Integrand size = 20, antiderivative size = 102 \[ \int (d+e x)^3 \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{3} b d^3 x^3+\frac {3}{4} b d^2 e x^4+\frac {1}{5} d \left (c d^2+3 b e^2\right ) x^5+\frac {1}{6} e \left (3 c d^2+b e^2\right ) x^6+\frac {3}{7} c d e^2 x^7+\frac {1}{8} c e^3 x^8+\frac {a (d+e x)^4}{4 e} \] Output:
1/3*b*d^3*x^3+3/4*b*d^2*e*x^4+1/5*d*(3*b*e^2+c*d^2)*x^5+1/6*e*(b*e^2+3*c*d ^2)*x^6+3/7*c*d*e^2*x^7+1/8*c*e^3*x^8+1/4*a*(e*x+d)^4/e
Time = 0.03 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.20 \[ \int (d+e x)^3 \left (a+b x^2+c x^4\right ) \, dx=a d^3 x+\frac {3}{2} a d^2 e x^2+\frac {1}{3} d \left (b d^2+3 a e^2\right ) x^3+\frac {1}{4} e \left (3 b d^2+a e^2\right ) x^4+\frac {1}{5} d \left (c d^2+3 b e^2\right ) x^5+\frac {1}{6} e \left (3 c d^2+b e^2\right ) x^6+\frac {3}{7} c d e^2 x^7+\frac {1}{8} c e^3 x^8 \] Input:
Integrate[(d + e*x)^3*(a + b*x^2 + c*x^4),x]
Output:
a*d^3*x + (3*a*d^2*e*x^2)/2 + (d*(b*d^2 + 3*a*e^2)*x^3)/3 + (e*(3*b*d^2 + a*e^2)*x^4)/4 + (d*(c*d^2 + 3*b*e^2)*x^5)/5 + (e*(3*c*d^2 + b*e^2)*x^6)/6 + (3*c*d*e^2*x^7)/7 + (c*e^3*x^8)/8
Time = 0.52 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.16, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2200, 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.
\(\displaystyle \int (d+e x)^3 \left (a+b x^2+c x^4\right ) \, dx\) |
\(\Big \downarrow \) 2200 |
\(\displaystyle \int \left (\frac {(d+e x)^3 \left (a e^4+b d^2 e^2+c d^4\right )}{e^4}-\frac {2 (d+e x)^4 \left (b d e^2+2 c d^3\right )}{e^4}+\frac {(d+e x)^5 \left (b e^2+6 c d^2\right )}{e^4}+\frac {c (d+e x)^7}{e^4}-\frac {4 c d (d+e x)^6}{e^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(d+e x)^4 \left (a e^4+b d^2 e^2+c d^4\right )}{4 e^5}+\frac {(d+e x)^6 \left (b e^2+6 c d^2\right )}{6 e^5}-\frac {2 d (d+e x)^5 \left (b e^2+2 c d^2\right )}{5 e^5}+\frac {c (d+e x)^8}{8 e^5}-\frac {4 c d (d+e x)^7}{7 e^5}\) |
Input:
Int[(d + e*x)^3*(a + b*x^2 + c*x^4),x]
Output:
((c*d^4 + b*d^2*e^2 + a*e^4)*(d + e*x)^4)/(4*e^5) - (2*d*(2*c*d^2 + b*e^2) *(d + e*x)^5)/(5*e^5) + ((6*c*d^2 + b*e^2)*(d + e*x)^6)/(6*e^5) - (4*c*d*( d + e*x)^7)/(7*e^5) + (c*(d + e*x)^8)/(8*e^5)
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[Expa ndIntegrand[Px*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c}, x] && Poly Q[Px, x] && IGtQ[p, 0]
Time = 0.16 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.06
method | result | size |
norman | \(\frac {c \,e^{3} x^{8}}{8}+\frac {3 c d \,e^{2} x^{7}}{7}+\left (\frac {1}{6} b \,e^{3}+\frac {1}{2} d^{2} e c \right ) x^{6}+\left (\frac {3}{5} b d \,e^{2}+\frac {1}{5} c \,d^{3}\right ) x^{5}+\left (\frac {1}{4} e^{3} a +\frac {3}{4} b \,d^{2} e \right ) x^{4}+\left (a d \,e^{2}+\frac {1}{3} b \,d^{3}\right ) x^{3}+\frac {3 a \,d^{2} e \,x^{2}}{2}+a \,d^{3} x\) | \(108\) |
default | \(\frac {c \,e^{3} x^{8}}{8}+\frac {3 c d \,e^{2} x^{7}}{7}+\frac {\left (b \,e^{3}+3 d^{2} e c \right ) x^{6}}{6}+\frac {\left (3 b d \,e^{2}+c \,d^{3}\right ) x^{5}}{5}+\frac {\left (e^{3} a +3 b \,d^{2} e \right ) x^{4}}{4}+\frac {\left (3 a d \,e^{2}+b \,d^{3}\right ) x^{3}}{3}+\frac {3 a \,d^{2} e \,x^{2}}{2}+a \,d^{3} x\) | \(109\) |
