\(\int (d+e x)^m (b x+c x^2)^3 \, dx\) [238]

Optimal result

Integrand size = 19, antiderivative size = 267 \[ \int (d+e x)^m \left (b x+c x^2\right )^3 \, dx=\frac {d^3 (c d-b e)^3 (d+e x)^{1+m}}{e^7 (1+m)}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{2+m}}{e^7 (2+m)}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{3+m}}{e^7 (3+m)}-\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{4+m}}{e^7 (4+m)}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{5+m}}{e^7 (5+m)}-\frac {3 c^2 (2 c d-b e) (d+e x)^{6+m}}{e^7 (6+m)}+\frac {c^3 (d+e x)^{7+m}}{e^7 (7+m)} \] Output:

d^3*(-b*e+c*d)^3*(e*x+d)^(1+m)/e^7/(1+m)-3*d^2*(-b*e+c*d)^2*(-b*e+2*c*d)*( 
e*x+d)^(2+m)/e^7/(2+m)+3*d*(-b*e+c*d)*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)*(e*x+d 
)^(3+m)/e^7/(3+m)-(-b*e+2*c*d)*(b^2*e^2-10*b*c*d*e+10*c^2*d^2)*(e*x+d)^(4+ 
m)/e^7/(4+m)+3*c*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)*(e*x+d)^(5+m)/e^7/(5+m)-3*c 
^2*(-b*e+2*c*d)*(e*x+d)^(6+m)/e^7/(6+m)+c^3*(e*x+d)^(7+m)/e^7/(7+m)
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.88 \[ \int (d+e x)^m \left (b x+c x^2\right )^3 \, dx=\frac {(d+e x)^{1+m} \left (\frac {d^3 (c d-b e)^3}{1+m}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)}{2+m}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^2}{3+m}-\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^3}{4+m}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^4}{5+m}-\frac {3 c^2 (2 c d-b e) (d+e x)^5}{6+m}+\frac {c^3 (d+e x)^6}{7+m}\right )}{e^7} \] Input:

Integrate[(d + e*x)^m*(b*x + c*x^2)^3,x]
 

Output:

((d + e*x)^(1 + m)*((d^3*(c*d - b*e)^3)/(1 + m) - (3*d^2*(c*d - b*e)^2*(2* 
c*d - b*e)*(d + e*x))/(2 + m) + (3*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + 
b^2*e^2)*(d + e*x)^2)/(3 + m) - ((2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + 
b^2*e^2)*(d + e*x)^3)/(4 + m) + (3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d 
+ e*x)^4)/(5 + m) - (3*c^2*(2*c*d - b*e)*(d + e*x)^5)/(6 + m) + (c^3*(d + 
e*x)^6)/(7 + m)))/e^7
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1140, 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.

\(\displaystyle \int \left (b x+c x^2\right )^3 (d+e x)^m \, dx\)

\(\Big \downarrow \) 1140

\(\displaystyle \int \left (\frac {3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) (d+e x)^{m+2}}{e^6}+\frac {(2 c d-b e) \left (-b^2 e^2+10 b c d e-10 c^2 d^2\right ) (d+e x)^{m+3}}{e^6}+\frac {3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) (d+e x)^{m+4}}{e^6}-\frac {3 c^2 (2 c d-b e) (d+e x)^{m+5}}{e^6}+\frac {d^3 (c d-b e)^3 (d+e x)^m}{e^6}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{m+1}}{e^6}+\frac {c^3 (d+e x)^{m+6}}{e^6}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) (d+e x)^{m+3}}{e^7 (m+3)}-\frac {(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right ) (d+e x)^{m+4}}{e^7 (m+4)}+\frac {3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) (d+e x)^{m+5}}{e^7 (m+5)}-\frac {3 c^2 (2 c d-b e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac {d^3 (c d-b e)^3 (d+e x)^{m+1}}{e^7 (m+1)}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{m+2}}{e^7 (m+2)}+\frac {c^3 (d+e x)^{m+7}}{e^7 (m+7)}\)

