Integrand size = 20, antiderivative size = 121 \[ \int (e x)^m (c+d x) \left (a+b x^2\right )^2 \, dx=\frac {a^2 c (e x)^{1+m}}{e (1+m)}+\frac {a^2 d (e x)^{2+m}}{e^2 (2+m)}+\frac {2 a b c (e x)^{3+m}}{e^3 (3+m)}+\frac {2 a b d (e x)^{4+m}}{e^4 (4+m)}+\frac {b^2 c (e x)^{5+m}}{e^5 (5+m)}+\frac {b^2 d (e x)^{6+m}}{e^6 (6+m)} \] Output:
a^2*c*(e*x)^(1+m)/e/(1+m)+a^2*d*(e*x)^(2+m)/e^2/(2+m)+2*a*b*c*(e*x)^(3+m)/ e^3/(3+m)+2*a*b*d*(e*x)^(4+m)/e^4/(4+m)+b^2*c*(e*x)^(5+m)/e^5/(5+m)+b^2*d* (e*x)^(6+m)/e^6/(6+m)
Time = 0.16 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.61 \[ \int (e x)^m (c+d x) \left (a+b x^2\right )^2 \, dx=x (e x)^m \left (a^2 \left (\frac {c}{1+m}+\frac {d x}{2+m}\right )+2 a b x^2 \left (\frac {c}{3+m}+\frac {d x}{4+m}\right )+b^2 x^4 \left (\frac {c}{5+m}+\frac {d x}{6+m}\right )\right ) \] Input:
Integrate[(e*x)^m*(c + d*x)*(a + b*x^2)^2,x]
Output:
x*(e*x)^m*(a^2*(c/(1 + m) + (d*x)/(2 + m)) + 2*a*b*x^2*(c/(3 + m) + (d*x)/ (4 + m)) + b^2*x^4*(c/(5 + m) + (d*x)/(6 + m)))
Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {522, 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.
| \(\displaystyle \int \left (a+b x^2\right )^2 (c+d x) (e x)^m \, dx\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \int \left (a^2 c (e x)^m+\frac {a^2 d (e x)^{m+1}}{e}+\frac {2 a b c (e x)^{m+2}}{e^2}+\frac {2 a b d (e x)^{m+3}}{e^3}+\frac {b^2 c (e x)^{m+4}}{e^4}+\frac {b^2 d (e x)^{m+5}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 c (e x)^{m+1}}{e (m+1)}+\frac {a^2 d (e x)^{m+2}}{e^2 (m+2)}+\frac {2 a b c (e x)^{m+3}}{e^3 (m+3)}+\frac {2 a b d (e x)^{m+4}}{e^4 (m+4)}+\frac {b^2 c (e x)^{m+5}}{e^5 (m+5)}+\frac {b^2 d (e x)^{m+6}}{e^6 (m+6)}\) |
Input:
Int[(e*x)^m*(c + d*x)*(a + b*x^2)^2,x]
Output:
(a^2*c*(e*x)^(1 + m))/(e*(1 + m)) + (a^2*d*(e*x)^(2 + m))/(e^2*(2 + m)) + (2*a*b*c*(e*x)^(3 + m))/(e^3*(3 + m)) + (2*a*b*d*(e*x)^(4 + m))/(e^4*(4 + m)) + (b^2*c*(e*x)^(5 + m))/(e^5*(5 + m)) + (b^2*d*(e*x)^(6 + m))/(e^6*(6 + m))
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Time = 0.23 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.99
| method | result | size |
| norman | \(\frac {a^{2} c x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}+\frac {a^{2} d \,x^{2} {\mathrm e}^{m \ln \left (e x \right )}}{2+m}+\frac {b^{2} c \,x^{5} {\mathrm e}^{m \ln \left (e x \right )}}{5+m}+\frac {b^{2} d \,x^{6} {\mathrm e}^{m \ln \left (e x \right )}}{6+m}+\frac {2 a b c \,x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}+\frac {2 d a b \,x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}\) | \(120\) |
