^4
Time = 0.33 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {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.
\(\displaystyle \int \frac {1}{x^4 \left (a+b x^3\right )^3} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int \frac {1}{x^6 \left (b x^3+a\right )^3}dx^3\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {1}{3} \int \left (\frac {3 b^2}{a^4 \left (b x^3+a\right )}+\frac {2 b^2}{a^3 \left (b x^3+a\right )^2}+\frac {b^2}{a^2 \left (b x^3+a\right )^3}-\frac {3 b}{a^4 x^3}+\frac {1}{a^3 x^6}\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-\frac {3 b \log \left (x^3\right )}{a^4}+\frac {3 b \log \left (a+b x^3\right )}{a^4}-\frac {2 b}{a^3 \left (a+b x^3\right )}-\frac {1}{a^3 x^3}-\frac {b}{2 a^2 \left (a+b x^3\right )^2}\right )\) |
Input:
Int[1/(x^4*(a + b*x^3)^3),x]
Output:
(-(1/(a^3*x^3)) - b/(2*a^2*(a + b*x^3)^2) - (2*b)/(a^3*(a + b*x^3)) - (3*b *Log[x^3])/a^4 + (3*b*Log[a + b*x^3])/a^4)/3
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]]
Time = 0.45 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.97
method | result | size |
norman | \(\frac {-\frac {1}{3 a}+\frac {2 b^{2} x^{6}}{a^{3}}+\frac {3 b^{3} x^{9}}{2 a^{4}}}{x^{3} \left (b \,x^{3}+a \right )^{2}}+\frac {b \ln \left (b \,x^{3}+a \right )}{a^{4}}-\frac {3 b \ln \left (x \right )}{a^{4}}\) | \(64\) |
risch | \(\frac {-\frac {b^{2} x^{6}}{a^{3}}-\frac {3 b \,x^{3}}{2 a^{2}}-\frac {1}{3 a}}{x^{3} \left (b \,x^{3}+a \right )^{2}}-\frac {3 b \ln \left (x \right )}{a^{4}}+\frac {b \ln \left (-b \,x^{3}-a \right )}{a^{4}}\) | \(65\) |
default | \(-\frac {1}{3 a^{3} x^{3}}-\frac {3 b \ln \left (x \right )}{a^{4}}+\frac {b^{2} \left (-\frac {a^{2}}{2 b \left (b \,x^{3}+a \right )^{2}}-\frac {2 a}{b \left (b \,x^{3}+a \right )}+\frac {3 \ln \left (b \,x^{3}+a \right )}{b}\right )}{3 a^{4}}\) | \(72\) |
parallelrisch | \(-\frac {18 b^{3} \ln \left (x \right ) x^{9}-6 \ln \left (b \,x^{3}+a \right ) x^{9} b^{3}-9 b^{3} x^{9}+36 a \,b^{2} \ln \left (x \right ) x^{6}-12 \ln \left (b \,x^{3}+a \right ) x^{6} a \,b^{2}-12 a \,b^{2} x^{6}+18 \ln \left (x \right ) x^{3} a^{2} b -6 \ln \left (b \,x^{3}+a \right ) x^{3} a^{2} b +2 a^{3}}{6 a^{4} x^{3} \left (b \,x^{3}+a \right )^{2}}\) | \(123\) |
Input:
int(1/x^4/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
Output:
(-1/3/a+2*b^2/a^3*x^6+3/2*b^3/a^4*x^9)/x^3/(b*x^3+a)^2+b*ln(b*x^3+a)/a^4-3 *b*ln(x)/a^4
Time = 0.07 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.80 \[ \int \frac {1}{x^4 \left (a+b x^3\right )^3} \, dx=-\frac {6 \, a b^{2} x^{6} + 9 \, a^{2} b x^{3} + 2 \, a^{3} - 6 \, {\left (b^{3} x^{9} + 2 \, a b^{2} x^{6} + a^{2} b x^{3}\right )} \log \left (b x^{3} + a\right ) + 18 \, {\left (b^{3} x^{9} + 2 \, a b^{2} x^{6} + a^{2} b x^{3}\right )} \log \left (x\right )}{6 \, {\left (a^{4} b^{2} x^{9} + 2 \, a^{5} b x^{6} + a^{6} x^{3}\right )}} \] Input:
integrate(1/x^4/(b*x^3+a)^3,x, algorithm="fricas")
Output:
-1/6*(6*a*b^2*x^6 + 9*a^2*b*x^3 + 2*a^3 - 6*(b^3*x^9 + 2*a*b^2*x^6 + a^2*b *x^3)*log(b*x^3 + a) + 18*(b^3*x^9 + 2*a*b^2*x^6 + a^2*b*x^3)*log(x))/(a^4 *b^2*x^9 + 2*a^5*b*x^6 + a^6*x^3)
Time = 0.32 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x^4 \left (a+b x^3\right )^3} \, dx=\frac {- 2 a^{2} - 9 a b x^{3} - 6 b^{2} x^{6}}{6 a^{5} x^{3} + 12 a^{4} b x^{6} + 6 a^{3} b^{2} x^{9}} - \frac {3 b \log {\left (x \right )}}{a^{4}} + \frac {b \log {\left (\frac {a}{b} + x^{3} \right )}}{a^{4}} \] Input:
integrate(1/x**4/(b*x**3+a)**3,x)
Output:
(-2*a**2 - 9*a*b*x**3 - 6*b**2*x**6)/(6*a**5*x**3 + 12*a**4*b*x**6 + 6*a** 3*b**2*x**9) - 3*b*log(x)/a**4 + b*log(a/b + x**3)/a**4
Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x^4 \left (a+b x^3\right )^3} \, dx=-\frac {6 \, b^{2} x^{6} + 9 \, a b x^{3} + 2 \, a^{2}}{6 \, {\left (a^{3} b^{2} x^{9} + 2 \, a^{4} b x^{6} + a^{5} x^{3}\right )}} + \frac {b \log \left (b x^{3} + a\right )}{a^{4}} - \frac {b \log \left (x^{3}\right )}{a^{4}} \] Input:
integrate(1/x^4/(b*x^3+a)^3,x, algorithm="maxima")
Output:
-1/6*(6*b^2*x^6 + 9*a*b*x^3 + 2*a^2)/(a^3*b^2*x^9 + 2*a^4*b*x^6 + a^5*x^3) + b*log(b*x^3 + a)/a^4 - b*log(x^3)/a^4
Time = 0.13 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.21 \[ \int \frac {1}{x^4 \left (a+b x^3\right )^3} \, dx=\frac {b \log \left ({\left | b x^{3} + a \right |}\right )}{a^{4}} - \frac {3 \, b \log \left ({\left | x \right |}\right )}{a^{4}} - \frac {9 \, b^{3} x^{6} + 22 \, a b^{2} x^{3} + 14 \, a^{2} b}{6 \, {\left (b x^{3} + a\right )}^{2} a^{4}} + \frac {3 \, b x^{3} - a}{3 \, a^{4} x^{3}} \] Input:
integrate(1/x^4/(b*x^3+a)^3,x, algorithm="giac")
Output:
b*log(abs(b*x^3 + a))/a^4 - 3*b*log(abs(x))/a^4 - 1/6*(9*b^3*x^6 + 22*a*b^ 2*x^3 + 14*a^2*b)/((b*x^3 + a)^2*a^4) + 1/3*(3*b*x^3 - a)/(a^4*x^3)
Time = 0.18 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^4 \left (a+b x^3\right )^3} \, dx=\frac {b\,\ln \left (b\,x^3+a\right )}{a^4}-\frac {\frac {1}{3\,a}+\frac {3\,b\,x^3}{2\,a^2}+\frac {b^2\,x^6}{a^3}}{a^2\,x^3+2\,a\,b\,x^6+b^2\,x^9}-\frac {3\,b\,\ln \left (x\right )}{a^4} \] Input:
int(1/(x^4*(a + b*x^3)^3),x)
Output:
(b*log(a + b*x^3))/a^4 - (1/(3*a) + (3*b*x^3)/(2*a^2) + (b^2*x^6)/a^3)/(a^ 2*x^3 + b^2*x^9 + 2*a*b*x^6) - (3*b*log(x))/a^4
Time = 0.24 (sec) , antiderivative size = 228, normalized size of antiderivative = 3.45 \[ \int \frac {1}{x^4 \left (a+b x^3\right )^3} \, dx=\frac {6 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{2} b \,x^{3}+12 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a \,b^{2} x^{6}+6 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b^{3} x^{9}+6 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{2} b \,x^{3}+12 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a \,b^{2} x^{6}+6 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b^{3} x^{9}-18 \,\mathrm {log}\left (x \right ) a^{2} b \,x^{3}-36 \,\mathrm {log}\left (x \right ) a \,b^{2} x^{6}-18 \,\mathrm {log}\left (x \right ) b^{3} x^{9}-2 a^{3}-6 a^{2} b \,x^{3}+3 b^{3} x^{9}}{6 a^{4} x^{3} \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )} \] Input:
int(1/x^4/(b*x^3+a)^3,x)
Output:
(6*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*b*x**3 + 12*lo g(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a*b**2*x**6 + 6*log(a**( 2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*b**3*x**9 + 6*log(a**(1/3) + b **(1/3)*x)*a**2*b*x**3 + 12*log(a**(1/3) + b**(1/3)*x)*a*b**2*x**6 + 6*log (a**(1/3) + b**(1/3)*x)*b**3*x**9 - 18*log(x)*a**2*b*x**3 - 36*log(x)*a*b* *2*x**6 - 18*log(x)*b**3*x**9 - 2*a**3 - 6*a**2*b*x**3 + 3*b**3*x**9)/(6*a **4*x**3*(a**2 + 2*a*b*x**3 + b**2*x**6))