Integrand size = 22, antiderivative size = 84 \[ \int (a+b x)^m \left (a^2-b^2 x^2\right )^3 \, dx=\frac {8 a^3 (a+b x)^{4+m}}{b (4+m)}-\frac {12 a^2 (a+b x)^{5+m}}{b (5+m)}+\frac {6 a (a+b x)^{6+m}}{b (6+m)}-\frac {(a+b x)^{7+m}}{b (7+m)} \]
8*a^3*(b*x+a)^(4+m)/b/(4+m)-12*a^2*(b*x+a)^(5+m)/b/(5+m)+6*a*(b*x+a)^(6+m) /b/(6+m)-(b*x+a)^(7+m)/b/(7+m)
Time = 0.25 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.81 \[ \int (a+b x)^m \left (a^2-b^2 x^2\right )^3 \, dx=\frac {(a+b x)^{4+m} \left (\frac {8 a^3}{4+m}-\frac {12 a^2 (a+b x)}{5+m}+\frac {6 a (a+b x)^2}{6+m}-\frac {(a+b x)^3}{7+m}\right )}{b} \]
((a + b*x)^(4 + m)*((8*a^3)/(4 + m) - (12*a^2*(a + b*x))/(5 + m) + (6*a*(a + b*x)^2)/(6 + m) - (a + b*x)^3/(7 + m)))/b
Time = 0.23 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {456, 53, 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.
| \(\displaystyle \int \left (a^2-b^2 x^2\right )^3 (a+b x)^m \, dx\) |
| \(\Big \downarrow \) 456 |
| \(\displaystyle \int (a-b x)^3 (a+b x)^{m+3}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (8 a^3 (a+b x)^{m+3}-12 a^2 (a+b x)^{m+4}+6 a (a+b x)^{m+5}-(a+b x)^{m+6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {8 a^3 (a+b x)^{m+4}}{b (m+4)}-\frac {12 a^2 (a+b x)^{m+5}}{b (m+5)}+\frac {6 a (a+b x)^{m+6}}{b (m+6)}-\frac {(a+b x)^{m+7}}{b (m+7)}\) |
(8*a^3*(a + b*x)^(4 + m))/(b*(4 + m)) - (12*a^2*(a + b*x)^(5 + m))/(b*(5 + m)) + (6*a*(a + b*x)^(6 + m))/(b*(6 + m)) - (a + b*x)^(7 + m)/(b*(7 + m))
3.10.45.3.1 Defintions of rubi rules used
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, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ (c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && !Integ erQ[n]))
Leaf count of result is larger than twice the leaf count of optimal. \(177\) vs. \(2(84)=168\).
Time = 2.36 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.12
| method | result | size |
| gosper | \(\frac {\left (b x +a \right )^{4+m} \left (-b^{3} m^{3} x^{3}+3 a \,b^{2} m^{3} x^{2}-15 b^{3} m^{2} x^{3}-3 a^{2} b \,m^{3} x +51 a \,b^{2} m^{2} x^{2}-74 b^{3} m \,x^{3}+a^{3} m^{3}-57 a^{2} b \,m^{2} x +276 a \,b^{2} m \,x^{2}-120 b^{3} x^{3}+21 a^{3} m^{2}-354 a^{2} b m x +480 a \,b^{2} x^{2}+152 a^{3} m -696 a^{2} b x +384 a^{3}\right )}{b \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}\) | \(178\) |
| norman | \(\frac {a^{6} \left (m^{3}+27 m^{2}+254 m +840\right ) x \,{\mathrm e}^{m \ln \left (b x +a \right )}}{m^{4}+22 m^{3}+179 m^{2}+638 m +840}+\frac {a^{7} \left (m^{3}+21 m^{2}+152 m +384\right ) {\mathrm e}^{m \ln \left (b x +a \right )}}{b \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}-\frac {b^{6} x^{7} {\mathrm e}^{m \ln \left (b x +a \right )}}{7+m}-\frac {a m \,b^{5} x^{6} {\mathrm e}^{m \ln \left (b x +a \right )}}{m^{2}+13 m +42}-\frac {3 a^{4} b^{2} \left (m^{3}+23 m^{2}+162 m +280\right ) x^{3} {\mathrm e}^{m \ln \left (b x +a \right )}}{m^{4}+22 m^{3}+179 m^{2}+638 m +840}+\frac {3 \left (m^{2}+15 m +42\right ) a^{2} b^{4} x^{5} {\mathrm e}^{m \ln \left (b x +a \right )}}{m^{3}+18 m^{2}+107 m +210}+\frac {3 a^{3} b^{3} m \left (m^{2}+13 m +32\right ) x^{4} {\mathrm e}^{m \ln \left (b x +a \right )}}{m^{4}+22 m^{3}+179 m^{2}+638 m +840}-\frac {3 b m \,a^{5} \left (m^{2}+17 m +76\right ) x^{2} {\mathrm e}^{m \ln \left (b x +a \right )}}{m^{4}+22 m^{3}+179 m^{2}+638 m +840}\) | \(341\) |