| gosper | \(\frac {x \left (b^{2} d \,m^{5} x^{5}+b^{2} c \,m^{5} x^{4}+15 b^{2} d \,m^{4} x^{5}+2 a b d \,m^{5} x^{3}+16 b^{2} c \,m^{4} x^{4}+85 b^{2} d \,m^{3} x^{5}+2 a b c \,m^{5} x^{2}+34 a b d \,m^{4} x^{3}+95 b^{2} c \,m^{3} x^{4}+225 b^{2} d \,m^{2} x^{5}+a^{2} d \,m^{5} x +36 a b c \,m^{4} x^{2}+214 a b d \,m^{3} x^{3}+260 b^{2} c \,m^{2} x^{4}+274 m \,x^{5} b^{2} d +a^{2} c \,m^{5}+19 a^{2} d \,m^{4} x +242 a b c \,m^{3} x^{2}+614 a b d \,m^{2} x^{3}+324 m \,x^{4} b^{2} c +120 b^{2} d \,x^{5}+20 a^{2} c \,m^{4}+137 a^{2} d \,m^{3} x +744 a b c \,m^{2} x^{2}+792 a b d \,x^{3} m +144 b^{2} c \,x^{4}+155 a^{2} c \,m^{3}+461 a^{2} d \,m^{2} x +1016 a b c \,x^{2} m +360 a b d \,x^{3}+580 a^{2} c \,m^{2}+702 a^{2} d x m +480 a b c \,x^{2}+1044 a^{2} c m +360 a^{2} d x +720 a^{2} c \right ) \left (e x \right )^{m}}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(395\) |
| risch | \(\frac {x \left (b^{2} d \,m^{5} x^{5}+b^{2} c \,m^{5} x^{4}+15 b^{2} d \,m^{4} x^{5}+2 a b d \,m^{5} x^{3}+16 b^{2} c \,m^{4} x^{4}+85 b^{2} d \,m^{3} x^{5}+2 a b c \,m^{5} x^{2}+34 a b d \,m^{4} x^{3}+95 b^{2} c \,m^{3} x^{4}+225 b^{2} d \,m^{2} x^{5}+a^{2} d \,m^{5} x +36 a b c \,m^{4} x^{2}+214 a b d \,m^{3} x^{3}+260 b^{2} c \,m^{2} x^{4}+274 m \,x^{5} b^{2} d +a^{2} c \,m^{5}+19 a^{2} d \,m^{4} x +242 a b c \,m^{3} x^{2}+614 a b d \,m^{2} x^{3}+324 m \,x^{4} b^{2} c +120 b^{2} d \,x^{5}+20 a^{2} c \,m^{4}+137 a^{2} d \,m^{3} x +744 a b c \,m^{2} x^{2}+792 a b d \,x^{3} m +144 b^{2} c \,x^{4}+155 a^{2} c \,m^{3}+461 a^{2} d \,m^{2} x +1016 a b c \,x^{2} m +360 a b d \,x^{3}+580 a^{2} c \,m^{2}+702 a^{2} d x m +480 a b c \,x^{2}+1044 a^{2} c m +360 a^{2} d x +720 a^{2} c \right ) \left (e x \right )^{m}}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(395\) |
| orering | \(\frac {x \left (b^{2} d \,m^{5} x^{5}+b^{2} c \,m^{5} x^{4}+15 b^{2} d \,m^{4} x^{5}+2 a b d \,m^{5} x^{3}+16 b^{2} c \,m^{4} x^{4}+85 b^{2} d \,m^{3} x^{5}+2 a b c \,m^{5} x^{2}+34 a b d \,m^{4} x^{3}+95 b^{2} c \,m^{3} x^{4}+225 b^{2} d \,m^{2} x^{5}+a^{2} d \,m^{5} x +36 a b c \,m^{4} x^{2}+214 a b d \,m^{3} x^{3}+260 b^{2} c \,m^{2} x^{4}+274 m \,x^{5} b^{2} d +a^{2} c \,m^{5}+19 a^{2} d \,m^{4} x +242 a b c \,m^{3} x^{2}+614 a b d \,m^{2} x^{3}+324 m \,x^{4} b^{2} c +120 b^{2} d \,x^{5}+20 a^{2} c \,m^{4}+137 a^{2} d \,m^{3} x +744 a b c \,m^{2} x^{2}+792 a b d \,x^{3} m +144 b^{2} c \,x^{4}+155 a^{2} c \,m^{3}+461 a^{2} d \,m^{2} x +1016 a b c \,x^{2} m +360 a b d \,x^{3}+580 a^{2} c \,m^{2}+702 a^{2} d x m +480 a b c \,x^{2}+1044 a^{2} c m +360 a^{2} d x +720 a^{2} c \right ) \left (e x \right )^{m}}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(395\) |
