Integrand size = 27, antiderivative size = 72 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{10}} \, dx=-\frac {A (a+b x)^7}{9 a x^9}+\frac {(2 A b-9 a B) (a+b x)^7}{72 a^2 x^8}-\frac {b (2 A b-9 a B) (a+b x)^7}{504 a^3 x^7} \] Output:
-1/9*A*(b*x+a)^7/a/x^9+1/72*(2*A*b-9*B*a)*(b*x+a)^7/a^2/x^8-1/504*b*(2*A*b -9*B*a)*(b*x+a)^7/a^3/x^7
Time = 0.04 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.75 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{10}} \, dx=-\frac {84 b^6 x^6 (2 A+3 B x)+252 a b^5 x^5 (3 A+4 B x)+378 a^2 b^4 x^4 (4 A+5 B x)+336 a^3 b^3 x^3 (5 A+6 B x)+180 a^4 b^2 x^2 (6 A+7 B x)+54 a^5 b x (7 A+8 B x)+7 a^6 (8 A+9 B x)}{504 x^9} \] Input:
Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^10,x]
Output:
-1/504*(84*b^6*x^6*(2*A + 3*B*x) + 252*a*b^5*x^5*(3*A + 4*B*x) + 378*a^2*b ^4*x^4*(4*A + 5*B*x) + 336*a^3*b^3*x^3*(5*A + 6*B*x) + 180*a^4*b^2*x^2*(6* A + 7*B*x) + 54*a^5*b*x*(7*A + 8*B*x) + 7*a^6*(8*A + 9*B*x))/x^9
Time = 0.32 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1184, 27, 87, 55, 48}
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.
\(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3 (A+B x)}{x^{10}} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \frac {\int \frac {b^6 (a+b x)^6 (A+B x)}{x^{10}}dx}{b^6}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {(a+b x)^6 (A+B x)}{x^{10}}dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {(2 A b-9 a B) \int \frac {(a+b x)^6}{x^9}dx}{9 a}-\frac {A (a+b x)^7}{9 a x^9}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {(2 A b-9 a B) \left (-\frac {b \int \frac {(a+b x)^6}{x^8}dx}{8 a}-\frac {(a+b x)^7}{8 a x^8}\right )}{9 a}-\frac {A (a+b x)^7}{9 a x^9}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {\left (\frac {b (a+b x)^7}{56 a^2 x^7}-\frac {(a+b x)^7}{8 a x^8}\right ) (2 A b-9 a B)}{9 a}-\frac {A (a+b x)^7}{9 a x^9}\) |
Input:
Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^10,x]
Output:
-1/9*(A*(a + b*x)^7)/(a*x^9) - ((2*A*b - 9*a*B)*(-1/8*(a + b*x)^7/(a*x^8) + (b*(a + b*x)^7)/(56*a^2*x^7)))/(9*a)
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] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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 = 0.89 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.78
method | result | size |
default | \(-\frac {a^{2} b^{3} \left (3 A b +4 B a \right )}{x^{5}}-\frac {5 a^{3} b^{2} \left (4 A b +3 B a \right )}{6 x^{6}}-\frac {b^{5} \left (A b +6 B a \right )}{3 x^{3}}-\frac {B \,b^{6}}{2 x^{2}}-\frac {a^{5} \left (6 A b +B a \right )}{8 x^{8}}-\frac {3 a \,b^{4} \left (2 A b +5 B a \right )}{4 x^{4}}-\frac {A \,a^{6}}{9 x^{9}}-\frac {3 a^{4} b \left (5 A b +2 B a \right )}{7 x^{7}}\) | \(128\) |
norman | \(\frac {-\frac {b^{6} B \,x^{7}}{2}+\left (-\frac {1}{3} A \,b^{6}-2 B a \,b^{5}\right ) x^{6}+\left (-\frac {3}{2} A a \,b^{5}-\frac {15}{4} B \,a^{2} b^{4}\right ) x^{5}+\left (-3 A \,a^{2} b^{4}-4 B \,a^{3} b^{3}\right ) x^{4}+\left (-\frac {10}{3} A \,a^{3} b^{3}-\frac {5}{2} B \,a^{4} b^{2}\right ) x^{3}+\left (-\frac {15}{7} A \,a^{4} b^{2}-\frac {6}{7} B \,a^{5} b \right ) x^{2}+\left (-\frac {3}{4} A \,a^{5} b -\frac {1}{8} B \,a^{6}\right ) x -\frac {A \,a^{6}}{9}}{x^{9}}\) | \(143\) |
risch | \(\frac {-\frac {b^{6} B \,x^{7}}{2}+\left (-\frac {1}{3} A \,b^{6}-2 B a \,b^{5}\right ) x^{6}+\left (-\frac {3}{2} A a \,b^{5}-\frac {15}{4} B \,a^{2} b^{4}\right ) x^{5}+\left (-3 A \,a^{2} b^{4}-4 B \,a^{3} b^{3}\right ) x^{4}+\left (-\frac {10}{3} A \,a^{3} b^{3}-\frac {5}{2} B \,a^{4} b^{2}\right ) x^{3}+\left (-\frac {15}{7} A \,a^{4} b^{2}-\frac {6}{7} B \,a^{5} b \right ) x^{2}+\left (-\frac {3}{4} A \,a^{5} b -\frac {1}{8} B \,a^{6}\right ) x -\frac {A \,a^{6}}{9}}{x^{9}}\) | \(143\) |
