Integrand size = 24, antiderivative size = 143 \[ \int (e x)^m (a+b x)^3 (a d-b d x)^2 \, dx=\frac {a^5 d^2 (e x)^{1+m}}{e (1+m)}+\frac {a^4 b d^2 (e x)^{2+m}}{e^2 (2+m)}-\frac {2 a^3 b^2 d^2 (e x)^{3+m}}{e^3 (3+m)}-\frac {2 a^2 b^3 d^2 (e x)^{4+m}}{e^4 (4+m)}+\frac {a b^4 d^2 (e x)^{5+m}}{e^5 (5+m)}+\frac {b^5 d^2 (e x)^{6+m}}{e^6 (6+m)} \]
a^5*d^2*(e*x)^(1+m)/e/(1+m)+a^4*b*d^2*(e*x)^(2+m)/e^2/(2+m)-2*a^3*b^2*d^2* (e*x)^(3+m)/e^3/(3+m)-2*a^2*b^3*d^2*(e*x)^(4+m)/e^4/(4+m)+a*b^4*d^2*(e*x)^ (5+m)/e^5/(5+m)+b^5*d^2*(e*x)^(6+m)/e^6/(6+m)
Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.62 \[ \int (e x)^m (a+b x)^3 (a d-b d x)^2 \, dx=d^2 x (e x)^m \left (\frac {a^5}{1+m}+\frac {a^4 b x}{2+m}-\frac {2 a^3 b^2 x^2}{3+m}-\frac {2 a^2 b^3 x^3}{4+m}+\frac {a b^4 x^4}{5+m}+\frac {b^5 x^5}{6+m}\right ) \]
d^2*x*(e*x)^m*(a^5/(1 + m) + (a^4*b*x)/(2 + m) - (2*a^3*b^2*x^2)/(3 + m) - (2*a^2*b^3*x^3)/(4 + m) + (a*b^4*x^4)/(5 + m) + (b^5*x^5)/(6 + m))
Time = 0.29 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {99, 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.
| \(\displaystyle \int (a+b x)^3 (e x)^m (a d-b d x)^2 \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (a^5 d^2 (e x)^m+\frac {a^4 b d^2 (e x)^{m+1}}{e}-\frac {2 a^3 b^2 d^2 (e x)^{m+2}}{e^2}-\frac {2 a^2 b^3 d^2 (e x)^{m+3}}{e^3}+\frac {a b^4 d^2 (e x)^{m+4}}{e^4}+\frac {b^5 d^2 (e x)^{m+5}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^5 d^2 (e x)^{m+1}}{e (m+1)}+\frac {a^4 b d^2 (e x)^{m+2}}{e^2 (m+2)}-\frac {2 a^3 b^2 d^2 (e x)^{m+3}}{e^3 (m+3)}-\frac {2 a^2 b^3 d^2 (e x)^{m+4}}{e^4 (m+4)}+\frac {a b^4 d^2 (e x)^{m+5}}{e^5 (m+5)}+\frac {b^5 d^2 (e x)^{m+6}}{e^6 (m+6)}\) |
(a^5*d^2*(e*x)^(1 + m))/(e*(1 + m)) + (a^4*b*d^2*(e*x)^(2 + m))/(e^2*(2 + m)) - (2*a^3*b^2*d^2*(e*x)^(3 + m))/(e^3*(3 + m)) - (2*a^2*b^3*d^2*(e*x)^( 4 + m))/(e^4*(4 + m)) + (a*b^4*d^2*(e*x)^(5 + m))/(e^5*(5 + m)) + (b^5*d^2 *(e*x)^(6 + m))/(e^6*(6 + m))
3.1.63.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Time = 0.42 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.99
| method | result | size |
