Integrand size = 22, antiderivative size = 121 \[ \int (e x)^m (c+d x) \left (a x+b x^2\right )^3 \, dx=\frac {a^3 c (e x)^{4+m}}{e^4 (4+m)}+\frac {a^2 (3 b c+a d) (e x)^{5+m}}{e^5 (5+m)}+\frac {3 a b (b c+a d) (e x)^{6+m}}{e^6 (6+m)}+\frac {b^2 (b c+3 a d) (e x)^{7+m}}{e^7 (7+m)}+\frac {b^3 d (e x)^{8+m}}{e^8 (8+m)} \] Output:
a^3*c*(e*x)^(4+m)/e^4/(4+m)+a^2*(a*d+3*b*c)*(e*x)^(5+m)/e^5/(5+m)+3*a*b*(a *d+b*c)*(e*x)^(6+m)/e^6/(6+m)+b^2*(3*a*d+b*c)*(e*x)^(7+m)/e^7/(7+m)+b^3*d* (e*x)^(8+m)/e^8/(8+m)
Time = 0.14 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.74 \[ \int (e x)^m (c+d x) \left (a x+b x^2\right )^3 \, dx=\frac {x^4 (e x)^m \left (d (a+b x)^4+(-a d (4+m)+b c (8+m)) \left (\frac {a^3}{4+m}+\frac {3 a^2 b x}{5+m}+\frac {3 a b^2 x^2}{6+m}+\frac {b^3 x^3}{7+m}\right )\right )}{b (8+m)} \] Input:
Integrate[(e*x)^m*(c + d*x)*(a*x + b*x^2)^3,x]
Output:
(x^4*(e*x)^m*(d*(a + b*x)^4 + (-(a*d*(4 + m)) + b*c*(8 + m))*(a^3/(4 + m) + (3*a^2*b*x)/(5 + m) + (3*a*b^2*x^2)/(6 + m) + (b^3*x^3)/(7 + m))))/(b*(8 + m))
Time = 0.47 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {9, 85, 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.
| \(\displaystyle \int \left (a x+b x^2\right )^3 (c+d x) (e x)^m \, dx\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {\int (e x)^{m+3} (a+b x)^3 (c+d x)dx}{e^3}\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \frac {\int \left (a^3 c (e x)^{m+3}+\frac {a^2 (3 b c+a d) (e x)^{m+4}}{e}+\frac {3 a b (b c+a d) (e x)^{m+5}}{e^2}+\frac {b^2 (b c+3 a d) (e x)^{m+6}}{e^3}+\frac {b^3 d (e x)^{m+7}}{e^4}\right )dx}{e^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a^3 c (e x)^{m+4}}{e (m+4)}+\frac {a^2 (e x)^{m+5} (a d+3 b c)}{e^2 (m+5)}+\frac {b^2 (e x)^{m+7} (3 a d+b c)}{e^4 (m+7)}+\frac {3 a b (e x)^{m+6} (a d+b c)}{e^3 (m+6)}+\frac {b^3 d (e x)^{m+8}}{e^5 (m+8)}}{e^3}\) |
Input:
Int[(e*x)^m*(c + d*x)*(a*x + b*x^2)^3,x]
Output:
((a^3*c*(e*x)^(4 + m))/(e*(4 + m)) + (a^2*(3*b*c + a*d)*(e*x)^(5 + m))/(e^ 2*(5 + m)) + (3*a*b*(b*c + a*d)*(e*x)^(6 + m))/(e^3*(6 + m)) + (b^2*(b*c + 3*a*d)*(e*x)^(7 + m))/(e^4*(7 + m)) + (b^3*d*(e*x)^(8 + m))/(e^5*(8 + m)) )/e^3
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a* f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])
Time = 1.06 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.01
| method | result | size |