Input:

Int[(d + e*x)^m*(b*x + c*x^2)^3,x]
 

Output:

(d^3*(c*d - b*e)^3*(d + e*x)^(1 + m))/(e^7*(1 + m)) - (3*d^2*(c*d - b*e)^2 
*(2*c*d - b*e)*(d + e*x)^(2 + m))/(e^7*(2 + m)) + (3*d*(c*d - b*e)*(5*c^2* 
d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(3 + m))/(e^7*(3 + m)) - ((2*c*d - b* 
e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*(d + e*x)^(4 + m))/(e^7*(4 + m)) + 
(3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(5 + m))/(e^7*(5 + m)) - 
(3*c^2*(2*c*d - b*e)*(d + e*x)^(6 + m))/(e^7*(6 + m)) + (c^3*(d + e*x)^(7 
+ m))/(e^7*(7 + m))
 

Defintions of rubi rules used

rule 1140

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x 
_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; 
FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(903\) vs. \(2(267)=534\).

Time = 0.81 (sec) , antiderivative size = 904, normalized size of antiderivative = 3.39

method	result	size
norman	\(\frac {c^{3} x^{7} {\mathrm e}^{m \ln \left (e x +d \right )}}{7+m}+\frac {\left (b^{3} e^{3} m^{3}+3 b^{2} c d \,e^{2} m^{3}+18 b^{3} e^{3} m^{2}+39 b^{2} c d \,e^{2} m^{2}-15 b \,c^{2} d^{2} e \,m^{2}+107 b^{3} e^{3} m +126 b^{2} c d \,e^{2} m -105 b \,c^{2} d^{2} e m +30 c^{3} d^{3} m +210 b^{3} e^{3}\right ) x^{4} {\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 {c^{2} \left (3 b e m +c d m +21 b e \right ) x^{6} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+13 m +42\right )}+\frac {m d \left (b^{3} e^{3} m^{3}+18 b^{3} e^{3} m^{2}-12 b^{2} c d \,e^{2} m^{2}+107 b^{3} e^{3} m -156 b^{2} c d \,e^{2} m +60 b \,c^{2} d^{2} e m +210 b^{3} e^{3}-504 d \,e^{2} b^{2} c +420 d^{2} e b \,c^{2}-120 d^{3} c^{3}\right ) x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{4} \left (m^{5}+25 m^{4}+245 m^{3}+1175 m^{2}+2754 m +2520\right )}-\frac {6 d^{4} \left (b^{3} e^{3} m^{3}+18 b^{3} e^{3} m^{2}-12 b^{2} c d \,e^{2} m^{2}+107 b^{3} e^{3} m -156 b^{2} c d \,e^{2} m +60 b \,c^{2} d^{2} e m +210 b^{3} e^{3}-504 d \,e^{2} b^{2} c +420 d^{2} e b \,c^{2}-120 d^{3} c^{3}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{7} \left (m^{7}+28 m^{6}+322 m^{5}+1960 m^{4}+6769 m^{3}+13132 m^{2}+13068 m +5040\right )}+\frac {3 \left (b^{2} e^{2} m^{2}+b c d e \,m^{2}+13 b^{2} e^{2} m +7 b c d e m -2 c^{2} d^{2} m +42 b^{2} e^{2}\right ) c \,x^{5} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+18 m^{2}+107 m +210\right )}+\frac {6 m \,d^{3} \left (b^{3} e^{3} m^{3}+18 b^{3} e^{3} m^{2}-12 b^{2} c d \,e^{2} m^{2}+107 b^{3} e^{3} m -156 b^{2} c d \,e^{2} m +60 b \,c^{2} d^{2} e m +210 b^{3} e^{3}-504 d \,e^{2} b^{2} c +420 d^{2} e b \,c^{2}-120 d^{3} c^{3}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{6} \left (m^{7}+28 m^{6}+322 m^{5}+1960 m^{4}+6769 m^{3}+13132 m^{2}+13068 m +5040\right )}-\frac {3 \left (b^{3} e^{3} m^{3}+18 b^{3} e^{3} m^{2}-12 b^{2} c d \,e^{2} m^{2}+107 b^{3} e^{3} m -156 b^{2} c d \,e^{2} m +60 b \,c^{2} d^{2} e m +210 b^{3} e^{3}-504 d \,e^{2} b^{2} c +420 d^{2} e b \,c^{2}-120 d^{3} c^{3}\right ) d^{2} m \,x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{5} \left (m^{6}+27 m^{5}+295 m^{4}+1665 m^{3}+5104 m^{2}+8028 m +5040\right )}\)	\(904\)
gosper	\(\text {Expression too large to display}\)	\(1528\)
orering	\(\text {Expression too large to display}\)	\(1552\)
risch	\(\text {Expression too large to display}\)	\(1745\)
parallelrisch	\(\text {Expression too large to display}\)	\(2517\)