| norman | \(\frac {a^{5} d^{2} x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}+\frac {b^{5} d^{2} x^{6} {\mathrm e}^{m \ln \left (e x \right )}}{6+m}+\frac {a \,b^{4} d^{2} x^{5} {\mathrm e}^{m \ln \left (e x \right )}}{5+m}+\frac {a^{4} b \,d^{2} x^{2} {\mathrm e}^{m \ln \left (e x \right )}}{2+m}-\frac {2 a^{2} b^{3} d^{2} x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}-\frac {2 a^{3} b^{2} d^{2} x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}\) | \(142\) |
| gosper | \(\frac {d^{2} \left (e x \right )^{m} \left (b^{5} m^{5} x^{5}+a \,b^{4} m^{5} x^{4}+15 b^{5} m^{4} x^{5}-2 a^{2} b^{3} m^{5} x^{3}+16 a \,b^{4} m^{4} x^{4}+85 b^{5} m^{3} x^{5}-2 a^{3} b^{2} m^{5} x^{2}-34 a^{2} b^{3} m^{4} x^{3}+95 a \,b^{4} m^{3} x^{4}+225 b^{5} m^{2} x^{5}+a^{4} b \,m^{5} x -36 a^{3} b^{2} m^{4} x^{2}-214 a^{2} b^{3} m^{3} x^{3}+260 a \,b^{4} m^{2} x^{4}+274 m \,x^{5} b^{5}+a^{5} m^{5}+19 a^{4} b \,m^{4} x -242 a^{3} b^{2} m^{3} x^{2}-614 a^{2} b^{3} m^{2} x^{3}+324 a \,b^{4} m \,x^{4}+120 b^{5} x^{5}+20 a^{5} m^{4}+137 a^{4} b \,m^{3} x -744 a^{3} b^{2} m^{2} x^{2}-792 a^{2} b^{3} m \,x^{3}+144 a \,b^{4} x^{4}+155 a^{5} m^{3}+461 a^{4} b \,m^{2} x -1016 a^{3} b^{2} m \,x^{2}-360 a^{2} b^{3} x^{3}+580 a^{5} m^{2}+702 a^{4} b m x -480 a^{3} b^{2} x^{2}+1044 a^{5} m +360 a^{4} b x +720 a^{5}\right ) x}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(422\) |
| risch | \(\frac {d^{2} \left (e x \right )^{m} \left (b^{5} m^{5} x^{5}+a \,b^{4} m^{5} x^{4}+15 b^{5} m^{4} x^{5}-2 a^{2} b^{3} m^{5} x^{3}+16 a \,b^{4} m^{4} x^{4}+85 b^{5} m^{3} x^{5}-2 a^{3} b^{2} m^{5} x^{2}-34 a^{2} b^{3} m^{4} x^{3}+95 a \,b^{4} m^{3} x^{4}+225 b^{5} m^{2} x^{5}+a^{4} b \,m^{5} x -36 a^{3} b^{2} m^{4} x^{2}-214 a^{2} b^{3} m^{3} x^{3}+260 a \,b^{4} m^{2} x^{4}+274 m \,x^{5} b^{5}+a^{5} m^{5}+19 a^{4} b \,m^{4} x -242 a^{3} b^{2} m^{3} x^{2}-614 a^{2} b^{3} m^{2} x^{3}+324 a \,b^{4} m \,x^{4}+120 b^{5} x^{5}+20 a^{5} m^{4}+137 a^{4} b \,m^{3} x -744 a^{3} b^{2} m^{2} x^{2}-792 a^{2} b^{3} m \,x^{3}+144 a \,b^{4} x^{4}+155 a^{5} m^{3}+461 a^{4} b \,m^{2} x -1016 a^{3} b^{2} m \,x^{2}-360 a^{2} b^{3} x^{3}+580 a^{5} m^{2}+702 a^{4} b m x -480 a^{3} b^{2} x^{2}+1044 a^{5} m +360 a^{4} b x +720 a^{5}\right ) x}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(422\) |
