Integrand size = 21, antiderivative size = 95 \[ \int (d+e x)^m \left (c d x+c e x^2\right )^3 \, dx=-\frac {c^3 d^3 (d+e x)^{4+m}}{e^4 (4+m)}+\frac {3 c^3 d^2 (d+e x)^{5+m}}{e^4 (5+m)}-\frac {3 c^3 d (d+e x)^{6+m}}{e^4 (6+m)}+\frac {c^3 (d+e x)^{7+m}}{e^4 (7+m)} \]
-c^3*d^3*(e*x+d)^(4+m)/e^4/(4+m)+3*c^3*d^2*(e*x+d)^(5+m)/e^4/(5+m)-3*c^3*d *(e*x+d)^(6+m)/e^4/(6+m)+c^3*(e*x+d)^(7+m)/e^4/(7+m)
Time = 0.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.74 \[ \int (d+e x)^m \left (c d x+c e x^2\right )^3 \, dx=\frac {c^3 (d+e x)^{4+m} \left (-\frac {d^3}{4+m}+\frac {3 d^2 (d+e x)}{5+m}-\frac {3 d (d+e x)^2}{6+m}+\frac {(d+e x)^3}{7+m}\right )}{e^4} \]
(c^3*(d + e*x)^(4 + m)*(-(d^3/(4 + m)) + (3*d^2*(d + e*x))/(5 + m) - (3*d* (d + e*x)^2)/(6 + m) + (d + e*x)^3/(7 + m)))/e^4
Time = 0.26 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1121, 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.
| \(\displaystyle \int \left (c d x+c e x^2\right )^3 (d+e x)^m \, dx\) |
| \(\Big \downarrow \) 1121 |
| \(\displaystyle \int \left (-\frac {c^3 d^3 (d+e x)^{m+3}}{e^3}+\frac {3 c^3 d^2 (d+e x)^{m+4}}{e^3}-\frac {3 c^3 d (d+e x)^{m+5}}{e^3}+\frac {c^3 (d+e x)^{m+6}}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {c^3 d^3 (d+e x)^{m+4}}{e^4 (m+4)}+\frac {3 c^3 d^2 (d+e x)^{m+5}}{e^4 (m+5)}-\frac {3 c^3 d (d+e x)^{m+6}}{e^4 (m+6)}+\frac {c^3 (d+e x)^{m+7}}{e^4 (m+7)}\) |
-((c^3*d^3*(d + e*x)^(4 + m))/(e^4*(4 + m))) + (3*c^3*d^2*(d + e*x)^(5 + m ))/(e^4*(5 + m)) - (3*c^3*d*(d + e*x)^(6 + m))/(e^4*(6 + m)) + (c^3*(d + e *x)^(7 + m))/(e^4*(7 + m))
3.5.34.3.1 Defintions of rubi rules used
Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (Int egerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Time = 2.04 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.36
| method | result | size |
| gosper | \(-\frac {c^{3} \left (e x +d \right )^{4+m} \left (-e^{3} m^{3} x^{3}-15 e^{3} m^{2} x^{3}+3 d \,e^{2} m^{2} x^{2}-74 e^{3} m \,x^{3}+27 d \,e^{2} m \,x^{2}-120 e^{3} x^{3}-6 d^{2} e m x +60 d \,e^{2} x^{2}-24 d^{2} e x +6 d^{3}\right )}{e^{4} \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}\) | \(129\) |
| risch | \(-\frac {c^{3} \left (-e^{7} m^{3} x^{7}-4 d \,e^{6} m^{3} x^{6}-15 e^{7} m^{2} x^{7}-6 d^{2} e^{5} m^{3} x^{5}-57 d \,e^{6} m^{2} x^{6}-74 e^{7} m \,x^{7}-4 d^{3} e^{4} m^{3} x^{4}-78 d^{2} e^{5} m^{2} x^{5}-269 d \,e^{6} m \,x^{6}-120 e^{7} x^{7}-d^{4} e^{3} m^{3} x^{3}-42 d^{3} e^{4} m^{2} x^{4}-342 d^{2} e^{5} m \,x^{5}-420 d \,e^{6} x^{6}-3 d^{4} e^{3} m^{2} x^{3}-158 d^{3} e^{4} m \,x^{4}-504 d^{2} e^{5} x^{5}+3 d^{5} e^{2} m^{2} x^{2}-2 d^{4} e^{3} m \,x^{3}-210 d^{3} e^{4} x^{4}+3 d^{5} e^{2} m \,x^{2}-6 x \,d^{6} m e +6 d^{7}\right ) \left (e x +d \right )^{m}}{\left (6+m \right ) \left (7+m \right ) \left (5+m \right ) \left (4+m \right ) e^{4}}\) | \(300\) |