| norman | \(\frac {a^{2} \left (a d +3 b c \right ) x^{5} {\mathrm e}^{m \ln \left (e x \right )}}{5+m}+\frac {b^{2} \left (3 a d +b c \right ) x^{7} {\mathrm e}^{m \ln \left (e x \right )}}{7+m}+\frac {b^{3} d \,x^{8} {\mathrm e}^{m \ln \left (e x \right )}}{8+m}+\frac {c \,a^{3} x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}+\frac {3 a b \left (a d +b c \right ) x^{6} {\mathrm e}^{m \ln \left (e x \right )}}{6+m}\) | \(122\) |
| gosper | \(\frac {\left (e x \right )^{m} \left (b^{3} d \,m^{4} x^{4}+3 a \,b^{2} d \,m^{4} x^{3}+b^{3} c \,m^{4} x^{3}+22 b^{3} d \,m^{3} x^{4}+3 a^{2} b d \,m^{4} x^{2}+3 a \,b^{2} c \,m^{4} x^{2}+69 a \,b^{2} d \,m^{3} x^{3}+23 b^{3} c \,m^{3} x^{3}+179 b^{3} d \,m^{2} x^{4}+a^{3} d \,m^{4} x +3 a^{2} b c \,m^{4} x +72 a^{2} b d \,m^{3} x^{2}+72 a \,b^{2} c \,m^{3} x^{2}+582 a \,b^{2} d \,m^{2} x^{3}+194 b^{3} c \,m^{2} x^{3}+638 m \,x^{4} b^{3} d +a^{3} c \,m^{4}+25 a^{3} d \,m^{3} x +75 a^{2} b c \,m^{3} x +633 a^{2} b d \,m^{2} x^{2}+633 a \,b^{2} c \,m^{2} x^{2}+2136 a \,b^{2} d m \,x^{3}+712 b^{3} c m \,x^{3}+840 b^{3} d \,x^{4}+26 a^{3} c \,m^{3}+230 a^{3} d \,m^{2} x +690 a^{2} b c \,m^{2} x +2412 a^{2} b d m \,x^{2}+2412 a \,b^{2} c m \,x^{2}+2880 a \,b^{2} d \,x^{3}+960 b^{3} c \,x^{3}+251 a^{3} c \,m^{2}+920 a^{3} d m x +2760 a^{2} b c m x +3360 a^{2} b d \,x^{2}+3360 a \,b^{2} c \,x^{2}+1066 a^{3} c m +1344 a^{3} d x +4032 a^{2} b c x +1680 c \,a^{3}\right ) x^{4}}{\left (8+m \right ) \left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right )}\) | \(457\) |
| risch | \(\frac {\left (e x \right )^{m} \left (b^{3} d \,m^{4} x^{4}+3 a \,b^{2} d \,m^{4} x^{3}+b^{3} c \,m^{4} x^{3}+22 b^{3} d \,m^{3} x^{4}+3 a^{2} b d \,m^{4} x^{2}+3 a \,b^{2} c \,m^{4} x^{2}+69 a \,b^{2} d \,m^{3} x^{3}+23 b^{3} c \,m^{3} x^{3}+179 b^{3} d \,m^{2} x^{4}+a^{3} d \,m^{4} x +3 a^{2} b c \,m^{4} x +72 a^{2} b d \,m^{3} x^{2}+72 a \,b^{2} c \,m^{3} x^{2}+582 a \,b^{2} d \,m^{2} x^{3}+194 b^{3} c \,m^{2} x^{3}+638 m \,x^{4} b^{3} d +a^{3} c \,m^{4}+25 a^{3} d \,m^{3} x +75 a^{2} b c \,m^{3} x +633 a^{2} b d \,m^{2} x^{2}+633 a \,b^{2} c \,m^{2} x^{2}+2136 a \,b^{2} d m \,x^{3}+712 b^{3} c m \,x^{3}+840 b^{3} d \,x^{4}+26 a^{3} c \,m^{3}+230 a^{3} d \,m^{2} x +690 a^{2} b c \,m^{2} x +2412 a^{2} b d m \,x^{2}+2412 a \,b^{2} c m \,x^{2}+2880 a \,b^{2} d \,x^{3}+960 b^{3} c \,x^{3}+251 a^{3} c \,m^{2}+920 a^{3} d m x +2760 a^{2} b c m x +3360 a^{2} b d \,x^{2}+3360 a \,b^{2} c \,x^{2}+1066 a^{3} c m +1344 a^{3} d x +4032 a^{2} b c x +1680 c \,a^{3}\right ) x^{4}}{\left (8+m \right ) \left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right )}\) | \(457\) |
