Integrand size = 27, antiderivative size = 96 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x} \, dx=6 a^5 A b x+\frac {15}{2} a^4 A b^2 x^2+\frac {20}{3} a^3 A b^3 x^3+\frac {15}{4} a^2 A b^4 x^4+\frac {6}{5} a A b^5 x^5+\frac {1}{6} A b^6 x^6+\frac {B (a+b x)^7}{7 b}+a^6 A \log (x) \] Output:
6*a^5*A*b*x+15/2*a^4*A*b^2*x^2+20/3*a^3*A*b^3*x^3+15/4*a^2*A*b^4*x^4+6/5*a *A*b^5*x^5+1/6*A*b^6*x^6+1/7*B*(b*x+a)^7/b+a^6*A*ln(x)
Time = 0.04 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.33 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x} \, dx=a^6 B x+3 a^5 b x (2 A+B x)+\frac {5}{2} a^4 b^2 x^2 (3 A+2 B x)+\frac {5}{3} a^3 b^3 x^3 (4 A+3 B x)+\frac {3}{4} a^2 b^4 x^4 (5 A+4 B x)+\frac {1}{5} a b^5 x^5 (6 A+5 B x)+\frac {1}{42} b^6 x^6 (7 A+6 B x)+a^6 A \log (x) \] Input:
Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x,x]
Output:
a^6*B*x + 3*a^5*b*x*(2*A + B*x) + (5*a^4*b^2*x^2*(3*A + 2*B*x))/2 + (5*a^3 *b^3*x^3*(4*A + 3*B*x))/3 + (3*a^2*b^4*x^4*(5*A + 4*B*x))/4 + (a*b^5*x^5*( 6*A + 5*B*x))/5 + (b^6*x^6*(7*A + 6*B*x))/42 + a^6*A*Log[x]
Time = 0.40 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1184, 27, 90, 49, 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.
\(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3 (A+B x)}{x} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \frac {\int \frac {b^6 (a+b x)^6 (A+B x)}{x}dx}{b^6}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {(a+b x)^6 (A+B x)}{x}dx\) |
\(\Big \downarrow \) 90 |
\(\displaystyle A \int \frac {(a+b x)^6}{x}dx+\frac {B (a+b x)^7}{7 b}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle A \int \left (\frac {a^6}{x}+6 b a^5+15 b^2 x a^4+20 b^3 x^2 a^3+15 b^4 x^3 a^2+6 b^5 x^4 a+b^6 x^5\right )dx+\frac {B (a+b x)^7}{7 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle A \left (a^6 \log (x)+6 a^5 b x+\frac {15}{2} a^4 b^2 x^2+\frac {20}{3} a^3 b^3 x^3+\frac {15}{4} a^2 b^4 x^4+\frac {6}{5} a b^5 x^5+\frac {b^6 x^6}{6}\right )+\frac {B (a+b x)^7}{7 b}\) |
Input:
Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x,x]
Output:
(B*(a + b*x)^7)/(7*b) + A*(6*a^5*b*x + (15*a^4*b^2*x^2)/2 + (20*a^3*b^3*x^ 3)/3 + (15*a^2*b^4*x^4)/4 + (6*a*b^5*x^5)/5 + (b^6*x^6)/6 + a^6*Log[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_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 1.00 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.44
method | result | size |
norman | \(\left (\frac {1}{6} A \,b^{6}+B a \,b^{5}\right ) x^{6}+\left (\frac {6}{5} A a \,b^{5}+3 B \,a^{2} b^{4}\right ) x^{5}+\left (\frac {15}{4} A \,a^{2} b^{4}+5 B \,a^{3} b^{3}\right ) x^{4}+\left (\frac {20}{3} A \,a^{3} b^{3}+5 B \,a^{4} b^{2}\right ) x^{3}+\left (\frac {15}{2} A \,a^{4} b^{2}+3 B \,a^{5} b \right ) x^{2}+\left (6 A \,a^{5} b +B \,a^{6}\right ) x +\frac {b^{6} B \,x^{7}}{7}+a^{6} A \ln \left (x \right )\) | \(138\) |
default | \(\frac {b^{6} B \,x^{7}}{7}+\frac {A \,b^{6} x^{6}}{6}+B a \,b^{5} x^{6}+\frac {6 A a \,b^{5} x^{5}}{5}+3 B \,a^{2} b^{4} x^{5}+\frac {15 A \,a^{2} b^{4} x^{4}}{4}+5 B \,a^{3} b^{3} x^{4}+\frac {20 A \,a^{3} b^{3} x^{3}}{3}+5 B \,a^{4} b^{2} x^{3}+\frac {15 A \,a^{4} b^{2} x^{2}}{2}+3 B \,a^{5} b \,x^{2}+6 A \,a^{5} b x +B \,a^{6} x +a^{6} A \ln \left (x \right )\) | \(142\) |