| norman | \(\frac {c^{3} e^{3} x^{7} {\mathrm e}^{m \ln \left (e x +d \right )}}{7+m}+\frac {c^{3} d \,e^{2} \left (4 m +21\right ) x^{6} {\mathrm e}^{m \ln \left (e x +d \right )}}{m^{2}+13 m +42}+\frac {\left (2+m \right ) m \left (1+m \right ) c^{3} d^{4} x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}-\frac {6 c^{3} d^{7} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{4} \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}+\frac {2 c^{3} d^{3} \left (2 m^{3}+21 m^{2}+79 m +105\right ) x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{m^{4}+22 m^{3}+179 m^{2}+638 m +840}+\frac {6 m \,c^{3} d^{6} x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}+\frac {6 \left (m^{2}+9 m +21\right ) c^{3} d^{2} e \,x^{5} {\mathrm e}^{m \ln \left (e x +d \right )}}{m^{3}+18 m^{2}+107 m +210}-\frac {3 m \left (1+m \right ) c^{3} d^{5} x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}\) | \(338\) |
| parallelrisch | \(\frac {-6 c^{3} d^{8} \left (e x +d \right )^{m}+78 x^{5} \left (e x +d \right )^{m} c^{3} d^{3} e^{5} m^{2}+4 x^{4} \left (e x +d \right )^{m} c^{3} d^{4} e^{4} m^{3}+342 x^{5} \left (e x +d \right )^{m} c^{3} d^{3} e^{5} m +42 x^{4} \left (e x +d \right )^{m} c^{3} d^{4} e^{4} m^{2}+x^{3} \left (e x +d \right )^{m} c^{3} d^{5} e^{3} m^{3}+158 x^{4} \left (e x +d \right )^{m} c^{3} d^{4} e^{4} m +3 x^{3} \left (e x +d \right )^{m} c^{3} d^{5} e^{3} m^{2}+2 x^{3} \left (e x +d \right )^{m} c^{3} d^{5} e^{3} m -3 x^{2} \left (e x +d \right )^{m} c^{3} d^{6} e^{2} m^{2}-3 x^{2} \left (e x +d \right )^{m} c^{3} d^{6} e^{2} m +x^{7} \left (e x +d \right )^{m} c^{3} d \,e^{7} m^{3}+15 x^{7} \left (e x +d \right )^{m} c^{3} d \,e^{7} m^{2}+4 x^{6} \left (e x +d \right )^{m} c^{3} d^{2} e^{6} m^{3}+74 x^{7} \left (e x +d \right )^{m} c^{3} d \,e^{7} m +57 x^{6} \left (e x +d \right )^{m} c^{3} d^{2} e^{6} m^{2}+6 x^{5} \left (e x +d \right )^{m} c^{3} d^{3} e^{5} m^{3}+269 x^{6} \left (e x +d \right )^{m} c^{3} d^{2} e^{6} m +6 x \left (e x +d \right )^{m} c^{3} d^{7} e m +120 x^{7} \left (e x +d \right )^{m} c^{3} d \,e^{7}+420 x^{6} \left (e x +d \right )^{m} c^{3} d^{2} e^{6}+504 x^{5} \left (e x +d \right )^{m} c^{3} d^{3} e^{5}+210 x^{4} \left (e x +d \right )^{m} c^{3} d^{4} e^{4}}{\left (m^{3}+18 m^{2}+107 m +210\right ) d \left (4+m \right ) e^{4}}\) | \(532\) |
-c^3/e^4*(e*x+d)^(4+m)/(m^4+22*m^3+179*m^2+638*m+840)*(-e^3*m^3*x^3-15*e^3 *m^2*x^3+3*d*e^2*m^2*x^2-74*e^3*m*x^3+27*d*e^2*m*x^2-120*e^3*x^3-6*d^2*e*m *x+60*d*e^2*x^2-24*d^2*e*x+6*d^3)
Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (95) = 190\).