Input:

int((e*x+d)^m*(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)
 

Output:

c^3/(7+m)*x^7*exp(m*ln(e*x+d))+(b^3*e^3*m^3+3*b^2*c*d*e^2*m^3+18*b^3*e^3*m 
^2+39*b^2*c*d*e^2*m^2-15*b*c^2*d^2*e*m^2+107*b^3*e^3*m+126*b^2*c*d*e^2*m-1 
05*b*c^2*d^2*e*m+30*c^3*d^3*m+210*b^3*e^3)/e^3/(m^4+22*m^3+179*m^2+638*m+8 
40)*x^4*exp(m*ln(e*x+d))+c^2*(3*b*e*m+c*d*m+21*b*e)/e/(m^2+13*m+42)*x^6*ex 
p(m*ln(e*x+d))+m*d*(b^3*e^3*m^3+18*b^3*e^3*m^2-12*b^2*c*d*e^2*m^2+107*b^3* 
e^3*m-156*b^2*c*d*e^2*m+60*b*c^2*d^2*e*m+210*b^3*e^3-504*b^2*c*d*e^2+420*b 
*c^2*d^2*e-120*c^3*d^3)/e^4/(m^5+25*m^4+245*m^3+1175*m^2+2754*m+2520)*x^3* 
exp(m*ln(e*x+d))-6*d^4*(b^3*e^3*m^3+18*b^3*e^3*m^2-12*b^2*c*d*e^2*m^2+107* 
b^3*e^3*m-156*b^2*c*d*e^2*m+60*b*c^2*d^2*e*m+210*b^3*e^3-504*b^2*c*d*e^2+4 
20*b*c^2*d^2*e-120*c^3*d^3)/e^7/(m^7+28*m^6+322*m^5+1960*m^4+6769*m^3+1313 
2*m^2+13068*m+5040)*exp(m*ln(e*x+d))+3*(b^2*e^2*m^2+b*c*d*e*m^2+13*b^2*e^2 
*m+7*b*c*d*e*m-2*c^2*d^2*m+42*b^2*e^2)/e^2*c/(m^3+18*m^2+107*m+210)*x^5*ex 
p(m*ln(e*x+d))+6/e^6*m*d^3*(b^3*e^3*m^3+18*b^3*e^3*m^2-12*b^2*c*d*e^2*m^2+ 
107*b^3*e^3*m-156*b^2*c*d*e^2*m+60*b*c^2*d^2*e*m+210*b^3*e^3-504*b^2*c*d*e 
^2+420*b*c^2*d^2*e-120*c^3*d^3)/(m^7+28*m^6+322*m^5+1960*m^4+6769*m^3+1313 
2*m^2+13068*m+5040)*x*exp(m*ln(e*x+d))-3*(b^3*e^3*m^3+18*b^3*e^3*m^2-12*b^ 
2*c*d*e^2*m^2+107*b^3*e^3*m-156*b^2*c*d*e^2*m+60*b*c^2*d^2*e*m+210*b^3*e^3 
-504*b^2*c*d*e^2+420*b*c^2*d^2*e-120*c^3*d^3)*d^2/e^5*m/(m^6+27*m^5+295*m^ 
4+1665*m^3+5104*m^2+8028*m+5040)*x^2*exp(m*ln(e*x+d))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1449 vs. \(2 (267) = 534\).