| risch | \(\frac {\left (-b^{7} m^{3} x^{7}-a \,b^{6} m^{3} x^{6}-15 b^{7} m^{2} x^{7}+3 a^{2} b^{5} m^{3} x^{5}-9 m^{2} b^{6} a \,x^{6}-74 m \,b^{7} x^{7}+3 a^{3} b^{4} m^{3} x^{4}+57 a^{2} b^{5} m^{2} x^{5}-20 a \,b^{6} m \,x^{6}-120 b^{7} x^{7}-3 a^{4} b^{3} m^{3} x^{3}+39 a^{3} b^{4} m^{2} x^{4}+306 a^{2} b^{5} m \,x^{5}-3 a^{5} b^{2} m^{3} x^{2}-69 a^{4} b^{3} m^{2} x^{3}+96 a^{3} b^{4} m \,x^{4}+504 a^{2} b^{5} x^{5}+a^{6} b \,m^{3} x -51 a^{5} b^{2} m^{2} x^{2}-486 a^{4} b^{3} m \,x^{3}+a^{7} m^{3}+27 a^{6} b \,m^{2} x -228 a^{5} b^{2} m \,x^{2}-840 a^{4} b^{3} x^{3}+21 a^{7} m^{2}+254 a^{6} b m x +152 a^{7} m +840 a^{6} b x +384 a^{7}\right ) \left (b x +a \right )^{m}}{\left (6+m \right ) \left (7+m \right ) \left (5+m \right ) b \left (4+m \right )}\) | \(348\) |
| parallelrisch | \(-\frac {120 x^{7} \left (b x +a \right )^{m} a \,b^{7}-504 x^{5} \left (b x +a \right )^{m} a^{3} b^{5}+840 x^{3} \left (b x +a \right )^{m} a^{5} b^{3}-840 x \left (b x +a \right )^{m} a^{7} b +x^{7} \left (b x +a \right )^{m} a \,b^{7} m^{3}+15 x^{7} \left (b x +a \right )^{m} a \,b^{7} m^{2}+x^{6} \left (b x +a \right )^{m} a^{2} b^{6} m^{3}+74 x^{7} \left (b x +a \right )^{m} a \,b^{7} m +9 x^{6} \left (b x +a \right )^{m} a^{2} b^{6} m^{2}-3 x^{5} \left (b x +a \right )^{m} a^{3} b^{5} m^{3}+20 x^{6} \left (b x +a \right )^{m} a^{2} b^{6} m -57 x^{5} \left (b x +a \right )^{m} a^{3} b^{5} m^{2}-3 x^{4} \left (b x +a \right )^{m} a^{4} b^{4} m^{3}-306 x^{5} \left (b x +a \right )^{m} a^{3} b^{5} m -39 x^{4} \left (b x +a \right )^{m} a^{4} b^{4} m^{2}+3 x^{3} \left (b x +a \right )^{m} a^{5} b^{3} m^{3}-96 x^{4} \left (b x +a \right )^{m} a^{4} b^{4} m +69 x^{3} \left (b x +a \right )^{m} a^{5} b^{3} m^{2}+3 x^{2} \left (b x +a \right )^{m} a^{6} b^{2} m^{3}+486 x^{3} \left (b x +a \right )^{m} a^{5} b^{3} m +51 x^{2} \left (b x +a \right )^{m} a^{6} b^{2} m^{2}-384 \left (b x +a \right )^{m} a^{8}-\left (b x +a \right )^{m} a^{8} m^{3}-21 \left (b x +a \right )^{m} a^{8} m^{2}-152 \left (b x +a \right )^{m} a^{8} m -x \left (b x +a \right )^{m} a^{7} b \,m^{3}+228 x^{2} \left (b x +a \right )^{m} a^{6} b^{2} m -27 x \left (b x +a \right )^{m} a^{7} b \,m^{2}-254 x \left (b x +a \right )^{m} a^{7} b m}{\left (m^{3}+18 m^{2}+107 m +210\right ) a b \left (4+m \right )}\) | \(558\) |
1/b*(b*x+a)^(4+m)/(m^4+22*m^3+179*m^2+638*m+840)*(-b^3*m^3*x^3+3*a*b^2*m^3 *x^2-15*b^3*m^2*x^3-3*a^2*b*m^3*x+51*a*b^2*m^2*x^2-74*b^3*m*x^3+a^3*m^3-57 *a^2*b*m^2*x+276*a*b^2*m*x^2-120*b^3*x^3+21*a^3*m^2-354*a^2*b*m*x+480*a*b^ 2*x^2+152*a^3*m-696*a^2*b*x+384*a^3)
Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (84) = 168\).