Time = 0.50 (sec) , antiderivative size = 343, normalized size of antiderivative = 3.61 \[ \int (d+e x)^m \left (c d x+c e x^2\right )^3 \, dx=\frac {{\left (6 \, c^{3} d^{6} e m x - 6 \, c^{3} d^{7} + {\left (c^{3} e^{7} m^{3} + 15 \, c^{3} e^{7} m^{2} + 74 \, c^{3} e^{7} m + 120 \, c^{3} e^{7}\right )} x^{7} + {\left (4 \, c^{3} d e^{6} m^{3} + 57 \, c^{3} d e^{6} m^{2} + 269 \, c^{3} d e^{6} m + 420 \, c^{3} d e^{6}\right )} x^{6} + 6 \, {\left (c^{3} d^{2} e^{5} m^{3} + 13 \, c^{3} d^{2} e^{5} m^{2} + 57 \, c^{3} d^{2} e^{5} m + 84 \, c^{3} d^{2} e^{5}\right )} x^{5} + 2 \, {\left (2 \, c^{3} d^{3} e^{4} m^{3} + 21 \, c^{3} d^{3} e^{4} m^{2} + 79 \, c^{3} d^{3} e^{4} m + 105 \, c^{3} d^{3} e^{4}\right )} x^{4} + {\left (c^{3} d^{4} e^{3} m^{3} + 3 \, c^{3} d^{4} e^{3} m^{2} + 2 \, c^{3} d^{4} e^{3} m\right )} x^{3} - 3 \, {\left (c^{3} d^{5} e^{2} m^{2} + c^{3} d^{5} e^{2} m\right )} x^{2}\right )} {\left (e x + d\right )}^{m}}{e^{4} m^{4} + 22 \, e^{4} m^{3} + 179 \, e^{4} m^{2} + 638 \, e^{4} m + 840 \, e^{4}} \]
(6*c^3*d^6*e*m*x - 6*c^3*d^7 + (c^3*e^7*m^3 + 15*c^3*e^7*m^2 + 74*c^3*e^7* m + 120*c^3*e^7)*x^7 + (4*c^3*d*e^6*m^3 + 57*c^3*d*e^6*m^2 + 269*c^3*d*e^6 *m + 420*c^3*d*e^6)*x^6 + 6*(c^3*d^2*e^5*m^3 + 13*c^3*d^2*e^5*m^2 + 57*c^3 *d^2*e^5*m + 84*c^3*d^2*e^5)*x^5 + 2*(2*c^3*d^3*e^4*m^3 + 21*c^3*d^3*e^4*m ^2 + 79*c^3*d^3*e^4*m + 105*c^3*d^3*e^4)*x^4 + (c^3*d^4*e^3*m^3 + 3*c^3*d^ 4*e^3*m^2 + 2*c^3*d^4*e^3*m)*x^3 - 3*(c^3*d^5*e^2*m^2 + c^3*d^5*e^2*m)*x^2 )*(e*x + d)^m/(e^4*m^4 + 22*e^4*m^3 + 179*e^4*m^2 + 638*e^4*m + 840*e^4)
Leaf count of result is larger than twice the leaf count of optimal. 2218 vs. \(2 (85) = 170\).