| parallelrisch | \(\frac {20 x \left (e x \right )^{m} a^{2} c \,m^{4}+360 x^{4} \left (e x \right )^{m} a b d +461 x^{2} \left (e x \right )^{m} a^{2} d \,m^{2}+155 x \left (e x \right )^{m} a^{2} c \,m^{3}+480 x^{3} \left (e x \right )^{m} a b c +702 x^{2} \left (e x \right )^{m} a^{2} d m +580 x \left (e x \right )^{m} a^{2} c \,m^{2}+1044 x \left (e x \right )^{m} a^{2} c m +1016 x^{3} \left (e x \right )^{m} a b c m +214 x^{4} \left (e x \right )^{m} a b d \,m^{3}+2 x^{4} \left (e x \right )^{m} a b d \,m^{5}+36 x^{3} \left (e x \right )^{m} a b c \,m^{4}+614 x^{4} \left (e x \right )^{m} a b d \,m^{2}+242 x^{3} \left (e x \right )^{m} a b c \,m^{3}+792 x^{4} \left (e x \right )^{m} a b d m +744 x^{3} \left (e x \right )^{m} a b c \,m^{2}+34 x^{4} \left (e x \right )^{m} a b d \,m^{4}+2 x^{3} \left (e x \right )^{m} a b c \,m^{5}+324 x^{5} \left (e x \right )^{m} b^{2} c m +19 x^{2} \left (e x \right )^{m} a^{2} d \,m^{4}+x \left (e x \right )^{m} a^{2} c \,m^{5}+137 x^{2} \left (e x \right )^{m} a^{2} d \,m^{3}+95 x^{5} \left (e x \right )^{m} b^{2} c \,m^{3}+274 x^{6} \left (e x \right )^{m} b^{2} d m +260 x^{5} \left (e x \right )^{m} b^{2} c \,m^{2}+x^{2} \left (e x \right )^{m} a^{2} d \,m^{5}+x^{5} \left (e x \right )^{m} b^{2} c \,m^{5}+85 x^{6} \left (e x \right )^{m} b^{2} d \,m^{3}+16 x^{5} \left (e x \right )^{m} b^{2} c \,m^{4}+225 x^{6} \left (e x \right )^{m} b^{2} d \,m^{2}+15 x^{6} \left (e x \right )^{m} b^{2} d \,m^{4}+x^{6} \left (e x \right )^{m} b^{2} d \,m^{5}+120 x^{6} \left (e x \right )^{m} b^{2} d +144 x^{5} \left (e x \right )^{m} b^{2} c +360 x^{2} \left (e x \right )^{m} a^{2} d +720 x \left (e x \right )^{m} a^{2} c}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(587\) |
Input:
int((e*x)^m*(d*x+c)*(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
a^2*c/(1+m)*x*exp(m*ln(e*x))+a^2*d/(2+m)*x^2*exp(m*ln(e*x))+b^2*c/(5+m)*x^ 5*exp(m*ln(e*x))+b^2*d/(6+m)*x^6*exp(m*ln(e*x))+2*a*b*c/(3+m)*x^3*exp(m*ln (e*x))+2*d*a*b/(4+m)*x^4*exp(m*ln(e*x))
Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (121) = 242\).