Time = 0.30 (sec) , antiderivative size = 317, normalized size of antiderivative = 3.77 \[ \int (a+b x)^m \left (a^2-b^2 x^2\right )^3 \, dx=\frac {{\left (a^{7} m^{3} + 21 \, a^{7} m^{2} + 152 \, a^{7} m - {\left (b^{7} m^{3} + 15 \, b^{7} m^{2} + 74 \, b^{7} m + 120 \, b^{7}\right )} x^{7} + 384 \, a^{7} - {\left (a b^{6} m^{3} + 9 \, a b^{6} m^{2} + 20 \, a b^{6} m\right )} x^{6} + 3 \, {\left (a^{2} b^{5} m^{3} + 19 \, a^{2} b^{5} m^{2} + 102 \, a^{2} b^{5} m + 168 \, a^{2} b^{5}\right )} x^{5} + 3 \, {\left (a^{3} b^{4} m^{3} + 13 \, a^{3} b^{4} m^{2} + 32 \, a^{3} b^{4} m\right )} x^{4} - 3 \, {\left (a^{4} b^{3} m^{3} + 23 \, a^{4} b^{3} m^{2} + 162 \, a^{4} b^{3} m + 280 \, a^{4} b^{3}\right )} x^{3} - 3 \, {\left (a^{5} b^{2} m^{3} + 17 \, a^{5} b^{2} m^{2} + 76 \, a^{5} b^{2} m\right )} x^{2} + {\left (a^{6} b m^{3} + 27 \, a^{6} b m^{2} + 254 \, a^{6} b m + 840 \, a^{6} b\right )} x\right )} {\left (b x + a\right )}^{m}}{b m^{4} + 22 \, b m^{3} + 179 \, b m^{2} + 638 \, b m + 840 \, b} \]
(a^7*m^3 + 21*a^7*m^2 + 152*a^7*m - (b^7*m^3 + 15*b^7*m^2 + 74*b^7*m + 120 *b^7)*x^7 + 384*a^7 - (a*b^6*m^3 + 9*a*b^6*m^2 + 20*a*b^6*m)*x^6 + 3*(a^2* b^5*m^3 + 19*a^2*b^5*m^2 + 102*a^2*b^5*m + 168*a^2*b^5)*x^5 + 3*(a^3*b^4*m ^3 + 13*a^3*b^4*m^2 + 32*a^3*b^4*m)*x^4 - 3*(a^4*b^3*m^3 + 23*a^4*b^3*m^2 + 162*a^4*b^3*m + 280*a^4*b^3)*x^3 - 3*(a^5*b^2*m^3 + 17*a^5*b^2*m^2 + 76* a^5*b^2*m)*x^2 + (a^6*b*m^3 + 27*a^6*b*m^2 + 254*a^6*b*m + 840*a^6*b)*x)*( b*x + a)^m/(b*m^4 + 22*b*m^3 + 179*b*m^2 + 638*b*m + 840*b)
Leaf count of result is larger than twice the leaf count of optimal. 2059 vs. \(2 (66) = 132\).