Time = 0.11 (sec) , antiderivative size = 1449, normalized size of antiderivative = 5.43 \[ \int (d+e x)^m \left (b x+c x^2\right )^3 \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^m*(c*x^2+b*x)^3,x, algorithm="fricas")
 

Output:

-(6*b^3*d^4*e^3*m^3 - 720*c^3*d^7 + 2520*b*c^2*d^6*e - 3024*b^2*c*d^5*e^2 
+ 1260*b^3*d^4*e^3 - (c^3*e^7*m^6 + 21*c^3*e^7*m^5 + 175*c^3*e^7*m^4 + 735 
*c^3*e^7*m^3 + 1624*c^3*e^7*m^2 + 1764*c^3*e^7*m + 720*c^3*e^7)*x^7 - (252 
0*b*c^2*e^7 + (c^3*d*e^6 + 3*b*c^2*e^7)*m^6 + 3*(5*c^3*d*e^6 + 22*b*c^2*e^ 
7)*m^5 + 5*(17*c^3*d*e^6 + 114*b*c^2*e^7)*m^4 + 15*(15*c^3*d*e^6 + 164*b*c 
^2*e^7)*m^3 + (274*c^3*d*e^6 + 5547*b*c^2*e^7)*m^2 + 6*(20*c^3*d*e^6 + 101 
9*b*c^2*e^7)*m)*x^6 - 3*(1008*b^2*c*e^7 + (b*c^2*d*e^6 + b^2*c*e^7)*m^6 - 
(2*c^3*d^2*e^5 - 17*b*c^2*d*e^6 - 23*b^2*c*e^7)*m^5 - (20*c^3*d^2*e^5 - 10 
5*b*c^2*d*e^6 - 207*b^2*c*e^7)*m^4 - 5*(14*c^3*d^2*e^5 - 59*b*c^2*d*e^6 - 
185*b^2*c*e^7)*m^3 - 2*(50*c^3*d^2*e^5 - 187*b*c^2*d*e^6 - 1072*b^2*c*e^7) 
*m^2 - 12*(4*c^3*d^2*e^5 - 14*b*c^2*d*e^6 - 201*b^2*c*e^7)*m)*x^5 - (1260* 
b^3*e^7 + (3*b^2*c*d*e^6 + b^3*e^7)*m^6 - 3*(5*b*c^2*d^2*e^5 - 19*b^2*c*d* 
e^6 - 8*b^3*e^7)*m^5 + (30*c^3*d^3*e^4 - 195*b*c^2*d^2*e^5 + 393*b^2*c*d*e 
^6 + 226*b^3*e^7)*m^4 + 3*(60*c^3*d^3*e^4 - 265*b*c^2*d^2*e^5 + 401*b^2*c* 
d*e^6 + 352*b^3*e^7)*m^3 + 5*(66*c^3*d^3*e^4 - 249*b*c^2*d^2*e^5 + 324*b^2 
*c*d*e^6 + 509*b^3*e^7)*m^2 + 18*(10*c^3*d^3*e^4 - 35*b*c^2*d^2*e^5 + 42*b 
^2*c*d*e^6 + 164*b^3*e^7)*m)*x^4 - (b^3*d*e^6*m^6 - 3*(4*b^2*c*d^2*e^5 - 7 
*b^3*d*e^6)*m^5 + (60*b*c^2*d^3*e^4 - 192*b^2*c*d^2*e^5 + 163*b^3*d*e^6)*m 
^4 - 3*(40*c^3*d^4*e^3 - 200*b*c^2*d^3*e^4 + 332*b^2*c*d^2*e^5 - 189*b^3*d 
*e^6)*m^3 - 4*(90*c^3*d^4*e^3 - 345*b*c^2*d^3*e^4 + 456*b^2*c*d^2*e^5 -...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 21005 vs. \(2 (250) = 500\).