| parallelrisch | \(\frac {702 x^{2} \left (e x \right )^{m} a^{4} b \,d^{2} m +137 x^{2} \left (e x \right )^{m} a^{4} b \,d^{2} m^{3}-1016 x^{3} \left (e x \right )^{m} a^{3} b^{2} d^{2} m +461 x^{2} \left (e x \right )^{m} a^{4} b \,d^{2} m^{2}-34 x^{4} \left (e x \right )^{m} a^{2} b^{3} d^{2} m^{4}-2 x^{3} \left (e x \right )^{m} a^{3} b^{2} d^{2} m^{5}+260 x^{5} \left (e x \right )^{m} a \,b^{4} d^{2} m^{2}-214 x^{4} \left (e x \right )^{m} a^{2} b^{3} d^{2} m^{3}-36 x^{3} \left (e x \right )^{m} a^{3} b^{2} d^{2} m^{4}+x^{2} \left (e x \right )^{m} a^{4} b \,d^{2} m^{5}+324 x^{5} \left (e x \right )^{m} a \,b^{4} d^{2} m -614 x^{4} \left (e x \right )^{m} a^{2} b^{3} d^{2} m^{2}-242 x^{3} \left (e x \right )^{m} a^{3} b^{2} d^{2} m^{3}+19 x^{2} \left (e x \right )^{m} a^{4} b \,d^{2} m^{4}-792 x^{4} \left (e x \right )^{m} a^{2} b^{3} d^{2} m -744 x^{3} \left (e x \right )^{m} a^{3} b^{2} d^{2} m^{2}+x^{5} \left (e x \right )^{m} a \,b^{4} d^{2} m^{5}+16 x^{5} \left (e x \right )^{m} a \,b^{4} d^{2} m^{4}-2 x^{4} \left (e x \right )^{m} a^{2} b^{3} d^{2} m^{5}+95 x^{5} \left (e x \right )^{m} a \,b^{4} d^{2} m^{3}+120 x^{6} \left (e x \right )^{m} b^{5} d^{2}+720 x \left (e x \right )^{m} a^{5} d^{2}+x^{6} \left (e x \right )^{m} b^{5} d^{2} m^{5}+15 x^{6} \left (e x \right )^{m} b^{5} d^{2} m^{4}+85 x^{6} \left (e x \right )^{m} b^{5} d^{2} m^{3}+225 x^{6} \left (e x \right )^{m} b^{5} d^{2} m^{2}+274 x^{6} \left (e x \right )^{m} b^{5} d^{2} m +x \left (e x \right )^{m} a^{5} d^{2} m^{5}+144 x^{5} \left (e x \right )^{m} a \,b^{4} d^{2}+20 x \left (e x \right )^{m} a^{5} d^{2} m^{4}-360 x^{4} \left (e x \right )^{m} a^{2} b^{3} d^{2}+155 x \left (e x \right )^{m} a^{5} d^{2} m^{3}-480 x^{3} \left (e x \right )^{m} a^{3} b^{2} d^{2}+580 x \left (e x \right )^{m} a^{5} d^{2} m^{2}+360 x^{2} \left (e x \right )^{m} a^{4} b \,d^{2}+1044 x \left (e x \right )^{m} a^{5} d^{2} m}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(719\) |
a^5*d^2/(1+m)*x*exp(m*ln(e*x))+b^5*d^2/(6+m)*x^6*exp(m*ln(e*x))+a*b^4*d^2/ (5+m)*x^5*exp(m*ln(e*x))+a^4*b*d^2/(2+m)*x^2*exp(m*ln(e*x))-2*a^2*b^3*d^2/ (4+m)*x^4*exp(m*ln(e*x))-2*a^3*b^2*d^2/(3+m)*x^3*exp(m*ln(e*x))
Leaf count of result is larger than twice the leaf count of optimal. 475 vs. \(2 (143) = 286\).