Time = 1.30 (sec) , antiderivative size = 2059, normalized size of antiderivative = 24.51 \[ \int (a+b x)^m \left (a^2-b^2 x^2\right )^3 \, dx=\text {Too large to display} \]
Piecewise((a**6*a**m*x, Eq(b, 0)), (-3*a**3*log(a/b + x)/(3*a**3*b + 9*a** 2*b**2*x + 9*a*b**3*x**2 + 3*b**4*x**3) - 8*a**3/(3*a**3*b + 9*a**2*b**2*x + 9*a*b**3*x**2 + 3*b**4*x**3) - 9*a**2*b*x*log(a/b + x)/(3*a**3*b + 9*a* *2*b**2*x + 9*a*b**3*x**2 + 3*b**4*x**3) - 18*a**2*b*x/(3*a**3*b + 9*a**2* b**2*x + 9*a*b**3*x**2 + 3*b**4*x**3) - 9*a*b**2*x**2*log(a/b + x)/(3*a**3 *b + 9*a**2*b**2*x + 9*a*b**3*x**2 + 3*b**4*x**3) - 18*a*b**2*x**2/(3*a**3 *b + 9*a**2*b**2*x + 9*a*b**3*x**2 + 3*b**4*x**3) - 3*b**3*x**3*log(a/b + x)/(3*a**3*b + 9*a**2*b**2*x + 9*a*b**3*x**2 + 3*b**4*x**3), Eq(m, -7)), ( 6*a**3*log(a/b + x)/(a**2*b + 2*a*b**2*x + b**3*x**2) + 13*a**3/(a**2*b + 2*a*b**2*x + b**3*x**2) + 12*a**2*b*x*log(a/b + x)/(a**2*b + 2*a*b**2*x + b**3*x**2) + 21*a**2*b*x/(a**2*b + 2*a*b**2*x + b**3*x**2) + 6*a*b**2*x**2 *log(a/b + x)/(a**2*b + 2*a*b**2*x + b**3*x**2) + 3*a*b**2*x**2/(a**2*b + 2*a*b**2*x + b**3*x**2) - b**3*x**3/(a**2*b + 2*a*b**2*x + b**3*x**2), Eq( m, -6)), (-24*a**3*log(a/b + x)/(2*a*b + 2*b**2*x) - 50*a**3/(2*a*b + 2*b* *2*x) - 24*a**2*b*x*log(a/b + x)/(2*a*b + 2*b**2*x) - 24*a**2*b*x/(2*a*b + 2*b**2*x) + 9*a*b**2*x**2/(2*a*b + 2*b**2*x) - b**3*x**3/(2*a*b + 2*b**2* x), Eq(m, -5)), (8*a**3*log(a/b + x)/b - 7*a**2*x + 2*a*b*x**2 - b**2*x**3 /3, Eq(m, -4)), (a**7*m**3*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*b*m**2 + 638*b*m + 840*b) + 21*a**7*m**2*(a + b*x)**m/(b*m**4 + 22*b*m**3 + 179*b* m**2 + 638*b*m + 840*b) + 152*a**7*m*(a + b*x)**m/(b*m**4 + 22*b*m**3 +...
Leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (84) = 168\).
Time = 0.21 (sec) , antiderivative size = 468, normalized size of antiderivative = 5.57 \[ \int (a+b x)^m \left (a^2-b^2 x^2\right )^3 \, dx=\frac {{\left (b x + a\right )}^{m + 1} a^{6}}{b {\left (m + 1\right )}} - \frac {3 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} b^{3} x^{3} + {\left (m^{2} + m\right )} a b^{2} x^{2} - 2 \, a^{2} b m x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{m} a^{4}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b} + \frac {3 \, {\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} b^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a b^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{2} b^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b m x + 24 \, a^{5}\right )} {\left (b x + a\right )}^{m} a^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} b} - \frac {{\left ({\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} b^{7} x^{7} + {\left (m^{6} + 15 \, m^{5} + 85 \, m^{4} + 225 \, m^{3} + 274 \, m^{2} + 120 \, m\right )} a b^{6} x^{6} - 6 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} a^{2} b^{5} x^{5} + 30 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a^{3} b^{4} x^{4} - 120 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{4} b^{3} x^{3} + 360 \, {\left (m^{2} + m\right )} a^{5} b^{2} x^{2} - 720 \, a^{6} b m x + 720 \, a^{7}\right )} {\left (b x + a\right )}^{m}}{{\left (m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040\right )} b} \]
(b*x + a)^(m + 1)*a^6/(b*(m + 1)) - 3*((m^2 + 3*m + 2)*b^3*x^3 + (m^2 + m) *a*b^2*x^2 - 2*a^2*b*m*x + 2*a^3)*(b*x + a)^m*a^4/((m^3 + 6*m^2 + 11*m + 6 )*b) + 3*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*b^5*x^5 + (m^4 + 6*m^3 + 11* m^2 + 6*m)*a*b^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*a^2*b^3*x^3 + 12*(m^2 + m)*a^ 3*b^2*x^2 - 24*a^4*b*m*x + 24*a^5)*(b*x + a)^m*a^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*b) - ((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m ^2 + 1764*m + 720)*b^7*x^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2 + 120*m)*a*b^6*x^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*a^2*b^5*x^5 + 30*(m^4 + 6*m^3 + 11*m^2 + 6*m)*a^3*b^4*x^4 - 120*(m^3 + 3*m^2 + 2*m)*a^4 *b^3*x^3 + 360*(m^2 + m)*a^5*b^2*x^2 - 720*a^6*b*m*x + 720*a^7)*(b*x + a)^ m/((m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5 040)*b)
Leaf count of result is larger than twice the leaf count of optimal. 548 vs. \(2 (84) = 168\).