| orering | \(\frac {\left (b^{3} d \,m^{4} x^{4}+3 a \,b^{2} d \,m^{4} x^{3}+b^{3} c \,m^{4} x^{3}+22 b^{3} d \,m^{3} x^{4}+3 a^{2} b d \,m^{4} x^{2}+3 a \,b^{2} c \,m^{4} x^{2}+69 a \,b^{2} d \,m^{3} x^{3}+23 b^{3} c \,m^{3} x^{3}+179 b^{3} d \,m^{2} x^{4}+a^{3} d \,m^{4} x +3 a^{2} b c \,m^{4} x +72 a^{2} b d \,m^{3} x^{2}+72 a \,b^{2} c \,m^{3} x^{2}+582 a \,b^{2} d \,m^{2} x^{3}+194 b^{3} c \,m^{2} x^{3}+638 m \,x^{4} b^{3} d +a^{3} c \,m^{4}+25 a^{3} d \,m^{3} x +75 a^{2} b c \,m^{3} x +633 a^{2} b d \,m^{2} x^{2}+633 a \,b^{2} c \,m^{2} x^{2}+2136 a \,b^{2} d m \,x^{3}+712 b^{3} c m \,x^{3}+840 b^{3} d \,x^{4}+26 a^{3} c \,m^{3}+230 a^{3} d \,m^{2} x +690 a^{2} b c \,m^{2} x +2412 a^{2} b d m \,x^{2}+2412 a \,b^{2} c m \,x^{2}+2880 a \,b^{2} d \,x^{3}+960 b^{3} c \,x^{3}+251 a^{3} c \,m^{2}+920 a^{3} d m x +2760 a^{2} b c m x +3360 a^{2} b d \,x^{2}+3360 a \,b^{2} c \,x^{2}+1066 a^{3} c m +1344 a^{3} d x +4032 a^{2} b c x +1680 c \,a^{3}\right ) x \left (e x \right )^{m} \left (b \,x^{2}+a x \right )^{3}}{\left (8+m \right ) \left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (b x +a \right )^{3}}\) | \(473\) |
| parallelrisch | \(\frac {960 x^{7} \left (e x \right )^{m} b^{3} c +690 x^{5} \left (e x \right )^{m} a^{2} b c \,m^{2}+2760 x^{5} \left (e x \right )^{m} a^{2} b c m +23 x^{7} \left (e x \right )^{m} b^{3} c \,m^{3}+638 x^{8} \left (e x \right )^{m} b^{3} d m +194 x^{7} \left (e x \right )^{m} b^{3} c \,m^{2}+x^{5} \left (e x \right )^{m} a^{3} d \,m^{4}+712 x^{7} \left (e x \right )^{m} b^{3} c m +1344 x^{5} \left (e x \right )^{m} a^{3} d +1680 x^{4} \left (e x \right )^{m} a^{3} c +3 x^{7} \left (e x \right )^{m} a \,b^{2} d \,m^{4}+69 x^{7} \left (e x \right )^{m} a \,b^{2} d \,m^{3}+3 x^{6} \left (e x \right )^{m} a^{2} b d \,m^{4}+3 x^{6} \left (e x \right )^{m} a \,b^{2} c \,m^{4}+582 x^{7} \left (e x \right )^{m} a \,b^{2} d \,m^{2}+633 x^{6} \left (e x \right )^{m} a^{2} b d \,m^{2}+633 x^{6} \left (e x \right )^{m} a \,b^{2} c \,m^{2}+75 x^{5} \left (e x \right )^{m} a^{2} b c \,m^{3}+2412 x^{6} \left (e x \right )^{m} a^{2} b d m +2412 x^{6} \left (e x \right )^{m} a \,b^{2} c m +251 x^{4} \left (e x \right )^{m} a^{3} c \,m^{2}+4032 x^{5} \left (e x \right )^{m} a^{2} b c +72 x^{6} \left (e x \right )^{m} a^{2} b d \,m^{3}+72 x^{6} \left (e x \right )^{m} a \,b^{2} c \,m^{3}+3 x^{5} \left (e x \right )^{m} a^{2} b c \,m^{4}+2136 x^{7} \left (e x \right )^{m} a \,b^{2} d m +x^{8} \left (e x \right )^{m} b^{3} d \,m^{4}+22 x^{8} \left (e x \right )^{m} b^{3} d \,m^{3}+x^{7} \left (e x \right )^{m} b^{3} c \,m^{4}+179 x^{8} \left (e x \right )^{m} b^{3} d \,m^{2}+1066 x^{4} \left (e x \right )^{m} a^{3} c m +3360 x^{6} \left (e x \right )^{m} a^{2} b d +3360 x^{6} \left (e x \right )^{m} a \,b^{2} c +25 x^{5} \left (e x \right )^{m} a^{3} d \,m^{3}+x^{4} \left (e x \right )^{m} a^{3} c \,m^{4}+2880 x^{7} \left (e x \right )^{m} a \,b^{2} d +230 x^{5} \left (e x \right )^{m} a^{3} d \,m^{2}+26 x^{4} \left (e x \right )^{m} a^{3} c \,m^{3}+920 x^{5} \left (e x \right )^{m} a^{3} d m +840 x^{8} \left (e x \right )^{m} b^{3} d}{\left (8+m \right ) \left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right )}\) | \(684\) |
Input:
int((e*x)^m*(d*x+c)*(b*x^2+a*x)^3,x,method=_RETURNVERBOSE)
Output:
a^2*(a*d+3*b*c)/(5+m)*x^5*exp(m*ln(e*x))+b^2*(3*a*d+b*c)/(7+m)*x^7*exp(m*l n(e*x))+b^3*d/(8+m)*x^8*exp(m*ln(e*x))+c*a^3/(4+m)*x^4*exp(m*ln(e*x))+3*a* b*(a*d+b*c)/(6+m)*x^6*exp(m*ln(e*x))
Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (121) = 242\).
Time = 0.08 (sec) , antiderivative size = 383, normalized size of antiderivative = 3.17 \[ \int (e x)^m (c+d x) \left (a x+b x^2\right )^3 \, dx=\frac {{\left ({\left (b^{3} d m^{4} + 22 \, b^{3} d m^{3} + 179 \, b^{3} d m^{2} + 638 \, b^{3} d m + 840 \, b^{3} d\right )} x^{8} + {\left ({\left (b^{3} c + 3 \, a b^{2} d\right )} m^{4} + 960 \, b^{3} c + 2880 \, a b^{2} d + 23 \, {\left (b^{3} c + 3 \, a b^{2} d\right )} m^{3} + 194 \, {\left (b^{3} c + 3 \, a b^{2} d\right )} m^{2} + 712 \, {\left (b^{3} c + 3 \, a b^{2} d\right )} m\right )} x^{7} + 3 \, {\left ({\left (a b^{2} c + a^{2} b d\right )} m^{4} + 1120 \, a b^{2} c + 1120 \, a^{2} b d + 24 \, {\left (a b^{2} c + a^{2} b d\right )} m^{3} + 211 \, {\left (a b^{2} c + a^{2} b d\right )} m^{2} + 804 \, {\left (a b^{2} c + a^{2} b d\right )} m\right )} x^{6} + {\left ({\left (3 \, a^{2} b c + a^{3} d\right )} m^{4} + 4032 \, a^{2} b c + 1344 \, a^{3} d + 25 \, {\left (3 \, a^{2} b c + a^{3} d\right )} m^{3} + 230 \, {\left (3 \, a^{2} b c + a^{3} d\right )} m^{2} + 920 \, {\left (3 \, a^{2} b c + a^{3} d\right )} m\right )} x^{5} + {\left (a^{3} c m^{4} + 26 \, a^{3} c m^{3} + 251 \, a^{3} c m^{2} + 1066 \, a^{3} c m + 1680 \, a^{3} c\right )} x^{4}\right )} \left (e x\right )^{m}}{m^{5} + 30 \, m^{4} + 355 \, m^{3} + 2070 \, m^{2} + 5944 \, m + 6720} \] Input:
integrate((e*x)^m*(d*x+c)*(b*x^2+a*x)^3,x, algorithm="fricas")
Output:
((b^3*d*m^4 + 22*b^3*d*m^3 + 179*b^3*d*m^2 + 638*b^3*d*m + 840*b^3*d)*x^8 + ((b^3*c + 3*a*b^2*d)*m^4 + 960*b^3*c + 2880*a*b^2*d + 23*(b^3*c + 3*a*b^ 2*d)*m^3 + 194*(b^3*c + 3*a*b^2*d)*m^2 + 712*(b^3*c + 3*a*b^2*d)*m)*x^7 + 3*((a*b^2*c + a^2*b*d)*m^4 + 1120*a*b^2*c + 1120*a^2*b*d + 24*(a*b^2*c + a ^2*b*d)*m^3 + 211*(a*b^2*c + a^2*b*d)*m^2 + 804*(a*b^2*c + a^2*b*d)*m)*x^6 + ((3*a^2*b*c + a^3*d)*m^4 + 4032*a^2*b*c + 1344*a^3*d + 25*(3*a^2*b*c + a^3*d)*m^3 + 230*(3*a^2*b*c + a^3*d)*m^2 + 920*(3*a^2*b*c + a^3*d)*m)*x^5 + (a^3*c*m^4 + 26*a^3*c*m^3 + 251*a^3*c*m^2 + 1066*a^3*c*m + 1680*a^3*c)*x ^4)*(e*x)^m/(m^5 + 30*m^4 + 355*m^3 + 2070*m^2 + 5944*m + 6720)
Leaf count of result is larger than twice the leaf count of optimal. 2111 vs. \(2 (112) = 224\).
Time = 0.58 (sec) , antiderivative size = 2111, normalized size of antiderivative = 17.45 \[ \int (e x)^m (c+d x) \left (a x+b x^2\right )^3 \, dx=\text {Too large to display} \] Input:
integrate((e*x)**m*(d*x+c)*(b*x**2+a*x)**3,x)
Output:
Piecewise(((-a**3*c/(4*x**4) - a**3*d/(3*x**3) - a**2*b*c/x**3 - 3*a**2*b* d/(2*x**2) - 3*a*b**2*c/(2*x**2) - 3*a*b**2*d/x - b**3*c/x + b**3*d*log(x) )/e**8, Eq(m, -8)), ((-a**3*c/(3*x**3) - a**3*d/(2*x**2) - 3*a**2*b*c/(2*x **2) - 3*a**2*b*d/x - 3*a*b**2*c/x + 3*a*b**2*d*log(x) + b**3*c*log(x) + b **3*d*x)/e**7, Eq(m, -7)), ((-a**3*c/(2*x**2) - a**3*d/x - 3*a**2*b*c/x + 3*a**2*b*d*log(x) + 3*a*b**2*c*log(x) + 3*a*b**2*d*x + b**3*c*x + b**3*d*x **2/2)/e**6, Eq(m, -6)), ((-a**3*c/x + a**3*d*log(x) + 3*a**2*b*c*log(x) + 3*a**2*b*d*x + 3*a*b**2*c*x + 3*a*b**2*d*x**2/2 + b**3*c*x**2/2 + b**3*d* x**3/3)/e**5, Eq(m, -5)), ((a**3*c*log(x) + a**3*d*x + 3*a**2*b*c*x + 3*a* *2*b*d*x**2/2 + 3*a*b**2*c*x**2/2 + a*b**2*d*x**3 + b**3*c*x**3/3 + b**3*d *x**4/4)/e**4, Eq(m, -4)), (a**3*c*m**4*x**4*(e*x)**m/(m**5 + 30*m**4 + 35 5*m**3 + 2070*m**2 + 5944*m + 6720) + 26*a**3*c*m**3*x**4*(e*x)**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 251*a**3*c*m**2*x**4*(e *x)**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 1066*a**3 *c*m*x**4*(e*x)**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 1680*a**3*c*x**4*(e*x)**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944 *m + 6720) + a**3*d*m**4*x**5*(e*x)**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m **2 + 5944*m + 6720) + 25*a**3*d*m**3*x**5*(e*x)**m/(m**5 + 30*m**4 + 355* m**3 + 2070*m**2 + 5944*m + 6720) + 230*a**3*d*m**2*x**5*(e*x)**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 920*a**3*d*m*x**5*(e*...
