Integrand size = 20, antiderivative size = 122 \[ \int \frac {A+B x^3}{x^7 \left (a+b x^3\right )^3} \, dx=-\frac {A}{6 a^3 x^6}+\frac {3 A b-a B}{3 a^4 x^3}+\frac {b (A b-a B)}{6 a^3 \left (a+b x^3\right )^2}+\frac {b (3 A b-2 a B)}{3 a^4 \left (a+b x^3\right )}+\frac {3 b (2 A b-a B) \log (x)}{a^5}-\frac {b (2 A b-a B) \log \left (a+b x^3\right )}{a^5} \] Output:
-1/6*A/a^3/x^6+1/3*(3*A*b-B*a)/a^4/x^3+1/6*b*(A*b-B*a)/a^3/(b*x^3+a)^2+1/3 *b*(3*A*b-2*B*a)/a^4/(b*x^3+a)+3*b*(2*A*b-B*a)*ln(x)/a^5-b*(2*A*b-B*a)*ln( b*x^3+a)/a^5
Time = 0.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.89 \[ \int \frac {A+B x^3}{x^7 \left (a+b x^3\right )^3} \, dx=\frac {-\frac {a^2 A}{x^6}-\frac {2 a (-3 A b+a B)}{x^3}+\frac {a^2 b (A b-a B)}{\left (a+b x^3\right )^2}+\frac {2 a b (3 A b-2 a B)}{a+b x^3}+18 b (2 A b-a B) \log (x)+6 b (-2 A b+a B) \log \left (a+b x^3\right )}{6 a^5} \] Input:
Integrate[(A + B*x^3)/(x^7*(a + b*x^3)^3),x]
Output:
(-((a^2*A)/x^6) - (2*a*(-3*A*b + a*B))/x^3 + (a^2*b*(A*b - a*B))/(a + b*x^ 3)^2 + (2*a*b*(3*A*b - 2*a*B))/(a + b*x^3) + 18*b*(2*A*b - a*B)*Log[x] + 6 *b*(-2*A*b + a*B)*Log[a + b*x^3])/(6*a^5)
Time = 0.52 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {948, 86, 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.
\(\displaystyle \int \frac {A+B x^3}{x^7 \left (a+b x^3\right )^3} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{3} \int \frac {B x^3+A}{x^9 \left (b x^3+a\right )^3}dx^3\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \frac {1}{3} \int \left (\frac {3 (a B-2 A b) b^2}{a^5 \left (b x^3+a\right )}+\frac {(2 a B-3 A b) b^2}{a^4 \left (b x^3+a\right )^2}+\frac {(a B-A b) b^2}{a^3 \left (b x^3+a\right )^3}-\frac {3 (a B-2 A b) b}{a^5 x^3}+\frac {a B-3 A b}{a^4 x^6}+\frac {A}{a^3 x^9}\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (\frac {3 b \log \left (x^3\right ) (2 A b-a B)}{a^5}-\frac {3 b (2 A b-a B) \log \left (a+b x^3\right )}{a^5}+\frac {b (3 A b-2 a B)}{a^4 \left (a+b x^3\right )}+\frac {3 A b-a B}{a^4 x^3}+\frac {b (A b-a B)}{2 a^3 \left (a+b x^3\right )^2}-\frac {A}{2 a^3 x^6}\right )\) |
Input:
Int[(A + B*x^3)/(x^7*(a + b*x^3)^3),x]
Output:
(-1/2*A/(a^3*x^6) + (3*A*b - a*B)/(a^4*x^3) + (b*(A*b - a*B))/(2*a^3*(a + b*x^3)^2) + (b*(3*A*b - 2*a*B))/(a^4*(a + b*x^3)) + (3*b*(2*A*b - a*B)*Log [x^3])/a^5 - (3*b*(2*A*b - a*B)*Log[a + b*x^3])/a^5)/3
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.74 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.01