Time = 0.08 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.83 \[ \int (e x)^m (c+d x) \left (a+b x^2\right )^2 \, dx=\frac {{\left ({\left (b^{2} d m^{5} + 15 \, b^{2} d m^{4} + 85 \, b^{2} d m^{3} + 225 \, b^{2} d m^{2} + 274 \, b^{2} d m + 120 \, b^{2} d\right )} x^{6} + {\left (b^{2} c m^{5} + 16 \, b^{2} c m^{4} + 95 \, b^{2} c m^{3} + 260 \, b^{2} c m^{2} + 324 \, b^{2} c m + 144 \, b^{2} c\right )} x^{5} + 2 \, {\left (a b d m^{5} + 17 \, a b d m^{4} + 107 \, a b d m^{3} + 307 \, a b d m^{2} + 396 \, a b d m + 180 \, a b d\right )} x^{4} + 2 \, {\left (a b c m^{5} + 18 \, a b c m^{4} + 121 \, a b c m^{3} + 372 \, a b c m^{2} + 508 \, a b c m + 240 \, a b c\right )} x^{3} + {\left (a^{2} d m^{5} + 19 \, a^{2} d m^{4} + 137 \, a^{2} d m^{3} + 461 \, a^{2} d m^{2} + 702 \, a^{2} d m + 360 \, a^{2} d\right )} x^{2} + {\left (a^{2} c m^{5} + 20 \, a^{2} c m^{4} + 155 \, a^{2} c m^{3} + 580 \, a^{2} c m^{2} + 1044 \, a^{2} c m + 720 \, a^{2} c\right )} x\right )} \left (e x\right )^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \] Input:
integrate((e*x)^m*(d*x+c)*(b*x^2+a)^2,x, algorithm="fricas")
Output:
((b^2*d*m^5 + 15*b^2*d*m^4 + 85*b^2*d*m^3 + 225*b^2*d*m^2 + 274*b^2*d*m + 120*b^2*d)*x^6 + (b^2*c*m^5 + 16*b^2*c*m^4 + 95*b^2*c*m^3 + 260*b^2*c*m^2 + 324*b^2*c*m + 144*b^2*c)*x^5 + 2*(a*b*d*m^5 + 17*a*b*d*m^4 + 107*a*b*d*m ^3 + 307*a*b*d*m^2 + 396*a*b*d*m + 180*a*b*d)*x^4 + 2*(a*b*c*m^5 + 18*a*b* c*m^4 + 121*a*b*c*m^3 + 372*a*b*c*m^2 + 508*a*b*c*m + 240*a*b*c)*x^3 + (a^ 2*d*m^5 + 19*a^2*d*m^4 + 137*a^2*d*m^3 + 461*a^2*d*m^2 + 702*a^2*d*m + 360 *a^2*d)*x^2 + (a^2*c*m^5 + 20*a^2*c*m^4 + 155*a^2*c*m^3 + 580*a^2*c*m^2 + 1044*a^2*c*m + 720*a^2*c)*x)*(e*x)^m/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1 624*m^2 + 1764*m + 720)
Leaf count of result is larger than twice the leaf count of optimal. 2014 vs. \(2 (112) = 224\).
Time = 0.46 (sec) , antiderivative size = 2014, normalized size of antiderivative = 16.64 \[ \int (e x)^m (c+d x) \left (a+b x^2\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((e*x)**m*(d*x+c)*(b*x**2+a)**2,x)
Output:
Piecewise(((-a**2*c/(5*x**5) - a**2*d/(4*x**4) - 2*a*b*c/(3*x**3) - a*b*d/ x**2 - b**2*c/x + b**2*d*log(x))/e**6, Eq(m, -6)), ((-a**2*c/(4*x**4) - a* *2*d/(3*x**3) - a*b*c/x**2 - 2*a*b*d/x + b**2*c*log(x) + b**2*d*x)/e**5, E q(m, -5)), ((-a**2*c/(3*x**3) - a**2*d/(2*x**2) - 2*a*b*c/x + 2*a*b*d*log( x) + b**2*c*x + b**2*d*x**2/2)/e**4, Eq(m, -4)), ((-a**2*c/(2*x**2) - a**2 *d/x + 2*a*b*c*log(x) + 2*a*b*d*x + b**2*c*x**2/2 + b**2*d*x**3/3)/e**3, E q(m, -3)), ((-a**2*c/x + a**2*d*log(x) + 2*a*b*c*x + a*b*d*x**2 + b**2*c*x **3/3 + b**2*d*x**4/4)/e**2, Eq(m, -2)), ((a**2*c*log(x) + a**2*d*x + a*b* c*x**2 + 2*a*b*d*x**3/3 + b**2*c*x**4/4 + b**2*d*x**5/5)/e, Eq(m, -1)), (a **2*c*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**2*c*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**2*c*m**3*x*(e*x)**m/(m**6 + 21* m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 580*a**2*c*m**2*x *(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720 ) + 1044*a**2*c*m*x*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624* m**2 + 1764*m + 720) + 720*a**2*c*x*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + a**2*d*m**5*x**2*(e*x)**m/(m**6 + 2 1*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 19*a**2*d*m**4* x**2*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 137*a**2*d*m**3*x**2*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*...