risch | \(\frac {b^{6} B \,x^{7}}{7}+\frac {A \,b^{6} x^{6}}{6}+B a \,b^{5} x^{6}+\frac {6 A a \,b^{5} x^{5}}{5}+3 B \,a^{2} b^{4} x^{5}+\frac {15 A \,a^{2} b^{4} x^{4}}{4}+5 B \,a^{3} b^{3} x^{4}+\frac {20 A \,a^{3} b^{3} x^{3}}{3}+5 B \,a^{4} b^{2} x^{3}+\frac {15 A \,a^{4} b^{2} x^{2}}{2}+3 B \,a^{5} b \,x^{2}+6 A \,a^{5} b x +B \,a^{6} x +a^{6} A \ln \left (x \right )\) | \(142\) |
parallelrisch | \(\frac {b^{6} B \,x^{7}}{7}+\frac {A \,b^{6} x^{6}}{6}+B a \,b^{5} x^{6}+\frac {6 A a \,b^{5} x^{5}}{5}+3 B \,a^{2} b^{4} x^{5}+\frac {15 A \,a^{2} b^{4} x^{4}}{4}+5 B \,a^{3} b^{3} x^{4}+\frac {20 A \,a^{3} b^{3} x^{3}}{3}+5 B \,a^{4} b^{2} x^{3}+\frac {15 A \,a^{4} b^{2} x^{2}}{2}+3 B \,a^{5} b \,x^{2}+6 A \,a^{5} b x +B \,a^{6} x +a^{6} A \ln \left (x \right )\) | \(142\) |
Input:
int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x,x,method=_RETURNVERBOSE)
Output:
(1/6*A*b^6+B*a*b^5)*x^6+(6/5*A*a*b^5+3*B*a^2*b^4)*x^5+(15/4*A*a^2*b^4+5*B* a^3*b^3)*x^4+(20/3*A*a^3*b^3+5*B*a^4*b^2)*x^3+(15/2*A*a^4*b^2+3*B*a^5*b)*x ^2+(6*A*a^5*b+B*a^6)*x+1/7*b^6*B*x^7+a^6*A*ln(x)
Time = 0.07 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.48 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x} \, dx=\frac {1}{7} \, B b^{6} x^{7} + A a^{6} \log \left (x\right ) + \frac {1}{6} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + \frac {3}{5} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + \frac {5}{4} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + \frac {5}{3} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + \frac {3}{2} \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + {\left (B a^{6} + 6 \, A a^{5} b\right )} x \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x,x, algorithm="fricas")
Output:
1/7*B*b^6*x^7 + A*a^6*log(x) + 1/6*(6*B*a*b^5 + A*b^6)*x^6 + 3/5*(5*B*a^2* b^4 + 2*A*a*b^5)*x^5 + 5/4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 5/3*(3*B*a^4* b^2 + 4*A*a^3*b^3)*x^3 + 3/2*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + (B*a^6 + 6*A* a^5*b)*x
Time = 0.13 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.54 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x} \, dx=A a^{6} \log {\left (x \right )} + \frac {B b^{6} x^{7}}{7} + x^{6} \left (\frac {A b^{6}}{6} + B a b^{5}\right ) + x^{5} \cdot \left (\frac {6 A a b^{5}}{5} + 3 B a^{2} b^{4}\right ) + x^{4} \cdot \left (\frac {15 A a^{2} b^{4}}{4} + 5 B a^{3} b^{3}\right ) + x^{3} \cdot \left (\frac {20 A a^{3} b^{3}}{3} + 5 B a^{4} b^{2}\right ) + x^{2} \cdot \left (\frac {15 A a^{4} b^{2}}{2} + 3 B a^{5} b\right ) + x \left (6 A a^{5} b + B a^{6}\right ) \] Input:
integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3/x,x)
Output:
A*a**6*log(x) + B*b**6*x**7/7 + x**6*(A*b**6/6 + B*a*b**5) + x**5*(6*A*a*b **5/5 + 3*B*a**2*b**4) + x**4*(15*A*a**2*b**4/4 + 5*B*a**3*b**3) + x**3*(2 0*A*a**3*b**3/3 + 5*B*a**4*b**2) + x**2*(15*A*a**4*b**2/2 + 3*B*a**5*b) + x*(6*A*a**5*b + B*a**6)