Time = 0.23 (sec) , antiderivative size = 475, normalized size of antiderivative = 3.32 \[ \int (e x)^m (a+b x)^3 (a d-b d x)^2 \, dx=\frac {{\left ({\left (b^{5} d^{2} m^{5} + 15 \, b^{5} d^{2} m^{4} + 85 \, b^{5} d^{2} m^{3} + 225 \, b^{5} d^{2} m^{2} + 274 \, b^{5} d^{2} m + 120 \, b^{5} d^{2}\right )} x^{6} + {\left (a b^{4} d^{2} m^{5} + 16 \, a b^{4} d^{2} m^{4} + 95 \, a b^{4} d^{2} m^{3} + 260 \, a b^{4} d^{2} m^{2} + 324 \, a b^{4} d^{2} m + 144 \, a b^{4} d^{2}\right )} x^{5} - 2 \, {\left (a^{2} b^{3} d^{2} m^{5} + 17 \, a^{2} b^{3} d^{2} m^{4} + 107 \, a^{2} b^{3} d^{2} m^{3} + 307 \, a^{2} b^{3} d^{2} m^{2} + 396 \, a^{2} b^{3} d^{2} m + 180 \, a^{2} b^{3} d^{2}\right )} x^{4} - 2 \, {\left (a^{3} b^{2} d^{2} m^{5} + 18 \, a^{3} b^{2} d^{2} m^{4} + 121 \, a^{3} b^{2} d^{2} m^{3} + 372 \, a^{3} b^{2} d^{2} m^{2} + 508 \, a^{3} b^{2} d^{2} m + 240 \, a^{3} b^{2} d^{2}\right )} x^{3} + {\left (a^{4} b d^{2} m^{5} + 19 \, a^{4} b d^{2} m^{4} + 137 \, a^{4} b d^{2} m^{3} + 461 \, a^{4} b d^{2} m^{2} + 702 \, a^{4} b d^{2} m + 360 \, a^{4} b d^{2}\right )} x^{2} + {\left (a^{5} d^{2} m^{5} + 20 \, a^{5} d^{2} m^{4} + 155 \, a^{5} d^{2} m^{3} + 580 \, a^{5} d^{2} m^{2} + 1044 \, a^{5} d^{2} m + 720 \, a^{5} d^{2}\right )} x\right )} \left (e x\right )^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \]
((b^5*d^2*m^5 + 15*b^5*d^2*m^4 + 85*b^5*d^2*m^3 + 225*b^5*d^2*m^2 + 274*b^ 5*d^2*m + 120*b^5*d^2)*x^6 + (a*b^4*d^2*m^5 + 16*a*b^4*d^2*m^4 + 95*a*b^4* d^2*m^3 + 260*a*b^4*d^2*m^2 + 324*a*b^4*d^2*m + 144*a*b^4*d^2)*x^5 - 2*(a^ 2*b^3*d^2*m^5 + 17*a^2*b^3*d^2*m^4 + 107*a^2*b^3*d^2*m^3 + 307*a^2*b^3*d^2 *m^2 + 396*a^2*b^3*d^2*m + 180*a^2*b^3*d^2)*x^4 - 2*(a^3*b^2*d^2*m^5 + 18* a^3*b^2*d^2*m^4 + 121*a^3*b^2*d^2*m^3 + 372*a^3*b^2*d^2*m^2 + 508*a^3*b^2* d^2*m + 240*a^3*b^2*d^2)*x^3 + (a^4*b*d^2*m^5 + 19*a^4*b*d^2*m^4 + 137*a^4 *b*d^2*m^3 + 461*a^4*b*d^2*m^2 + 702*a^4*b*d^2*m + 360*a^4*b*d^2)*x^2 + (a ^5*d^2*m^5 + 20*a^5*d^2*m^4 + 155*a^5*d^2*m^3 + 580*a^5*d^2*m^2 + 1044*a^5 *d^2*m + 720*a^5*d^2)*x)*(e*x)^m/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624* m^2 + 1764*m + 720)
Leaf count of result is larger than twice the leaf count of optimal. 2259 vs. \(2 (133) = 266\).