Time = 0.04 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.96 \[ \int (e x)^m (c+d x) \left (a+b x^2\right )^2 \, dx=\frac {b^{2} d e^{m} x^{6} x^{m}}{m + 6} + \frac {b^{2} c e^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, a b d e^{m} x^{4} x^{m}}{m + 4} + \frac {2 \, a b c e^{m} x^{3} x^{m}}{m + 3} + \frac {a^{2} d e^{m} x^{2} x^{m}}{m + 2} + \frac {\left (e x\right )^{m + 1} a^{2} c}{e {\left (m + 1\right )}} \] Input:
integrate((e*x)^m*(d*x+c)*(b*x^2+a)^2,x, algorithm="maxima")
Output:
b^2*d*e^m*x^6*x^m/(m + 6) + b^2*c*e^m*x^5*x^m/(m + 5) + 2*a*b*d*e^m*x^4*x^ m/(m + 4) + 2*a*b*c*e^m*x^3*x^m/(m + 3) + a^2*d*e^m*x^2*x^m/(m + 2) + (e*x )^(m + 1)*a^2*c/(e*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (121) = 242\).
Time = 0.14 (sec) , antiderivative size = 586, normalized size of antiderivative = 4.84 \[ \int (e x)^m (c+d x) \left (a+b x^2\right )^2 \, dx=\frac {\left (e x\right )^{m} b^{2} d m^{5} x^{6} + \left (e x\right )^{m} b^{2} c m^{5} x^{5} + 15 \, \left (e x\right )^{m} b^{2} d m^{4} x^{6} + 2 \, \left (e x\right )^{m} a b d m^{5} x^{4} + 16 \, \left (e x\right )^{m} b^{2} c m^{4} x^{5} + 85 \, \left (e x\right )^{m} b^{2} d m^{3} x^{6} + 2 \, \left (e x\right )^{m} a b c m^{5} x^{3} + 34 \, \left (e x\right )^{m} a b d m^{4} x^{4} + 95 \, \left (e x\right )^{m} b^{2} c m^{3} x^{5} + 225 \, \left (e x\right )^{m} b^{2} d m^{2} x^{6} + \left (e x\right )^{m} a^{2} d m^{5} x^{2} + 36 \, \left (e x\right )^{m} a b c m^{4} x^{3} + 214 \, \left (e x\right )^{m} a b d m^{3} x^{4} + 260 \, \left (e x\right )^{m} b^{2} c m^{2} x^{5} + 274 \, \left (e x\right )^{m} b^{2} d m x^{6} + \left (e x\right )^{m} a^{2} c m^{5} x + 19 \, \left (e x\right )^{m} a^{2} d m^{4} x^{2} + 242 \, \left (e x\right )^{m} a b c m^{3} x^{3} + 614 \, \left (e x\right )^{m} a b d m^{2} x^{4} + 324 \, \left (e x\right )^{m} b^{2} c m x^{5} + 120 \, \left (e x\right )^{m} b^{2} d x^{6} + 20 \, \left (e x\right )^{m} a^{2} c m^{4} x + 137 \, \left (e x\right )^{m} a^{2} d m^{3} x^{2} + 744 \, \left (e x\right )^{m} a b c m^{2} x^{3} + 792 \, \left (e x\right )^{m} a b d m x^{4} + 144 \, \left (e x\right )^{m} b^{2} c x^{5} + 155 \, \left (e x\right )^{m} a^{2} c m^{3} x + 461 \, \left (e x\right )^{m} a^{2} d m^{2} x^{2} + 1016 \, \left (e x\right )^{m} a b c m x^{3} + 360 \, \left (e x\right )^{m} a b d x^{4} + 580 \, \left (e x\right )^{m} a^{2} c m^{2} x + 702 \, \left (e x\right )^{m} a^{2} d m x^{2} + 480 \, \left (e x\right )^{m} a b c x^{3} + 1044 \, \left (e x\right )^{m} a^{2} c m x + 360 \, \left (e x\right )^{m} a^{2} d x^{2} + 720 \, \left (e x\right )^{m} a^{2} c x}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \] Input:
integrate((e*x)^m*(d*x+c)*(b*x^2+a)^2,x, algorithm="giac")
Output:
((e*x)^m*b^2*d*m^5*x^6 + (e*x)^m*b^2*c*m^5*x^5 + 15*(e*x)^m*b^2*d*m^4*x^6 + 2*(e*x)^m*a*b*d*m^5*x^4 + 16*(e*x)^m*b^2*c*m^4*x^5 + 85*(e*x)^m*b^2*d*m^ 3*x^6 + 2*(e*x)^m*a*b*c*m^5*x^3 + 34*(e*x)^m*a*b*d*m^4*x^4 + 95*(e*x)^m*b^ 2*c*m^3*x^5 + 225*(e*x)^m*b^2*d*m^2*x^6 + (e*x)^m*a^2*d*m^5*x^2 + 36*(e*x) ^m*a*b*c*m^4*x^3 + 214*(e*x)^m*a*b*d*m^3*x^4 + 260*(e*x)^m*b^2*c*m^2*x^5 + 274*(e*x)^m*b^2*d*m*x^6 + (e*x)^m*a^2*c*m^5*x + 19*(e*x)^m*a^2*d*m^4*x^2 + 242*(e*x)^m*a*b*c*m^3*x^3 + 614*(e*x)^m*a*b*d*m^2*x^4 + 324*(e*x)^m*b^2* c*m*x^5 + 120*(e*x)^m*b^2*d*x^6 + 20*(e*x)^m*a^2*c*m^4*x + 137*(e*x)^m*a^2 *d*m^3*x^2 + 744*(e*x)^m*a*b*c*m^2*x^3 + 792*(e*x)^m*a*b*d*m*x^4 + 144*(e* x)^m*b^2*c*x^5 + 155*(e*x)^m*a^2*c*m^3*x + 461*(e*x)^m*a^2*d*m^2*x^2 + 101 6*(e*x)^m*a*b*c*m*x^3 + 360*(e*x)^m*a*b*d*x^4 + 580*(e*x)^m*a^2*c*m^2*x + 702*(e*x)^m*a^2*d*m*x^2 + 480*(e*x)^m*a*b*c*x^3 + 1044*(e*x)^m*a^2*c*m*x + 360*(e*x)^m*a^2*d*x^2 + 720*(e*x)^m*a^2*c*x)/(m^6 + 21*m^5 + 175*m^4 + 73 5*m^3 + 1624*m^2 + 1764*m + 720)
Time = 9.17 (sec) , antiderivative size = 371, normalized size of antiderivative = 3.07 \[ \int (e x)^m (c+d x) \left (a+b x^2\right )^2 \, dx={\left (e\,x\right )}^m\,\left (\frac {a^2\,c\,x\,\left (m^5+20\,m^4+155\,m^3+580\,m^2+1044\,m+720\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {b^2\,c\,x^5\,\left (m^5+16\,m^4+95\,m^3+260\,m^2+324\,m+144\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a^2\,d\,x^2\,\left (m^5+19\,m^4+137\,m^3+461\,m^2+702\,m+360\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {b^2\,d\,x^6\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {2\,a\,b\,c\,x^3\,\left (m^5+18\,m^4+121\,m^3+372\,m^2+508\,m+240\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {2\,a\,b\,d\,x^4\,\left (m^5+17\,m^4+107\,m^3+307\,m^2+396\,m+180\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}\right ) \] Input:
int((e*x)^m*(a + b*x^2)^2*(c + d*x),x)
Output:
(e*x)^m*((a^2*c*x*(1044*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) + (b^2*c*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^2*d*x^2*(702*m + 461*m^2 + 137*m^3 + 1 9*m^4 + m^5 + 360))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (b^2*d*x^6*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120))/(1764 *m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (2*a*b*c*x^3*(50 8*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) + (2*a*b*d*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))
Time = 0.21 (sec) , antiderivative size = 395, normalized size of antiderivative = 3.26 \[ \int (e x)^m (c+d x) \left (a+b x^2\right )^2 \, dx=\frac {x^{m} e^{m} x \left (b^{2} d \,m^{5} x^{5}+b^{2} c \,m^{5} x^{4}+15 b^{2} d \,m^{4} x^{5}+2 a b d \,m^{5} x^{3}+16 b^{2} c \,m^{4} x^{4}+85 b^{2} d \,m^{3} x^{5}+2 a b c \,m^{5} x^{2}+34 a b d \,m^{4} x^{3}+95 b^{2} c \,m^{3} x^{4}+225 b^{2} d \,m^{2} x^{5}+a^{2} d \,m^{5} x +36 a b c \,m^{4} x^{2}+214 a b d \,m^{3} x^{3}+260 b^{2} c \,m^{2} x^{4}+274 b^{2} d m \,x^{5}+a^{2} c \,m^{5}+19 a^{2} d \,m^{4} x +242 a b c \,m^{3} x^{2}+614 a b d \,m^{2} x^{3}+324 b^{2} c m \,x^{4}+120 b^{2} d \,x^{5}+20 a^{2} c \,m^{4}+137 a^{2} d \,m^{3} x +744 a b c \,m^{2} x^{2}+792 a b d m \,x^{3}+144 b^{2} c \,x^{4}+155 a^{2} c \,m^{3}+461 a^{2} d \,m^{2} x +1016 a b c m \,x^{2}+360 a b d \,x^{3}+580 a^{2} c \,m^{2}+702 a^{2} d m x +480 a b c \,x^{2}+1044 a^{2} c m +360 a^{2} d x +720 a^{2} c \right )}{m^{6}+21 m^{5}+175 m^{4}+735 m^{3}+1624 m^{2}+1764 m +720} \] Input:
int((e*x)^m*(d*x+c)*(b*x^2+a)^2,x)
Output:
(x**m*e**m*x*(a**2*c*m**5 + 20*a**2*c*m**4 + 155*a**2*c*m**3 + 580*a**2*c* m**2 + 1044*a**2*c*m + 720*a**2*c + a**2*d*m**5*x + 19*a**2*d*m**4*x + 137 *a**2*d*m**3*x + 461*a**2*d*m**2*x + 702*a**2*d*m*x + 360*a**2*d*x + 2*a*b *c*m**5*x**2 + 36*a*b*c*m**4*x**2 + 242*a*b*c*m**3*x**2 + 744*a*b*c*m**2*x **2 + 1016*a*b*c*m*x**2 + 480*a*b*c*x**2 + 2*a*b*d*m**5*x**3 + 34*a*b*d*m* *4*x**3 + 214*a*b*d*m**3*x**3 + 614*a*b*d*m**2*x**3 + 792*a*b*d*m*x**3 + 3 60*a*b*d*x**3 + b**2*c*m**5*x**4 + 16*b**2*c*m**4*x**4 + 95*b**2*c*m**3*x* *4 + 260*b**2*c*m**2*x**4 + 324*b**2*c*m*x**4 + 144*b**2*c*x**4 + b**2*d*m **5*x**5 + 15*b**2*d*m**4*x**5 + 85*b**2*d*m**3*x**5 + 225*b**2*d*m**2*x** 5 + 274*b**2*d*m*x**5 + 120*b**2*d*x**5))/(m**6 + 21*m**5 + 175*m**4 + 735 *m**3 + 1624*m**2 + 1764*m + 720)