Time = 1.55 (sec) , antiderivative size = 2218, normalized size of antiderivative = 23.35 \[ \int (d+e x)^m \left (c d x+c e x^2\right )^3 \, dx=\text {Too large to display} \]
Piecewise((c**3*d**3*d**m*x**4/4, Eq(e, 0)), (6*c**3*d**3*log(d/e + x)/(6* d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 11*c**3*d**3/ (6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*c**3*d* *2*e*x*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e** 7*x**3) + 27*c**3*d**2*e*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*c**3*d*e**2*x**2*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e **5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*c**3*d*e**2*x**2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 6*c**3*e**3*x**3*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3), Eq(m, -7)), (-6*c**3*d**3*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 15*c**3*d**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 12*c**3*d**2*e* x*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 24*c**3*d**2*e*x /(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 6*c**3*d*e**2*x**2*log(d/e + x )/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 6*c**3*d*e**2*x**2/(2*d**2*e* *4 + 4*d*e**5*x + 2*e**6*x**2) + 2*c**3*e**3*x**3/(2*d**2*e**4 + 4*d*e**5* x + 2*e**6*x**2), Eq(m, -6)), (6*c**3*d**3*log(d/e + x)/(2*d*e**4 + 2*e**5 *x) + 12*c**3*d**3/(2*d*e**4 + 2*e**5*x) + 6*c**3*d**2*e*x*log(d/e + x)/(2 *d*e**4 + 2*e**5*x) + 6*c**3*d**2*e*x/(2*d*e**4 + 2*e**5*x) - 3*c**3*d*e** 2*x**2/(2*d*e**4 + 2*e**5*x) + c**3*e**3*x**3/(2*d*e**4 + 2*e**5*x), Eq(m, -5)), (-c**3*d**3*log(d/e + x)/e**4 + c**3*d**2*x/e**3 - c**3*d*x**2/(...
Leaf count of result is larger than twice the leaf count of optimal. 674 vs. \(2 (95) = 190\).
Time = 0.21 (sec) , antiderivative size = 674, normalized size of antiderivative = 7.09 \[ \int (d+e x)^m \left (c d x+c e x^2\right )^3 \, dx=\frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} c^{3} d^{3}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {3 \, {\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} c^{3} d^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{4}} + \frac {3 \, {\left ({\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{6} x^{6} + {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d e^{5} x^{5} - 5 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{2} e^{4} x^{4} + 20 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{3} e^{3} x^{3} - 60 \, {\left (m^{2} + m\right )} d^{4} e^{2} x^{2} + 120 \, d^{5} e m x - 120 \, d^{6}\right )} {\left (e x + d\right )}^{m} c^{3} d}{{\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} e^{4}} + \frac {{\left ({\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} e^{7} x^{7} + {\left (m^{6} + 15 \, m^{5} + 85 \, m^{4} + 225 \, m^{3} + 274 \, m^{2} + 120 \, m\right )} d e^{6} x^{6} - 6 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d^{2} e^{5} x^{5} + 30 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{3} e^{4} x^{4} - 120 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{4} e^{3} x^{3} + 360 \, {\left (m^{2} + m\right )} d^{5} e^{2} x^{2} - 720 \, d^{6} e m x + 720 \, d^{7}\right )} {\left (e x + d\right )}^{m} c^{3}}{{\left (m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040\right )} e^{4}} \]
((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3*e*m*x - 6*d^4)*(e*x + d)^m*c^3*d^3/((m^4 + 10*m^ 3 + 35*m^2 + 50*m + 24)*e^4) + 3*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^5* x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e ^3*x^3 + 12*(m^2 + m)*d^3*e^2*x^2 - 24*d^4*e*m*x + 24*d^5)*(e*x + d)^m*c^3 *d^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^4) + 3*((m^5 + 15* m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^6*x^6 + (m^5 + 10*m^4 + 35*m^3 + 5 0*m^2 + 24*m)*d*e^5*x^5 - 5*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^2*e^4*x^4 + 20* (m^3 + 3*m^2 + 2*m)*d^3*e^3*x^3 - 60*(m^2 + m)*d^4*e^2*x^2 + 120*d^5*e*m*x - 120*d^6)*(e*x + d)^m*c^3*d/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^ 2 + 1764*m + 720)*e^4) + ((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1 764*m + 720)*e^7*x^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2 + 120*m) *d*e^6*x^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d^2*e^5*x^5 + 30*(m ^4 + 6*m^3 + 11*m^2 + 6*m)*d^3*e^4*x^4 - 120*(m^3 + 3*m^2 + 2*m)*d^4*e^3*x ^3 + 360*(m^2 + m)*d^5*e^2*x^2 - 720*d^6*e*m*x + 720*d^7)*(e*x + d)^m*c^3/ ((m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 504 0)*e^4)
Leaf count of result is larger than twice the leaf count of optimal. 530 vs. \(2 (95) = 190\).