gosper | \(-\frac {252 b^{6} B \,x^{7}+168 A \,b^{6} x^{6}+1008 B a \,b^{5} x^{6}+756 A a \,b^{5} x^{5}+1890 B \,a^{2} b^{4} x^{5}+1512 A \,a^{2} b^{4} x^{4}+2016 B \,a^{3} b^{3} x^{4}+1680 A \,a^{3} b^{3} x^{3}+1260 B \,a^{4} b^{2} x^{3}+1080 A \,a^{4} b^{2} x^{2}+432 B \,a^{5} b \,x^{2}+378 A \,a^{5} b x +63 B \,a^{6} x +56 A \,a^{6}}{504 x^{9}}\) | \(148\) |
parallelrisch | \(-\frac {252 b^{6} B \,x^{7}+168 A \,b^{6} x^{6}+1008 B a \,b^{5} x^{6}+756 A a \,b^{5} x^{5}+1890 B \,a^{2} b^{4} x^{5}+1512 A \,a^{2} b^{4} x^{4}+2016 B \,a^{3} b^{3} x^{4}+1680 A \,a^{3} b^{3} x^{3}+1260 B \,a^{4} b^{2} x^{3}+1080 A \,a^{4} b^{2} x^{2}+432 B \,a^{5} b \,x^{2}+378 A \,a^{5} b x +63 B \,a^{6} x +56 A \,a^{6}}{504 x^{9}}\) | \(148\) |
orering | \(-\frac {\left (252 b^{6} B \,x^{7}+168 A \,b^{6} x^{6}+1008 B a \,b^{5} x^{6}+756 A a \,b^{5} x^{5}+1890 B \,a^{2} b^{4} x^{5}+1512 A \,a^{2} b^{4} x^{4}+2016 B \,a^{3} b^{3} x^{4}+1680 A \,a^{3} b^{3} x^{3}+1260 B \,a^{4} b^{2} x^{3}+1080 A \,a^{4} b^{2} x^{2}+432 B \,a^{5} b \,x^{2}+378 A \,a^{5} b x +63 B \,a^{6} x +56 A \,a^{6}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{3}}{504 x^{9} \left (b x +a \right )^{6}}\) | \(173\) |
Input:
int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^10,x,method=_RETURNVERBOSE)
Output:
-a^2*b^3*(3*A*b+4*B*a)/x^5-5/6*a^3*b^2*(4*A*b+3*B*a)/x^6-1/3*b^5*(A*b+6*B* a)/x^3-1/2*B*b^6/x^2-1/8*a^5*(6*A*b+B*a)/x^8-3/4*a*b^4*(2*A*b+5*B*a)/x^4-1 /9*A*a^6/x^9-3/7*a^4*b*(5*A*b+2*B*a)/x^7
Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (66) = 132\).
Time = 0.08 (sec) , antiderivative size = 147, normalized size of antiderivative = 2.04 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{10}} \, dx=-\frac {252 \, B b^{6} x^{7} + 56 \, A a^{6} + 168 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 378 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 504 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 420 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 216 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 63 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{504 \, x^{9}} \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^10,x, algorithm="fricas")
Output:
-1/504*(252*B*b^6*x^7 + 56*A*a^6 + 168*(6*B*a*b^5 + A*b^6)*x^6 + 378*(5*B* a^2*b^4 + 2*A*a*b^5)*x^5 + 504*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 420*(3*B* a^4*b^2 + 4*A*a^3*b^3)*x^3 + 216*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 63*(B*a^6 + 6*A*a^5*b)*x)/x^9
Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (65) = 130\).
Time = 3.91 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.19 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{10}} \, dx=\frac {- 56 A a^{6} - 252 B b^{6} x^{7} + x^{6} \left (- 168 A b^{6} - 1008 B a b^{5}\right ) + x^{5} \left (- 756 A a b^{5} - 1890 B a^{2} b^{4}\right ) + x^{4} \left (- 1512 A a^{2} b^{4} - 2016 B a^{3} b^{3}\right ) + x^{3} \left (- 1680 A a^{3} b^{3} - 1260 B a^{4} b^{2}\right ) + x^{2} \left (- 1080 A a^{4} b^{2} - 432 B a^{5} b\right ) + x \left (- 378 A a^{5} b - 63 B a^{6}\right )}{504 x^{9}} \] Input:
integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3/x**10,x)
Output:
(-56*A*a**6 - 252*B*b**6*x**7 + x**6*(-168*A*b**6 - 1008*B*a*b**5) + x**5* (-756*A*a*b**5 - 1890*B*a**2*b**4) + x**4*(-1512*A*a**2*b**4 - 2016*B*a**3 *b**3) + x**3*(-1680*A*a**3*b**3 - 1260*B*a**4*b**2) + x**2*(-1080*A*a**4* b**2 - 432*B*a**5*b) + x*(-378*A*a**5*b - 63*B*a**6))/(504*x**9)
Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (66) = 132\).