Time = 0.05 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.33 \[ \int (e x)^m (c+d x) \left (a x+b x^2\right )^3 \, dx=\frac {b^{3} d e^{m} x^{8} x^{m}}{m + 8} + \frac {b^{3} c e^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, a b^{2} d e^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, a b^{2} c e^{m} x^{6} x^{m}}{m + 6} + \frac {3 \, a^{2} b d e^{m} x^{6} x^{m}}{m + 6} + \frac {3 \, a^{2} b c e^{m} x^{5} x^{m}}{m + 5} + \frac {a^{3} d e^{m} x^{5} x^{m}}{m + 5} + \frac {a^{3} c e^{m} x^{4} x^{m}}{m + 4} \] Input:
integrate((e*x)^m*(d*x+c)*(b*x^2+a*x)^3,x, algorithm="maxima")
Output:
b^3*d*e^m*x^8*x^m/(m + 8) + b^3*c*e^m*x^7*x^m/(m + 7) + 3*a*b^2*d*e^m*x^7* x^m/(m + 7) + 3*a*b^2*c*e^m*x^6*x^m/(m + 6) + 3*a^2*b*d*e^m*x^6*x^m/(m + 6 ) + 3*a^2*b*c*e^m*x^5*x^m/(m + 5) + a^3*d*e^m*x^5*x^m/(m + 5) + a^3*c*e^m* x^4*x^m/(m + 4)
Leaf count of result is larger than twice the leaf count of optimal. 683 vs. \(2 (121) = 242\).
Time = 0.14 (sec) , antiderivative size = 683, normalized size of antiderivative = 5.64 \[ \int (e x)^m (c+d x) \left (a x+b x^2\right )^3 \, dx =\text {Too large to display} \] Input:
integrate((e*x)^m*(d*x+c)*(b*x^2+a*x)^3,x, algorithm="giac")
Output:
((e*x)^m*b^3*d*m^4*x^8 + (e*x)^m*b^3*c*m^4*x^7 + 3*(e*x)^m*a*b^2*d*m^4*x^7 + 22*(e*x)^m*b^3*d*m^3*x^8 + 3*(e*x)^m*a*b^2*c*m^4*x^6 + 3*(e*x)^m*a^2*b* d*m^4*x^6 + 23*(e*x)^m*b^3*c*m^3*x^7 + 69*(e*x)^m*a*b^2*d*m^3*x^7 + 179*(e *x)^m*b^3*d*m^2*x^8 + 3*(e*x)^m*a^2*b*c*m^4*x^5 + (e*x)^m*a^3*d*m^4*x^5 + 72*(e*x)^m*a*b^2*c*m^3*x^6 + 72*(e*x)^m*a^2*b*d*m^3*x^6 + 194*(e*x)^m*b^3* c*m^2*x^7 + 582*(e*x)^m*a*b^2*d*m^2*x^7 + 638*(e*x)^m*b^3*d*m*x^8 + (e*x)^ m*a^3*c*m^4*x^4 + 75*(e*x)^m*a^2*b*c*m^3*x^5 + 25*(e*x)^m*a^3*d*m^3*x^5 + 633*(e*x)^m*a*b^2*c*m^2*x^6 + 633*(e*x)^m*a^2*b*d*m^2*x^6 + 712*(e*x)^m*b^ 3*c*m*x^7 + 2136*(e*x)^m*a*b^2*d*m*x^7 + 840*(e*x)^m*b^3*d*x^8 + 26*(e*x)^ m*a^3*c*m^3*x^4 + 690*(e*x)^m*a^2*b*c*m^2*x^5 + 230*(e*x)^m*a^3*d*m^2*x^5 + 2412*(e*x)^m*a*b^2*c*m*x^6 + 2412*(e*x)^m*a^2*b*d*m*x^6 + 960*(e*x)^m*b^ 3*c*x^7 + 2880*(e*x)^m*a*b^2*d*x^7 + 251*(e*x)^m*a^3*c*m^2*x^4 + 2760*(e*x )^m*a^2*b*c*m*x^5 + 920*(e*x)^m*a^3*d*m*x^5 + 3360*(e*x)^m*a*b^2*c*x^6 + 3 360*(e*x)^m*a^2*b*d*x^6 + 1066*(e*x)^m*a^3*c*m*x^4 + 4032*(e*x)^m*a^2*b*c* x^5 + 1344*(e*x)^m*a^3*d*x^5 + 1680*(e*x)^m*a^3*c*x^4)/(m^5 + 30*m^4 + 355 *m^3 + 2070*m^2 + 5944*m + 6720)
Time = 8.86 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.33 \[ \int (e x)^m (c+d x) \left (a x+b x^2\right )^3 \, dx={\left (e\,x\right )}^m\,\left (\frac {a^3\,c\,x^4\,\left (m^4+26\,m^3+251\,m^2+1066\,m+1680\right )}{m^5+30\,m^4+355\,m^3+2070\,m^2+5944\,m+6720}+\frac {b^3\,d\,x^8\,\left (m^4+22\,m^3+179\,m^2+638\,m+840\right )}{m^5+30\,m^4+355\,m^3+2070\,m^2+5944\,m+6720}+\frac {a^2\,x^5\,\left (a\,d+3\,b\,c\right )\,\left (m^4+25\,m^3+230\,m^2+920\,m+1344\right )}{m^5+30\,m^4+355\,m^3+2070\,m^2+5944\,m+6720}+\frac {b^2\,x^7\,\left (3\,a\,d+b\,c\right )\,\left (m^4+23\,m^3+194\,m^2+712\,m+960\right )}{m^5+30\,m^4+355\,m^3+2070\,m^2+5944\,m+6720}+\frac {3\,a\,b\,x^6\,\left (a\,d+b\,c\right )\,\left (m^4+24\,m^3+211\,m^2+804\,m+1120\right )}{m^5+30\,m^4+355\,m^3+2070\,m^2+5944\,m+6720}\right ) \] Input:
int((a*x + b*x^2)^3*(e*x)^m*(c + d*x),x)
Output:
(e*x)^m*((a^3*c*x^4*(1066*m + 251*m^2 + 26*m^3 + m^4 + 1680))/(5944*m + 20 70*m^2 + 355*m^3 + 30*m^4 + m^5 + 6720) + (b^3*d*x^8*(638*m + 179*m^2 + 22 *m^3 + m^4 + 840))/(5944*m + 2070*m^2 + 355*m^3 + 30*m^4 + m^5 + 6720) + ( a^2*x^5*(a*d + 3*b*c)*(920*m + 230*m^2 + 25*m^3 + m^4 + 1344))/(5944*m + 2 070*m^2 + 355*m^3 + 30*m^4 + m^5 + 6720) + (b^2*x^7*(3*a*d + b*c)*(712*m + 194*m^2 + 23*m^3 + m^4 + 960))/(5944*m + 2070*m^2 + 355*m^3 + 30*m^4 + m^ 5 + 6720) + (3*a*b*x^6*(a*d + b*c)*(804*m + 211*m^2 + 24*m^3 + m^4 + 1120) )/(5944*m + 2070*m^2 + 355*m^3 + 30*m^4 + m^5 + 6720))