Time = 4.14 (sec) , antiderivative size = 21005, normalized size of antiderivative = 78.67 \[ \int (d+e x)^m \left (b x+c x^2\right )^3 \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)**m*(c*x**2+b*x)**3,x)
 

Output:

Piecewise((d**m*(b**3*x**4/4 + 3*b**2*c*x**5/5 + b*c**2*x**6/2 + c**3*x**7 
/7), Eq(e, 0)), (-b**3*d**3*e**3/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d** 
4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x** 
5 + 60*e**13*x**6) - 6*b**3*d**2*e**4*x/(60*d**6*e**7 + 360*d**5*e**8*x + 
900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e* 
*12*x**5 + 60*e**13*x**6) - 15*b**3*d*e**5*x**2/(60*d**6*e**7 + 360*d**5*e 
**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 
360*d*e**12*x**5 + 60*e**13*x**6) - 20*b**3*e**6*x**3/(60*d**6*e**7 + 360* 
d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x 
**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 6*b**2*c*d**4*e**2/(60*d**6*e**7 
 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2* 
e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 36*b**2*c*d**3*e**3*x/(60 
*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 
 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 90*b**2*c*d**2* 
e**4*x**2/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3 
*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 12 
0*b**2*c*d*e**5*x**3/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 
+ 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13 
*x**6) - 90*b**2*c*e**6*x**4/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e* 
*9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 669 vs. \(2 (267) = 534\).

Time = 0.05 (sec) , antiderivative size = 669, normalized size of antiderivative = 2.51 \[ \int (d+e x)^m \left (b x+c x^2\right )^3 \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^m*(c*x^2+b*x)^3,x, algorithm="maxima")
 

Output:

((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*b^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*b^2*c/( 
(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) + 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 + 50*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*b*c^2/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 17 
64*m + 720)*e^6) + ((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*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 + 3 
60*(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 + 5040)*e^7 
)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2539 vs. \(2 (267) = 534\).

Time = 0.15 (sec) , antiderivative size = 2539, normalized size of antiderivative = 9.51 \[ \int (d+e x)^m \left (b x+c x^2\right )^3 \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^m*(c*x^2+b*x)^3,x, algorithm="giac")
 

Output:

((e*x + d)^m*c^3*e^7*m^6*x^7 + (e*x + d)^m*c^3*d*e^6*m^6*x^6 + 3*(e*x + d) 
^m*b*c^2*e^7*m^6*x^6 + 21*(e*x + d)^m*c^3*e^7*m^5*x^7 + 3*(e*x + d)^m*b*c^ 
2*d*e^6*m^6*x^5 + 3*(e*x + d)^m*b^2*c*e^7*m^6*x^5 + 15*(e*x + d)^m*c^3*d*e 
^6*m^5*x^6 + 66*(e*x + d)^m*b*c^2*e^7*m^5*x^6 + 175*(e*x + d)^m*c^3*e^7*m^ 
4*x^7 + 3*(e*x + d)^m*b^2*c*d*e^6*m^6*x^4 + (e*x + d)^m*b^3*e^7*m^6*x^4 - 
6*(e*x + d)^m*c^3*d^2*e^5*m^5*x^5 + 51*(e*x + d)^m*b*c^2*d*e^6*m^5*x^5 + 6 
9*(e*x + d)^m*b^2*c*e^7*m^5*x^5 + 85*(e*x + d)^m*c^3*d*e^6*m^4*x^6 + 570*( 
e*x + d)^m*b*c^2*e^7*m^4*x^6 + 735*(e*x + d)^m*c^3*e^7*m^3*x^7 + (e*x + d) 
^m*b^3*d*e^6*m^6*x^3 - 15*(e*x + d)^m*b*c^2*d^2*e^5*m^5*x^4 + 57*(e*x + d) 
^m*b^2*c*d*e^6*m^5*x^4 + 24*(e*x + d)^m*b^3*e^7*m^5*x^4 - 60*(e*x + d)^m*c 
^3*d^2*e^5*m^4*x^5 + 315*(e*x + d)^m*b*c^2*d*e^6*m^4*x^5 + 621*(e*x + d)^m 
*b^2*c*e^7*m^4*x^5 + 225*(e*x + d)^m*c^3*d*e^6*m^3*x^6 + 2460*(e*x + d)^m* 
b*c^2*e^7*m^3*x^6 + 1624*(e*x + d)^m*c^3*e^7*m^2*x^7 - 12*(e*x + d)^m*b^2* 
c*d^2*e^5*m^5*x^3 + 21*(e*x + d)^m*b^3*d*e^6*m^5*x^3 + 30*(e*x + d)^m*c^3* 
d^3*e^4*m^4*x^4 - 195*(e*x + d)^m*b*c^2*d^2*e^5*m^4*x^4 + 393*(e*x + d)^m* 
b^2*c*d*e^6*m^4*x^4 + 226*(e*x + d)^m*b^3*e^7*m^4*x^4 - 210*(e*x + d)^m*c^ 
3*d^2*e^5*m^3*x^5 + 885*(e*x + d)^m*b*c^2*d*e^6*m^3*x^5 + 2775*(e*x + d)^m 
*b^2*c*e^7*m^3*x^5 + 274*(e*x + d)^m*c^3*d*e^6*m^2*x^6 + 5547*(e*x + d)^m* 
b*c^2*e^7*m^2*x^6 + 1764*(e*x + d)^m*c^3*e^7*m*x^7 - 3*(e*x + d)^m*b^3*d^2 
*e^5*m^5*x^2 + 60*(e*x + d)^m*b*c^2*d^3*e^4*m^4*x^3 - 192*(e*x + d)^m*b...
 

Mupad [B] (verification not implemented)

Time = 5.65 (sec) , antiderivative size = 1085, normalized size of antiderivative = 4.06 \[ \int (d+e x)^m \left (b x+c x^2\right )^3 \, dx =\text {Too large to display} \] Input:

int((b*x + c*x^2)^3*(d + e*x)^m,x)
 

Output:

(c^3*x^7*(d + e*x)^m*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 
 + 720))/(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m 
^7 + 5040) - (6*d^4*(d + e*x)^m*(210*b^3*e^3 - 120*c^3*d^3 + 107*b^3*e^3*m 
 + 18*b^3*e^3*m^2 + b^3*e^3*m^3 + 420*b*c^2*d^2*e - 504*b^2*c*d*e^2 + 60*b 
*c^2*d^2*e*m - 156*b^2*c*d*e^2*m - 12*b^2*c*d*e^2*m^2))/(e^7*(13068*m + 13 
132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (x^4*(d 
+ e*x)^m*(11*m + 6*m^2 + m^3 + 6)*(210*b^3*e^3 + 107*b^3*e^3*m + 30*c^3*d^ 
3*m + 18*b^3*e^3*m^2 + b^3*e^3*m^3 - 105*b*c^2*d^2*e*m + 126*b^2*c*d*e^2*m 
 - 15*b*c^2*d^2*e*m^2 + 39*b^2*c*d*e^2*m^2 + 3*b^2*c*d*e^2*m^3))/(e^3*(130 
68*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + 
 (3*c*x^5*(d + e*x)^m*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)*(42*b^2*e^2 + 13 
*b^2*e^2*m - 2*c^2*d^2*m + b^2*e^2*m^2 + 7*b*c*d*e*m + b*c*d*e*m^2))/(e^2* 
(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040 
)) + (6*d^3*m*x*(d + e*x)^m*(210*b^3*e^3 - 120*c^3*d^3 + 107*b^3*e^3*m + 1 
8*b^3*e^3*m^2 + b^3*e^3*m^3 + 420*b*c^2*d^2*e - 504*b^2*c*d*e^2 + 60*b*c^2 
*d^2*e*m - 156*b^2*c*d*e^2*m - 12*b^2*c*d*e^2*m^2))/(e^6*(13068*m + 13132* 
m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (c^2*x^6*(d 
+ e*x)^m*(21*b*e + 3*b*e*m + c*d*m)*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m 
^5 + 120))/(e*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^ 
6 + m^7 + 5040)) + (d*m*x^3*(d + e*x)^m*(3*m + m^2 + 2)*(210*b^3*e^3 - ...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1739, normalized size of antiderivative = 6.51 \[ \int (d+e x)^m \left (b x+c x^2\right )^3 \, dx =\text {Too large to display} \] Input:

int((e*x+d)^m*(c*x^2+b*x)^3,x)
 

Output:

((d + e*x)**m*( - 6*b**3*d**4*e**3*m**3 - 108*b**3*d**4*e**3*m**2 - 642*b* 
*3*d**4*e**3*m - 1260*b**3*d**4*e**3 + 6*b**3*d**3*e**4*m**4*x + 108*b**3* 
d**3*e**4*m**3*x + 642*b**3*d**3*e**4*m**2*x + 1260*b**3*d**3*e**4*m*x - 3 
*b**3*d**2*e**5*m**5*x**2 - 57*b**3*d**2*e**5*m**4*x**2 - 375*b**3*d**2*e* 
*5*m**3*x**2 - 951*b**3*d**2*e**5*m**2*x**2 - 630*b**3*d**2*e**5*m*x**2 + 
b**3*d*e**6*m**6*x**3 + 21*b**3*d*e**6*m**5*x**3 + 163*b**3*d*e**6*m**4*x* 
*3 + 567*b**3*d*e**6*m**3*x**3 + 844*b**3*d*e**6*m**2*x**3 + 420*b**3*d*e* 
*6*m*x**3 + b**3*e**7*m**6*x**4 + 24*b**3*e**7*m**5*x**4 + 226*b**3*e**7*m 
**4*x**4 + 1056*b**3*e**7*m**3*x**4 + 2545*b**3*e**7*m**2*x**4 + 2952*b**3 
*e**7*m*x**4 + 1260*b**3*e**7*x**4 + 72*b**2*c*d**5*e**2*m**2 + 936*b**2*c 
*d**5*e**2*m + 3024*b**2*c*d**5*e**2 - 72*b**2*c*d**4*e**3*m**3*x - 936*b* 
*2*c*d**4*e**3*m**2*x - 3024*b**2*c*d**4*e**3*m*x + 36*b**2*c*d**3*e**4*m* 
*4*x**2 + 504*b**2*c*d**3*e**4*m**3*x**2 + 1980*b**2*c*d**3*e**4*m**2*x**2 
 + 1512*b**2*c*d**3*e**4*m*x**2 - 12*b**2*c*d**2*e**5*m**5*x**3 - 192*b**2 
*c*d**2*e**5*m**4*x**3 - 996*b**2*c*d**2*e**5*m**3*x**3 - 1824*b**2*c*d**2 
*e**5*m**2*x**3 - 1008*b**2*c*d**2*e**5*m*x**3 + 3*b**2*c*d*e**6*m**6*x**4 
 + 57*b**2*c*d*e**6*m**5*x**4 + 393*b**2*c*d*e**6*m**4*x**4 + 1203*b**2*c* 
d*e**6*m**3*x**4 + 1620*b**2*c*d*e**6*m**2*x**4 + 756*b**2*c*d*e**6*m*x**4 
 + 3*b**2*c*e**7*m**6*x**5 + 69*b**2*c*e**7*m**5*x**5 + 621*b**2*c*e**7*m* 
*4*x**5 + 2775*b**2*c*e**7*m**3*x**5 + 6432*b**2*c*e**7*m**2*x**5 + 723...