Time = 0.03 (sec) , antiderivative size = 147, normalized size of antiderivative = 2.04 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{10}} \, dx=-\frac {252 \, B b^{6} x^{7} + 56 \, A a^{6} + 168 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 378 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 504 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 420 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 216 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 63 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{504 \, x^{9}} \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^10,x, algorithm="maxima")
Output:
-1/504*(252*B*b^6*x^7 + 56*A*a^6 + 168*(6*B*a*b^5 + A*b^6)*x^6 + 378*(5*B* a^2*b^4 + 2*A*a*b^5)*x^5 + 504*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 420*(3*B* a^4*b^2 + 4*A*a^3*b^3)*x^3 + 216*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 63*(B*a^6 + 6*A*a^5*b)*x)/x^9
Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (66) = 132\).
Time = 0.18 (sec) , antiderivative size = 147, normalized size of antiderivative = 2.04 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{10}} \, dx=-\frac {252 \, B b^{6} x^{7} + 1008 \, B a b^{5} x^{6} + 168 \, A b^{6} x^{6} + 1890 \, B a^{2} b^{4} x^{5} + 756 \, A a b^{5} x^{5} + 2016 \, B a^{3} b^{3} x^{4} + 1512 \, A a^{2} b^{4} x^{4} + 1260 \, B a^{4} b^{2} x^{3} + 1680 \, A a^{3} b^{3} x^{3} + 432 \, B a^{5} b x^{2} + 1080 \, A a^{4} b^{2} x^{2} + 63 \, B a^{6} x + 378 \, A a^{5} b x + 56 \, A a^{6}}{504 \, x^{9}} \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^10,x, algorithm="giac")
Output:
-1/504*(252*B*b^6*x^7 + 1008*B*a*b^5*x^6 + 168*A*b^6*x^6 + 1890*B*a^2*b^4* x^5 + 756*A*a*b^5*x^5 + 2016*B*a^3*b^3*x^4 + 1512*A*a^2*b^4*x^4 + 1260*B*a ^4*b^2*x^3 + 1680*A*a^3*b^3*x^3 + 432*B*a^5*b*x^2 + 1080*A*a^4*b^2*x^2 + 6 3*B*a^6*x + 378*A*a^5*b*x + 56*A*a^6)/x^9
Time = 0.07 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.99 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{10}} \, dx=-\frac {x\,\left (\frac {B\,a^6}{8}+\frac {3\,A\,b\,a^5}{4}\right )+\frac {A\,a^6}{9}+x^5\,\left (\frac {15\,B\,a^2\,b^4}{4}+\frac {3\,A\,a\,b^5}{2}\right )+x^2\,\left (\frac {6\,B\,a^5\,b}{7}+\frac {15\,A\,a^4\,b^2}{7}\right )+x^6\,\left (\frac {A\,b^6}{3}+2\,B\,a\,b^5\right )+x^4\,\left (4\,B\,a^3\,b^3+3\,A\,a^2\,b^4\right )+x^3\,\left (\frac {5\,B\,a^4\,b^2}{2}+\frac {10\,A\,a^3\,b^3}{3}\right )+\frac {B\,b^6\,x^7}{2}}{x^9} \] Input:
int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^3)/x^10,x)
Output:
-(x*((B*a^6)/8 + (3*A*a^5*b)/4) + (A*a^6)/9 + x^5*((15*B*a^2*b^4)/4 + (3*A *a*b^5)/2) + x^2*((15*A*a^4*b^2)/7 + (6*B*a^5*b)/7) + x^6*((A*b^6)/3 + 2*B *a*b^5) + x^4*(3*A*a^2*b^4 + 4*B*a^3*b^3) + x^3*((10*A*a^3*b^3)/3 + (5*B*a ^4*b^2)/2) + (B*b^6*x^7)/2)/x^9
Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.10 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{10}} \, dx=\frac {-36 b^{7} x^{7}-168 a \,b^{6} x^{6}-378 a^{2} b^{5} x^{5}-504 a^{3} b^{4} x^{4}-420 a^{4} b^{3} x^{3}-216 a^{5} b^{2} x^{2}-63 a^{6} b x -8 a^{7}}{72 x^{9}} \] Input:
int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^10,x)
Output:
( - 8*a**7 - 63*a**6*b*x - 216*a**5*b**2*x**2 - 420*a**4*b**3*x**3 - 504*a **3*b**4*x**4 - 378*a**2*b**5*x**5 - 168*a*b**6*x**6 - 36*b**7*x**7)/(72*x **9)