Time = 0.04 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.48 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x} \, dx=\frac {1}{7} \, B b^{6} x^{7} + A a^{6} \log \left (x\right ) + \frac {1}{6} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + \frac {3}{5} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + \frac {5}{4} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + \frac {5}{3} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + \frac {3}{2} \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + {\left (B a^{6} + 6 \, A a^{5} b\right )} x \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x,x, algorithm="maxima")
Output:
1/7*B*b^6*x^7 + A*a^6*log(x) + 1/6*(6*B*a*b^5 + A*b^6)*x^6 + 3/5*(5*B*a^2* b^4 + 2*A*a*b^5)*x^5 + 5/4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 5/3*(3*B*a^4* b^2 + 4*A*a^3*b^3)*x^3 + 3/2*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + (B*a^6 + 6*A* a^5*b)*x
Time = 0.21 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.48 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x} \, dx=\frac {1}{7} \, B b^{6} x^{7} + B a b^{5} x^{6} + \frac {1}{6} \, A b^{6} x^{6} + 3 \, B a^{2} b^{4} x^{5} + \frac {6}{5} \, A a b^{5} x^{5} + 5 \, B a^{3} b^{3} x^{4} + \frac {15}{4} \, A a^{2} b^{4} x^{4} + 5 \, B a^{4} b^{2} x^{3} + \frac {20}{3} \, A a^{3} b^{3} x^{3} + 3 \, B a^{5} b x^{2} + \frac {15}{2} \, A a^{4} b^{2} x^{2} + B a^{6} x + 6 \, A a^{5} b x + A a^{6} \log \left ({\left | x \right |}\right ) \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x,x, algorithm="giac")
Output:
1/7*B*b^6*x^7 + B*a*b^5*x^6 + 1/6*A*b^6*x^6 + 3*B*a^2*b^4*x^5 + 6/5*A*a*b^ 5*x^5 + 5*B*a^3*b^3*x^4 + 15/4*A*a^2*b^4*x^4 + 5*B*a^4*b^2*x^3 + 20/3*A*a^ 3*b^3*x^3 + 3*B*a^5*b*x^2 + 15/2*A*a^4*b^2*x^2 + B*a^6*x + 6*A*a^5*b*x + A *a^6*log(abs(x))
Time = 0.06 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.30 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x} \, dx=x\,\left (B\,a^6+6\,A\,b\,a^5\right )+x^6\,\left (\frac {A\,b^6}{6}+B\,a\,b^5\right )+\frac {B\,b^6\,x^7}{7}+A\,a^6\,\ln \left (x\right )+\frac {5\,a^3\,b^2\,x^3\,\left (4\,A\,b+3\,B\,a\right )}{3}+\frac {5\,a^2\,b^3\,x^4\,\left (3\,A\,b+4\,B\,a\right )}{4}+\frac {3\,a^4\,b\,x^2\,\left (5\,A\,b+2\,B\,a\right )}{2}+\frac {3\,a\,b^4\,x^5\,\left (2\,A\,b+5\,B\,a\right )}{5} \] Input:
int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^3)/x,x)
Output:
x*(B*a^6 + 6*A*a^5*b) + x^6*((A*b^6)/6 + B*a*b^5) + (B*b^6*x^7)/7 + A*a^6* log(x) + (5*a^3*b^2*x^3*(4*A*b + 3*B*a))/3 + (5*a^2*b^3*x^4*(3*A*b + 4*B*a ))/4 + (3*a^4*b*x^2*(5*A*b + 2*B*a))/2 + (3*a*b^4*x^5*(2*A*b + 5*B*a))/5
Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.78 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x} \, dx=\mathrm {log}\left (x \right ) a^{7}+7 a^{6} b x +\frac {21 a^{5} b^{2} x^{2}}{2}+\frac {35 a^{4} b^{3} x^{3}}{3}+\frac {35 a^{3} b^{4} x^{4}}{4}+\frac {21 a^{2} b^{5} x^{5}}{5}+\frac {7 a \,b^{6} x^{6}}{6}+\frac {b^{7} x^{7}}{7} \] Input:
int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x,x)
Output:
(420*log(x)*a**7 + 2940*a**6*b*x + 4410*a**5*b**2*x**2 + 4900*a**4*b**3*x* *3 + 3675*a**3*b**4*x**4 + 1764*a**2*b**5*x**5 + 490*a*b**6*x**6 + 60*b**7 *x**7)/420