method | result | size |
default | \(-\frac {A}{6 a^{3} x^{6}}-\frac {-3 A b +B a}{3 a^{4} x^{3}}+\frac {3 b \left (2 A b -B a \right ) \ln \left (x \right )}{a^{5}}-\frac {b^{2} \left (-\frac {a^{2} \left (A b -B a \right )}{2 b \left (b \,x^{3}+a \right )^{2}}-\frac {a \left (3 A b -2 B a \right )}{b \left (b \,x^{3}+a \right )}+\frac {\left (6 A b -3 B a \right ) \ln \left (b \,x^{3}+a \right )}{b}\right )}{3 a^{5}}\) | \(123\) |
norman | \(\frac {-\frac {A}{6 a}+\frac {\left (2 A b -B a \right ) x^{3}}{3 a^{2}}-\frac {2 b \left (2 b^{2} A -a b B \right ) x^{9}}{a^{4}}-\frac {b^{2} \left (6 b^{2} A -3 a b B \right ) x^{12}}{2 a^{5}}}{x^{6} \left (b \,x^{3}+a \right )^{2}}+\frac {3 b \left (2 A b -B a \right ) \ln \left (x \right )}{a^{5}}-\frac {b \left (2 A b -B a \right ) \ln \left (b \,x^{3}+a \right )}{a^{5}}\) | \(123\) |
risch | \(\frac {\frac {b^{2} \left (2 A b -B a \right ) x^{9}}{a^{4}}+\frac {3 b \left (2 A b -B a \right ) x^{6}}{2 a^{3}}+\frac {\left (2 A b -B a \right ) x^{3}}{3 a^{2}}-\frac {A}{6 a}}{x^{6} \left (b \,x^{3}+a \right )^{2}}+\frac {6 b^{2} \ln \left (x \right ) A}{a^{5}}-\frac {3 b \ln \left (x \right ) B}{a^{4}}-\frac {2 b^{2} \ln \left (b \,x^{3}+a \right ) A}{a^{5}}+\frac {b \ln \left (b \,x^{3}+a \right ) B}{a^{4}}\) | \(127\) |
parallelrisch | \(\frac {36 A \ln \left (x \right ) x^{12} b^{4}-12 A \ln \left (b \,x^{3}+a \right ) x^{12} b^{4}-18 B \ln \left (x \right ) x^{12} a \,b^{3}+6 B \ln \left (b \,x^{3}+a \right ) x^{12} a \,b^{3}-18 A \,x^{12} b^{4}+9 B \,x^{12} a \,b^{3}+72 A \ln \left (x \right ) x^{9} a \,b^{3}-24 A \ln \left (b \,x^{3}+a \right ) x^{9} a \,b^{3}-36 B \ln \left (x \right ) x^{9} a^{2} b^{2}+12 B \ln \left (b \,x^{3}+a \right ) x^{9} a^{2} b^{2}-24 A \,x^{9} a \,b^{3}+12 B \,x^{9} a^{2} b^{2}+36 A \ln \left (x \right ) x^{6} a^{2} b^{2}-12 A \ln \left (b \,x^{3}+a \right ) x^{6} a^{2} b^{2}-18 B \ln \left (x \right ) x^{6} a^{3} b +6 B \ln \left (b \,x^{3}+a \right ) x^{6} a^{3} b +4 A \,x^{3} a^{3} b -2 B \,x^{3} a^{4}-A \,a^{4}}{6 a^{5} x^{6} \left (b \,x^{3}+a \right )^{2}}\) | \(271\) |
Input:
int((B*x^3+A)/x^7/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
Output:
-1/6*A/a^3/x^6-1/3*(-3*A*b+B*a)/a^4/x^3+3*b*(2*A*b-B*a)*ln(x)/a^5-1/3/a^5* b^2*(-1/2*a^2*(A*b-B*a)/b/(b*x^3+a)^2-a*(3*A*b-2*B*a)/b/(b*x^3+a)+(6*A*b-3 *B*a)/b*ln(b*x^3+a))
Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (110) = 220\).