Time = 0.30 (sec) , antiderivative size = 530, normalized size of antiderivative = 5.58 \[ \int (d+e x)^m \left (c d x+c e x^2\right )^3 \, dx=\frac {{\left (e x + d\right )}^{m} c^{3} e^{7} m^{3} x^{7} + 4 \, {\left (e x + d\right )}^{m} c^{3} d e^{6} m^{3} x^{6} + 15 \, {\left (e x + d\right )}^{m} c^{3} e^{7} m^{2} x^{7} + 6 \, {\left (e x + d\right )}^{m} c^{3} d^{2} e^{5} m^{3} x^{5} + 57 \, {\left (e x + d\right )}^{m} c^{3} d e^{6} m^{2} x^{6} + 74 \, {\left (e x + d\right )}^{m} c^{3} e^{7} m x^{7} + 4 \, {\left (e x + d\right )}^{m} c^{3} d^{3} e^{4} m^{3} x^{4} + 78 \, {\left (e x + d\right )}^{m} c^{3} d^{2} e^{5} m^{2} x^{5} + 269 \, {\left (e x + d\right )}^{m} c^{3} d e^{6} m x^{6} + 120 \, {\left (e x + d\right )}^{m} c^{3} e^{7} x^{7} + {\left (e x + d\right )}^{m} c^{3} d^{4} e^{3} m^{3} x^{3} + 42 \, {\left (e x + d\right )}^{m} c^{3} d^{3} e^{4} m^{2} x^{4} + 342 \, {\left (e x + d\right )}^{m} c^{3} d^{2} e^{5} m x^{5} + 420 \, {\left (e x + d\right )}^{m} c^{3} d e^{6} x^{6} + 3 \, {\left (e x + d\right )}^{m} c^{3} d^{4} e^{3} m^{2} x^{3} + 158 \, {\left (e x + d\right )}^{m} c^{3} d^{3} e^{4} m x^{4} + 504 \, {\left (e x + d\right )}^{m} c^{3} d^{2} e^{5} x^{5} - 3 \, {\left (e x + d\right )}^{m} c^{3} d^{5} e^{2} m^{2} x^{2} + 2 \, {\left (e x + d\right )}^{m} c^{3} d^{4} e^{3} m x^{3} + 210 \, {\left (e x + d\right )}^{m} c^{3} d^{3} e^{4} x^{4} - 3 \, {\left (e x + d\right )}^{m} c^{3} d^{5} e^{2} m x^{2} + 6 \, {\left (e x + d\right )}^{m} c^{3} d^{6} e m x - 6 \, {\left (e x + d\right )}^{m} c^{3} d^{7}}{e^{4} m^{4} + 22 \, e^{4} m^{3} + 179 \, e^{4} m^{2} + 638 \, e^{4} m + 840 \, e^{4}} \]
((e*x + d)^m*c^3*e^7*m^3*x^7 + 4*(e*x + d)^m*c^3*d*e^6*m^3*x^6 + 15*(e*x + d)^m*c^3*e^7*m^2*x^7 + 6*(e*x + d)^m*c^3*d^2*e^5*m^3*x^5 + 57*(e*x + d)^m *c^3*d*e^6*m^2*x^6 + 74*(e*x + d)^m*c^3*e^7*m*x^7 + 4*(e*x + d)^m*c^3*d^3* e^4*m^3*x^4 + 78*(e*x + d)^m*c^3*d^2*e^5*m^2*x^5 + 269*(e*x + d)^m*c^3*d*e ^6*m*x^6 + 120*(e*x + d)^m*c^3*e^7*x^7 + (e*x + d)^m*c^3*d^4*e^3*m^3*x^3 + 42*(e*x + d)^m*c^3*d^3*e^4*m^2*x^4 + 342*(e*x + d)^m*c^3*d^2*e^5*m*x^5 + 420*(e*x + d)^m*c^3*d*e^6*x^6 + 3*(e*x + d)^m*c^3*d^4*e^3*m^2*x^3 + 158*(e *x + d)^m*c^3*d^3*e^4*m*x^4 + 504*(e*x + d)^m*c^3*d^2*e^5*x^5 - 3*(e*x + d )^m*c^3*d^5*e^2*m^2*x^2 + 2*(e*x + d)^m*c^3*d^4*e^3*m*x^3 + 210*(e*x + d)^ m*c^3*d^3*e^4*x^4 - 3*(e*x + d)^m*c^3*d^5*e^2*m*x^2 + 6*(e*x + d)^m*c^3*d^ 6*e*m*x - 6*(e*x + d)^m*c^3*d^7)/(e^4*m^4 + 22*e^4*m^3 + 179*e^4*m^2 + 638 *e^4*m + 840*e^4)
Time = 9.85 (sec) , antiderivative size = 333, normalized size of antiderivative = 3.51 \[ \int (d+e x)^m \left (c d x+c e x^2\right )^3 \, dx={\left (d+e\,x\right )}^m\,\left (\frac {c^3\,e^3\,x^7\,\left (m^3+15\,m^2+74\,m+120\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}-\frac {6\,c^3\,d^7}{e^4\,\left (m^4+22\,m^3+179\,m^2+638\,m+840\right )}+\frac {2\,c^3\,d^3\,x^4\,\left (2\,m^3+21\,m^2+79\,m+105\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}+\frac {6\,c^3\,d^6\,m\,x}{e^3\,\left (m^4+22\,m^3+179\,m^2+638\,m+840\right )}+\frac {c^3\,d\,e^2\,x^6\,\left (4\,m^3+57\,m^2+269\,m+420\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}+\frac {6\,c^3\,d^2\,e\,x^5\,\left (m^3+13\,m^2+57\,m+84\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}-\frac {3\,c^3\,d^5\,m\,x^2\,\left (m+1\right )}{e^2\,\left (m^4+22\,m^3+179\,m^2+638\,m+840\right )}+\frac {c^3\,d^4\,m\,x^3\,\left (m^2+3\,m+2\right )}{e\,\left (m^4+22\,m^3+179\,m^2+638\,m+840\right )}\right ) \]
(d + e*x)^m*((c^3*e^3*x^7*(74*m + 15*m^2 + m^3 + 120))/(638*m + 179*m^2 + 22*m^3 + m^4 + 840) - (6*c^3*d^7)/(e^4*(638*m + 179*m^2 + 22*m^3 + m^4 + 8 40)) + (2*c^3*d^3*x^4*(79*m + 21*m^2 + 2*m^3 + 105))/(638*m + 179*m^2 + 22 *m^3 + m^4 + 840) + (6*c^3*d^6*m*x)/(e^3*(638*m + 179*m^2 + 22*m^3 + m^4 + 840)) + (c^3*d*e^2*x^6*(269*m + 57*m^2 + 4*m^3 + 420))/(638*m + 179*m^2 + 22*m^3 + m^4 + 840) + (6*c^3*d^2*e*x^5*(57*m + 13*m^2 + m^3 + 84))/(638*m + 179*m^2 + 22*m^3 + m^4 + 840) - (3*c^3*d^5*m*x^2*(m + 1))/(e^2*(638*m + 179*m^2 + 22*m^3 + m^4 + 840)) + (c^3*d^4*m*x^3*(3*m + m^2 + 2))/(e*(638* m + 179*m^2 + 22*m^3 + m^4 + 840)))