Time = 0.19 (sec) , antiderivative size = 457, normalized size of antiderivative = 3.78 \[ \int (e x)^m (c+d x) \left (a x+b x^2\right )^3 \, dx=\frac {x^{m} e^{m} x^{4} \left (b^{3} d \,m^{4} x^{4}+3 a \,b^{2} d \,m^{4} x^{3}+b^{3} c \,m^{4} x^{3}+22 b^{3} d \,m^{3} x^{4}+3 a^{2} b d \,m^{4} x^{2}+3 a \,b^{2} c \,m^{4} x^{2}+69 a \,b^{2} d \,m^{3} x^{3}+23 b^{3} c \,m^{3} x^{3}+179 b^{3} d \,m^{2} x^{4}+a^{3} d \,m^{4} x +3 a^{2} b c \,m^{4} x +72 a^{2} b d \,m^{3} x^{2}+72 a \,b^{2} c \,m^{3} x^{2}+582 a \,b^{2} d \,m^{2} x^{3}+194 b^{3} c \,m^{2} x^{3}+638 b^{3} d m \,x^{4}+a^{3} c \,m^{4}+25 a^{3} d \,m^{3} x +75 a^{2} b c \,m^{3} x +633 a^{2} b d \,m^{2} x^{2}+633 a \,b^{2} c \,m^{2} x^{2}+2136 a \,b^{2} d m \,x^{3}+712 b^{3} c m \,x^{3}+840 b^{3} d \,x^{4}+26 a^{3} c \,m^{3}+230 a^{3} d \,m^{2} x +690 a^{2} b c \,m^{2} x +2412 a^{2} b d m \,x^{2}+2412 a \,b^{2} c m \,x^{2}+2880 a \,b^{2} d \,x^{3}+960 b^{3} c \,x^{3}+251 a^{3} c \,m^{2}+920 a^{3} d m x +2760 a^{2} b c m x +3360 a^{2} b d \,x^{2}+3360 a \,b^{2} c \,x^{2}+1066 a^{3} c m +1344 a^{3} d x +4032 a^{2} b c x +1680 a^{3} c \right )}{m^{5}+30 m^{4}+355 m^{3}+2070 m^{2}+5944 m +6720} \] Input:
int((e*x)^m*(d*x+c)*(b*x^2+a*x)^3,x)
Output:
(x**m*e**m*x**4*(a**3*c*m**4 + 26*a**3*c*m**3 + 251*a**3*c*m**2 + 1066*a** 3*c*m + 1680*a**3*c + a**3*d*m**4*x + 25*a**3*d*m**3*x + 230*a**3*d*m**2*x + 920*a**3*d*m*x + 1344*a**3*d*x + 3*a**2*b*c*m**4*x + 75*a**2*b*c*m**3*x + 690*a**2*b*c*m**2*x + 2760*a**2*b*c*m*x + 4032*a**2*b*c*x + 3*a**2*b*d* m**4*x**2 + 72*a**2*b*d*m**3*x**2 + 633*a**2*b*d*m**2*x**2 + 2412*a**2*b*d *m*x**2 + 3360*a**2*b*d*x**2 + 3*a*b**2*c*m**4*x**2 + 72*a*b**2*c*m**3*x** 2 + 633*a*b**2*c*m**2*x**2 + 2412*a*b**2*c*m*x**2 + 3360*a*b**2*c*x**2 + 3 *a*b**2*d*m**4*x**3 + 69*a*b**2*d*m**3*x**3 + 582*a*b**2*d*m**2*x**3 + 213 6*a*b**2*d*m*x**3 + 2880*a*b**2*d*x**3 + b**3*c*m**4*x**3 + 23*b**3*c*m**3 *x**3 + 194*b**3*c*m**2*x**3 + 712*b**3*c*m*x**3 + 960*b**3*c*x**3 + b**3* d*m**4*x**4 + 22*b**3*d*m**3*x**4 + 179*b**3*d*m**2*x**4 + 638*b**3*d*m*x* *4 + 840*b**3*d*x**4))/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6 720)