Time = 0.49 (sec) , antiderivative size = 2259, normalized size of antiderivative = 15.80 \[ \int (e x)^m (a+b x)^3 (a d-b d x)^2 \, dx=\text {Too large to display} \]
Piecewise(((-a**5*d**2/(5*x**5) - a**4*b*d**2/(4*x**4) + 2*a**3*b**2*d**2/ (3*x**3) + a**2*b**3*d**2/x**2 - a*b**4*d**2/x + b**5*d**2*log(x))/e**6, E q(m, -6)), ((-a**5*d**2/(4*x**4) - a**4*b*d**2/(3*x**3) + a**3*b**2*d**2/x **2 + 2*a**2*b**3*d**2/x + a*b**4*d**2*log(x) + b**5*d**2*x)/e**5, Eq(m, - 5)), ((-a**5*d**2/(3*x**3) - a**4*b*d**2/(2*x**2) + 2*a**3*b**2*d**2/x - 2 *a**2*b**3*d**2*log(x) + a*b**4*d**2*x + b**5*d**2*x**2/2)/e**4, Eq(m, -4) ), ((-a**5*d**2/(2*x**2) - a**4*b*d**2/x - 2*a**3*b**2*d**2*log(x) - 2*a** 2*b**3*d**2*x + a*b**4*d**2*x**2/2 + b**5*d**2*x**3/3)/e**3, Eq(m, -3)), ( (-a**5*d**2/x + a**4*b*d**2*log(x) - 2*a**3*b**2*d**2*x - a**2*b**3*d**2*x **2 + a*b**4*d**2*x**3/3 + b**5*d**2*x**4/4)/e**2, Eq(m, -2)), ((a**5*d**2 *log(x) + a**4*b*d**2*x - a**3*b**2*d**2*x**2 - 2*a**2*b**3*d**2*x**3/3 + a*b**4*d**2*x**4/4 + b**5*d**2*x**5/5)/e, Eq(m, -1)), (a**5*d**2*m**5*x*(e *x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 20*a**5*d**2*m**4*x*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624 *m**2 + 1764*m + 720) + 155*a**5*d**2*m**3*x*(e*x)**m/(m**6 + 21*m**5 + 17 5*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 580*a**5*d**2*m**2*x*(e*x) **m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 10 44*a**5*d**2*m*x*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m** 2 + 1764*m + 720) + 720*a**5*d**2*x*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + a**4*b*d**2*m**5*x**2*(e*x)**m/(...
Time = 0.27 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.97 \[ \int (e x)^m (a+b x)^3 (a d-b d x)^2 \, dx=\frac {b^{5} d^{2} e^{m} x^{6} x^{m}}{m + 6} + \frac {a b^{4} d^{2} e^{m} x^{5} x^{m}}{m + 5} - \frac {2 \, a^{2} b^{3} d^{2} e^{m} x^{4} x^{m}}{m + 4} - \frac {2 \, a^{3} b^{2} d^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {a^{4} b d^{2} e^{m} x^{2} x^{m}}{m + 2} + \frac {\left (e x\right )^{m + 1} a^{5} d^{2}}{e {\left (m + 1\right )}} \]
b^5*d^2*e^m*x^6*x^m/(m + 6) + a*b^4*d^2*e^m*x^5*x^m/(m + 5) - 2*a^2*b^3*d^ 2*e^m*x^4*x^m/(m + 4) - 2*a^3*b^2*d^2*e^m*x^3*x^m/(m + 3) + a^4*b*d^2*e^m* x^2*x^m/(m + 2) + (e*x)^(m + 1)*a^5*d^2/(e*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 718 vs. \(2 (143) = 286\).