Time = 0.09 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.88 \[ \int \frac {A+B x^3}{x^7 \left (a+b x^3\right )^3} \, dx=-\frac {6 \, {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{9} + 9 \, {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{6} + A a^{4} + 2 \, {\left (B a^{4} - 2 \, A a^{3} b\right )} x^{3} - 6 \, {\left ({\left (B a b^{3} - 2 \, A b^{4}\right )} x^{12} + 2 \, {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{9} + {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{6}\right )} \log \left (b x^{3} + a\right ) + 18 \, {\left ({\left (B a b^{3} - 2 \, A b^{4}\right )} x^{12} + 2 \, {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{9} + {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{6}\right )} \log \left (x\right )}{6 \, {\left (a^{5} b^{2} x^{12} + 2 \, a^{6} b x^{9} + a^{7} x^{6}\right )}} \] Input:
integrate((B*x^3+A)/x^7/(b*x^3+a)^3,x, algorithm="fricas")
Output:
-1/6*(6*(B*a^2*b^2 - 2*A*a*b^3)*x^9 + 9*(B*a^3*b - 2*A*a^2*b^2)*x^6 + A*a^ 4 + 2*(B*a^4 - 2*A*a^3*b)*x^3 - 6*((B*a*b^3 - 2*A*b^4)*x^12 + 2*(B*a^2*b^2 - 2*A*a*b^3)*x^9 + (B*a^3*b - 2*A*a^2*b^2)*x^6)*log(b*x^3 + a) + 18*((B*a *b^3 - 2*A*b^4)*x^12 + 2*(B*a^2*b^2 - 2*A*a*b^3)*x^9 + (B*a^3*b - 2*A*a^2* b^2)*x^6)*log(x))/(a^5*b^2*x^12 + 2*a^6*b*x^9 + a^7*x^6)
Time = 0.98 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.09 \[ \int \frac {A+B x^3}{x^7 \left (a+b x^3\right )^3} \, dx=\frac {- A a^{3} + x^{9} \cdot \left (12 A b^{3} - 6 B a b^{2}\right ) + x^{6} \cdot \left (18 A a b^{2} - 9 B a^{2} b\right ) + x^{3} \cdot \left (4 A a^{2} b - 2 B a^{3}\right )}{6 a^{6} x^{6} + 12 a^{5} b x^{9} + 6 a^{4} b^{2} x^{12}} - \frac {3 b \left (- 2 A b + B a\right ) \log {\left (x \right )}}{a^{5}} + \frac {b \left (- 2 A b + B a\right ) \log {\left (\frac {a}{b} + x^{3} \right )}}{a^{5}} \] Input:
integrate((B*x**3+A)/x**7/(b*x**3+a)**3,x)
Output:
(-A*a**3 + x**9*(12*A*b**3 - 6*B*a*b**2) + x**6*(18*A*a*b**2 - 9*B*a**2*b) + x**3*(4*A*a**2*b - 2*B*a**3))/(6*a**6*x**6 + 12*a**5*b*x**9 + 6*a**4*b* *2*x**12) - 3*b*(-2*A*b + B*a)*log(x)/a**5 + b*(-2*A*b + B*a)*log(a/b + x* *3)/a**5
Time = 0.03 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.11 \[ \int \frac {A+B x^3}{x^7 \left (a+b x^3\right )^3} \, dx=-\frac {6 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} x^{9} + 9 \, {\left (B a^{2} b - 2 \, A a b^{2}\right )} x^{6} + A a^{3} + 2 \, {\left (B a^{3} - 2 \, A a^{2} b\right )} x^{3}}{6 \, {\left (a^{4} b^{2} x^{12} + 2 \, a^{5} b x^{9} + a^{6} x^{6}\right )}} + \frac {{\left (B a b - 2 \, A b^{2}\right )} \log \left (b x^{3} + a\right )}{a^{5}} - \frac {{\left (B a b - 2 \, A b^{2}\right )} \log \left (x^{3}\right )}{a^{5}} \] Input:
integrate((B*x^3+A)/x^7/(b*x^3+a)^3,x, algorithm="maxima")
Output:
-1/6*(6*(B*a*b^2 - 2*A*b^3)*x^9 + 9*(B*a^2*b - 2*A*a*b^2)*x^6 + A*a^3 + 2* (B*a^3 - 2*A*a^2*b)*x^3)/(a^4*b^2*x^12 + 2*a^5*b*x^9 + a^6*x^6) + (B*a*b - 2*A*b^2)*log(b*x^3 + a)/a^5 - (B*a*b - 2*A*b^2)*log(x^3)/a^5
Time = 0.14 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.07 \[ \int \frac {A+B x^3}{x^7 \left (a+b x^3\right )^3} \, dx=-\frac {3 \, {\left (B a b - 2 \, A b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac {{\left (B a b^{2} - 2 \, A b^{3}\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{a^{5} b} - \frac {6 \, B a b^{2} x^{9} - 12 \, A b^{3} x^{9} + 9 \, B a^{2} b x^{6} - 18 \, A a b^{2} x^{6} + 2 \, B a^{3} x^{3} - 4 \, A a^{2} b x^{3} + A a^{3}}{6 \, {\left (b x^{6} + a x^{3}\right )}^{2} a^{4}} \] Input:
integrate((B*x^3+A)/x^7/(b*x^3+a)^3,x, algorithm="giac")
Output:
-3*(B*a*b - 2*A*b^2)*log(abs(x))/a^5 + (B*a*b^2 - 2*A*b^3)*log(abs(b*x^3 + a))/(a^5*b) - 1/6*(6*B*a*b^2*x^9 - 12*A*b^3*x^9 + 9*B*a^2*b*x^6 - 18*A*a* b^2*x^6 + 2*B*a^3*x^3 - 4*A*a^2*b*x^3 + A*a^3)/((b*x^6 + a*x^3)^2*a^4)
Time = 0.82 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.07 \[ \int \frac {A+B x^3}{x^7 \left (a+b x^3\right )^3} \, dx=\frac {\frac {x^3\,\left (2\,A\,b-B\,a\right )}{3\,a^2}-\frac {A}{6\,a}+\frac {b^2\,x^9\,\left (2\,A\,b-B\,a\right )}{a^4}+\frac {3\,b\,x^6\,\left (2\,A\,b-B\,a\right )}{2\,a^3}}{a^2\,x^6+2\,a\,b\,x^9+b^2\,x^{12}}-\frac {\ln \left (b\,x^3+a\right )\,\left (2\,A\,b^2-B\,a\,b\right )}{a^5}+\frac {\ln \left (x\right )\,\left (6\,A\,b^2-3\,B\,a\,b\right )}{a^5} \] Input:
int((A + B*x^3)/(x^7*(a + b*x^3)^3),x)
Output:
((x^3*(2*A*b - B*a))/(3*a^2) - A/(6*a) + (b^2*x^9*(2*A*b - B*a))/a^4 + (3* b*x^6*(2*A*b - B*a))/(2*a^3))/(a^2*x^6 + b^2*x^12 + 2*a*b*x^9) - (log(a + b*x^3)*(2*A*b^2 - B*a*b))/a^5 + (log(x)*(6*A*b^2 - 3*B*a*b))/a^5
Time = 0.23 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.29 \[ \int \frac {A+B x^3}{x^7 \left (a+b x^3\right )^3} \, dx=\frac {-6 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a \,b^{2} x^{6}-6 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b^{3} x^{9}-6 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a \,b^{2} x^{6}-6 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b^{3} x^{9}+18 \,\mathrm {log}\left (x \right ) a \,b^{2} x^{6}+18 \,\mathrm {log}\left (x \right ) b^{3} x^{9}-a^{3}+3 a^{2} b \,x^{3}-6 b^{3} x^{9}}{6 a^{4} x^{6} \left (b \,x^{3}+a \right )} \] Input:
int((B*x^3+A)/x^7/(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*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*b**2*x**6 - 6*log(a**(1/3) + b**(1/3)*x)*b**3*x**9 + 18*log(x)*a*b**2*x**6 + 18*log(x)*b**3*x**9 - a**3 + 3*a**2*b*x**3 - 6*b** 3*x**9)/(6*a**4*x**6*(a + b*x**3))