Time = 0.27 (sec) , antiderivative size = 548, normalized size of antiderivative = 6.52 \[ \int (a+b x)^m \left (a^2-b^2 x^2\right )^3 \, dx=-\frac {{\left (b x + a\right )}^{m} b^{7} m^{3} x^{7} + {\left (b x + a\right )}^{m} a b^{6} m^{3} x^{6} + 15 \, {\left (b x + a\right )}^{m} b^{7} m^{2} x^{7} - 3 \, {\left (b x + a\right )}^{m} a^{2} b^{5} m^{3} x^{5} + 9 \, {\left (b x + a\right )}^{m} a b^{6} m^{2} x^{6} + 74 \, {\left (b x + a\right )}^{m} b^{7} m x^{7} - 3 \, {\left (b x + a\right )}^{m} a^{3} b^{4} m^{3} x^{4} - 57 \, {\left (b x + a\right )}^{m} a^{2} b^{5} m^{2} x^{5} + 20 \, {\left (b x + a\right )}^{m} a b^{6} m x^{6} + 120 \, {\left (b x + a\right )}^{m} b^{7} x^{7} + 3 \, {\left (b x + a\right )}^{m} a^{4} b^{3} m^{3} x^{3} - 39 \, {\left (b x + a\right )}^{m} a^{3} b^{4} m^{2} x^{4} - 306 \, {\left (b x + a\right )}^{m} a^{2} b^{5} m x^{5} + 3 \, {\left (b x + a\right )}^{m} a^{5} b^{2} m^{3} x^{2} + 69 \, {\left (b x + a\right )}^{m} a^{4} b^{3} m^{2} x^{3} - 96 \, {\left (b x + a\right )}^{m} a^{3} b^{4} m x^{4} - 504 \, {\left (b x + a\right )}^{m} a^{2} b^{5} x^{5} - {\left (b x + a\right )}^{m} a^{6} b m^{3} x + 51 \, {\left (b x + a\right )}^{m} a^{5} b^{2} m^{2} x^{2} + 486 \, {\left (b x + a\right )}^{m} a^{4} b^{3} m x^{3} - {\left (b x + a\right )}^{m} a^{7} m^{3} - 27 \, {\left (b x + a\right )}^{m} a^{6} b m^{2} x + 228 \, {\left (b x + a\right )}^{m} a^{5} b^{2} m x^{2} + 840 \, {\left (b x + a\right )}^{m} a^{4} b^{3} x^{3} - 21 \, {\left (b x + a\right )}^{m} a^{7} m^{2} - 254 \, {\left (b x + a\right )}^{m} a^{6} b m x - 152 \, {\left (b x + a\right )}^{m} a^{7} m - 840 \, {\left (b x + a\right )}^{m} a^{6} b x - 384 \, {\left (b x + a\right )}^{m} a^{7}}{b m^{4} + 22 \, b m^{3} + 179 \, b m^{2} + 638 \, b m + 840 \, b} \]
-((b*x + a)^m*b^7*m^3*x^7 + (b*x + a)^m*a*b^6*m^3*x^6 + 15*(b*x + a)^m*b^7 *m^2*x^7 - 3*(b*x + a)^m*a^2*b^5*m^3*x^5 + 9*(b*x + a)^m*a*b^6*m^2*x^6 + 7 4*(b*x + a)^m*b^7*m*x^7 - 3*(b*x + a)^m*a^3*b^4*m^3*x^4 - 57*(b*x + a)^m*a ^2*b^5*m^2*x^5 + 20*(b*x + a)^m*a*b^6*m*x^6 + 120*(b*x + a)^m*b^7*x^7 + 3* (b*x + a)^m*a^4*b^3*m^3*x^3 - 39*(b*x + a)^m*a^3*b^4*m^2*x^4 - 306*(b*x + a)^m*a^2*b^5*m*x^5 + 3*(b*x + a)^m*a^5*b^2*m^3*x^2 + 69*(b*x + a)^m*a^4*b^ 3*m^2*x^3 - 96*(b*x + a)^m*a^3*b^4*m*x^4 - 504*(b*x + a)^m*a^2*b^5*x^5 - ( b*x + a)^m*a^6*b*m^3*x + 51*(b*x + a)^m*a^5*b^2*m^2*x^2 + 486*(b*x + a)^m* a^4*b^3*m*x^3 - (b*x + a)^m*a^7*m^3 - 27*(b*x + a)^m*a^6*b*m^2*x + 228*(b* x + a)^m*a^5*b^2*m*x^2 + 840*(b*x + a)^m*a^4*b^3*x^3 - 21*(b*x + a)^m*a^7* m^2 - 254*(b*x + a)^m*a^6*b*m*x - 152*(b*x + a)^m*a^7*m - 840*(b*x + a)^m* a^6*b*x - 384*(b*x + a)^m*a^7)/(b*m^4 + 22*b*m^3 + 179*b*m^2 + 638*b*m + 8 40*b)
Time = 10.57 (sec) , antiderivative size = 332, normalized size of antiderivative = 3.95 \[ \int (a+b x)^m \left (a^2-b^2 x^2\right )^3 \, dx={\left (a+b\,x\right )}^m\,\left (\frac {a^6\,x\,\left (m^3+27\,m^2+254\,m+840\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}+\frac {a^7\,\left (m^3+21\,m^2+152\,m+384\right )}{b\,\left (m^4+22\,m^3+179\,m^2+638\,m+840\right )}-\frac {b^6\,x^7\,\left (m^3+15\,m^2+74\,m+120\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}+\frac {3\,a^2\,b^4\,x^5\,\left (m^3+19\,m^2+102\,m+168\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}-\frac {3\,a^4\,b^2\,x^3\,\left (m^3+23\,m^2+162\,m+280\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}-\frac {a\,b^5\,m\,x^6\,\left (m^2+9\,m+20\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}-\frac {3\,a^5\,b\,m\,x^2\,\left (m^2+17\,m+76\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}+\frac {3\,a^3\,b^3\,m\,x^4\,\left (m^2+13\,m+32\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}\right ) \]
(a + b*x)^m*((a^6*x*(254*m + 27*m^2 + m^3 + 840))/(638*m + 179*m^2 + 22*m^ 3 + m^4 + 840) + (a^7*(152*m + 21*m^2 + m^3 + 384))/(b*(638*m + 179*m^2 + 22*m^3 + m^4 + 840)) - (b^6*x^7*(74*m + 15*m^2 + m^3 + 120))/(638*m + 179* m^2 + 22*m^3 + m^4 + 840) + (3*a^2*b^4*x^5*(102*m + 19*m^2 + m^3 + 168))/( 638*m + 179*m^2 + 22*m^3 + m^4 + 840) - (3*a^4*b^2*x^3*(162*m + 23*m^2 + m ^3 + 280))/(638*m + 179*m^2 + 22*m^3 + m^4 + 840) - (a*b^5*m*x^6*(9*m + m^ 2 + 20))/(638*m + 179*m^2 + 22*m^3 + m^4 + 840) - (3*a^5*b*m*x^2*(17*m + m ^2 + 76))/(638*m + 179*m^2 + 22*m^3 + m^4 + 840) + (3*a^3*b^3*m*x^4*(13*m + m^2 + 32))/(638*m + 179*m^2 + 22*m^3 + m^4 + 840))