Time = 0.29 (sec) , antiderivative size = 718, normalized size of antiderivative = 5.02 \[ \int (e x)^m (a+b x)^3 (a d-b d x)^2 \, dx=\frac {\left (e x\right )^{m} b^{5} d^{2} m^{5} x^{6} + \left (e x\right )^{m} a b^{4} d^{2} m^{5} x^{5} + 15 \, \left (e x\right )^{m} b^{5} d^{2} m^{4} x^{6} - 2 \, \left (e x\right )^{m} a^{2} b^{3} d^{2} m^{5} x^{4} + 16 \, \left (e x\right )^{m} a b^{4} d^{2} m^{4} x^{5} + 85 \, \left (e x\right )^{m} b^{5} d^{2} m^{3} x^{6} - 2 \, \left (e x\right )^{m} a^{3} b^{2} d^{2} m^{5} x^{3} - 34 \, \left (e x\right )^{m} a^{2} b^{3} d^{2} m^{4} x^{4} + 95 \, \left (e x\right )^{m} a b^{4} d^{2} m^{3} x^{5} + 225 \, \left (e x\right )^{m} b^{5} d^{2} m^{2} x^{6} + \left (e x\right )^{m} a^{4} b d^{2} m^{5} x^{2} - 36 \, \left (e x\right )^{m} a^{3} b^{2} d^{2} m^{4} x^{3} - 214 \, \left (e x\right )^{m} a^{2} b^{3} d^{2} m^{3} x^{4} + 260 \, \left (e x\right )^{m} a b^{4} d^{2} m^{2} x^{5} + 274 \, \left (e x\right )^{m} b^{5} d^{2} m x^{6} + \left (e x\right )^{m} a^{5} d^{2} m^{5} x + 19 \, \left (e x\right )^{m} a^{4} b d^{2} m^{4} x^{2} - 242 \, \left (e x\right )^{m} a^{3} b^{2} d^{2} m^{3} x^{3} - 614 \, \left (e x\right )^{m} a^{2} b^{3} d^{2} m^{2} x^{4} + 324 \, \left (e x\right )^{m} a b^{4} d^{2} m x^{5} + 120 \, \left (e x\right )^{m} b^{5} d^{2} x^{6} + 20 \, \left (e x\right )^{m} a^{5} d^{2} m^{4} x + 137 \, \left (e x\right )^{m} a^{4} b d^{2} m^{3} x^{2} - 744 \, \left (e x\right )^{m} a^{3} b^{2} d^{2} m^{2} x^{3} - 792 \, \left (e x\right )^{m} a^{2} b^{3} d^{2} m x^{4} + 144 \, \left (e x\right )^{m} a b^{4} d^{2} x^{5} + 155 \, \left (e x\right )^{m} a^{5} d^{2} m^{3} x + 461 \, \left (e x\right )^{m} a^{4} b d^{2} m^{2} x^{2} - 1016 \, \left (e x\right )^{m} a^{3} b^{2} d^{2} m x^{3} - 360 \, \left (e x\right )^{m} a^{2} b^{3} d^{2} x^{4} + 580 \, \left (e x\right )^{m} a^{5} d^{2} m^{2} x + 702 \, \left (e x\right )^{m} a^{4} b d^{2} m x^{2} - 480 \, \left (e x\right )^{m} a^{3} b^{2} d^{2} x^{3} + 1044 \, \left (e x\right )^{m} a^{5} d^{2} m x + 360 \, \left (e x\right )^{m} a^{4} b d^{2} x^{2} + 720 \, \left (e x\right )^{m} a^{5} d^{2} x}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \]
((e*x)^m*b^5*d^2*m^5*x^6 + (e*x)^m*a*b^4*d^2*m^5*x^5 + 15*(e*x)^m*b^5*d^2* m^4*x^6 - 2*(e*x)^m*a^2*b^3*d^2*m^5*x^4 + 16*(e*x)^m*a*b^4*d^2*m^4*x^5 + 8 5*(e*x)^m*b^5*d^2*m^3*x^6 - 2*(e*x)^m*a^3*b^2*d^2*m^5*x^3 - 34*(e*x)^m*a^2 *b^3*d^2*m^4*x^4 + 95*(e*x)^m*a*b^4*d^2*m^3*x^5 + 225*(e*x)^m*b^5*d^2*m^2* x^6 + (e*x)^m*a^4*b*d^2*m^5*x^2 - 36*(e*x)^m*a^3*b^2*d^2*m^4*x^3 - 214*(e* x)^m*a^2*b^3*d^2*m^3*x^4 + 260*(e*x)^m*a*b^4*d^2*m^2*x^5 + 274*(e*x)^m*b^5 *d^2*m*x^6 + (e*x)^m*a^5*d^2*m^5*x + 19*(e*x)^m*a^4*b*d^2*m^4*x^2 - 242*(e *x)^m*a^3*b^2*d^2*m^3*x^3 - 614*(e*x)^m*a^2*b^3*d^2*m^2*x^4 + 324*(e*x)^m* a*b^4*d^2*m*x^5 + 120*(e*x)^m*b^5*d^2*x^6 + 20*(e*x)^m*a^5*d^2*m^4*x + 137 *(e*x)^m*a^4*b*d^2*m^3*x^2 - 744*(e*x)^m*a^3*b^2*d^2*m^2*x^3 - 792*(e*x)^m *a^2*b^3*d^2*m*x^4 + 144*(e*x)^m*a*b^4*d^2*x^5 + 155*(e*x)^m*a^5*d^2*m^3*x + 461*(e*x)^m*a^4*b*d^2*m^2*x^2 - 1016*(e*x)^m*a^3*b^2*d^2*m*x^3 - 360*(e *x)^m*a^2*b^3*d^2*x^4 + 580*(e*x)^m*a^5*d^2*m^2*x + 702*(e*x)^m*a^4*b*d^2* m*x^2 - 480*(e*x)^m*a^3*b^2*d^2*x^3 + 1044*(e*x)^m*a^5*d^2*m*x + 360*(e*x) ^m*a^4*b*d^2*x^2 + 720*(e*x)^m*a^5*d^2*x)/(m^6 + 21*m^5 + 175*m^4 + 735*m^ 3 + 1624*m^2 + 1764*m + 720)
Time = 0.80 (sec) , antiderivative size = 393, normalized size of antiderivative = 2.75 \[ \int (e x)^m (a+b x)^3 (a d-b d x)^2 \, dx={\left (e\,x\right )}^m\,\left (\frac {b^5\,d^2\,x^6\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a^5\,d^2\,x\,\left (m^5+20\,m^4+155\,m^3+580\,m^2+1044\,m+720\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a\,b^4\,d^2\,x^5\,\left (m^5+16\,m^4+95\,m^3+260\,m^2+324\,m+144\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a^4\,b\,d^2\,x^2\,\left (m^5+19\,m^4+137\,m^3+461\,m^2+702\,m+360\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}-\frac {2\,a^2\,b^3\,d^2\,x^4\,\left (m^5+17\,m^4+107\,m^3+307\,m^2+396\,m+180\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}-\frac {2\,a^3\,b^2\,d^2\,x^3\,\left (m^5+18\,m^4+121\,m^3+372\,m^2+508\,m+240\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}\right ) \]
(e*x)^m*((b^5*d^2*x^6*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120))/(17 64*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (a^5*d^2*x*(10 44*m + 580*m^2 + 155*m^3 + 20*m^4 + m^5 + 720))/(1764*m + 1624*m^2 + 735*m ^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (a*b^4*d^2*x^5*(324*m + 260*m^2 + 95* m^3 + 16*m^4 + m^5 + 144))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (a^4*b*d^2*x^2*(702*m + 461*m^2 + 137*m^3 + 19*m^4 + m^5 + 360))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) - (2*a ^2*b^3*d^2*x^4*(396*m + 307*m^2 + 107*m^3 + 17*m^4 + m^5 + 180))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) - (2*a^3*b^2*d^2*x^3*( 508*m + 372*m^2 + 121*m^3 + 18*m^4 + m^5 + 240))/(1764*m + 1624*m^2 + 735* m^3 + 175*m^4 + 21*m^5 + m^6 + 720))