gosper | \(\frac {1}{8} c \,e^{3} x^{8}+\frac {3}{7} c d \,e^{2} x^{7}+\frac {1}{6} b \,e^{3} x^{6}+\frac {1}{2} c \,d^{2} e \,x^{6}+\frac {3}{5} x^{5} b d \,e^{2}+\frac {1}{5} c \,d^{3} x^{5}+\frac {1}{4} a \,e^{3} x^{4}+\frac {3}{4} b \,d^{2} e \,x^{4}+a d \,e^{2} x^{3}+\frac {1}{3} b \,d^{3} x^{3}+\frac {3}{2} a \,d^{2} e \,x^{2}+a \,d^{3} x\) | \(112\) |
risch | \(\frac {1}{8} c \,e^{3} x^{8}+\frac {3}{7} c d \,e^{2} x^{7}+\frac {1}{6} b \,e^{3} x^{6}+\frac {1}{2} c \,d^{2} e \,x^{6}+\frac {3}{5} x^{5} b d \,e^{2}+\frac {1}{5} c \,d^{3} x^{5}+\frac {1}{4} a \,e^{3} x^{4}+\frac {3}{4} b \,d^{2} e \,x^{4}+a d \,e^{2} x^{3}+\frac {1}{3} b \,d^{3} x^{3}+\frac {3}{2} a \,d^{2} e \,x^{2}+a \,d^{3} x\) | \(112\) |
parallelrisch | \(\frac {1}{8} c \,e^{3} x^{8}+\frac {3}{7} c d \,e^{2} x^{7}+\frac {1}{6} b \,e^{3} x^{6}+\frac {1}{2} c \,d^{2} e \,x^{6}+\frac {3}{5} x^{5} b d \,e^{2}+\frac {1}{5} c \,d^{3} x^{5}+\frac {1}{4} a \,e^{3} x^{4}+\frac {3}{4} b \,d^{2} e \,x^{4}+a d \,e^{2} x^{3}+\frac {1}{3} b \,d^{3} x^{3}+\frac {3}{2} a \,d^{2} e \,x^{2}+a \,d^{3} x\) | \(112\) |
orering | \(\frac {x \left (105 e^{3} c \,x^{7}+360 c d \,e^{2} x^{6}+140 b \,e^{3} x^{5}+420 d^{2} e c \,x^{5}+504 b d \,e^{2} x^{4}+168 c \,d^{3} x^{4}+210 e^{3} a \,x^{3}+630 b \,d^{2} e \,x^{3}+840 a d \,e^{2} x^{2}+280 b \,d^{3} x^{2}+1260 d^{2} e a x +840 d^{3} a \right )}{840}\) | \(114\) |
Input:
int((e*x+d)^3*(c*x^4+b*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/8*c*e^3*x^8+3/7*c*d*e^2*x^7+(1/6*b*e^3+1/2*d^2*e*c)*x^6+(3/5*b*d*e^2+1/5 *c*d^3)*x^5+(1/4*e^3*a+3/4*b*d^2*e)*x^4+(a*d*e^2+1/3*b*d^3)*x^3+3/2*a*d^2* e*x^2+a*d^3*x
Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.06 \[ \int (d+e x)^3 \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{8} \, c e^{3} x^{8} + \frac {3}{7} \, c d e^{2} x^{7} + \frac {1}{6} \, {\left (3 \, c d^{2} e + b e^{3}\right )} x^{6} + \frac {3}{2} \, a d^{2} e x^{2} + \frac {1}{5} \, {\left (c d^{3} + 3 \, b d e^{2}\right )} x^{5} + a d^{3} x + \frac {1}{4} \, {\left (3 \, b d^{2} e + a e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b d^{3} + 3 \, a d e^{2}\right )} x^{3} \] Input:
integrate((e*x+d)^3*(c*x^4+b*x^2+a),x, algorithm="fricas")
Output:
1/8*c*e^3*x^8 + 3/7*c*d*e^2*x^7 + 1/6*(3*c*d^2*e + b*e^3)*x^6 + 3/2*a*d^2* e*x^2 + 1/5*(c*d^3 + 3*b*d*e^2)*x^5 + a*d^3*x + 1/4*(3*b*d^2*e + a*e^3)*x^ 4 + 1/3*(b*d^3 + 3*a*d*e^2)*x^3
Time = 0.04 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.17 \[ \int (d+e x)^3 \left (a+b x^2+c x^4\right ) \, dx=a d^{3} x + \frac {3 a d^{2} e x^{2}}{2} + \frac {3 c d e^{2} x^{7}}{7} + \frac {c e^{3} x^{8}}{8} + x^{6} \left (\frac {b e^{3}}{6} + \frac {c d^{2} e}{2}\right ) + x^{5} \cdot \left (\frac {3 b d e^{2}}{5} + \frac {c d^{3}}{5}\right ) + x^{4} \left (\frac {a e^{3}}{4} + \frac {3 b d^{2} e}{4}\right ) + x^{3} \left (a d e^{2} + \frac {b d^{3}}{3}\right ) \] Input:
integrate((e*x+d)**3*(c*x**4+b*x**2+a),x)
Output:
a*d**3*x + 3*a*d**2*e*x**2/2 + 3*c*d*e**2*x**7/7 + c*e**3*x**8/8 + x**6*(b *e**3/6 + c*d**2*e/2) + x**5*(3*b*d*e**2/5 + c*d**3/5) + x**4*(a*e**3/4 + 3*b*d**2*e/4) + x**3*(a*d*e**2 + b*d**3/3)
Time = 0.03 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.06 \[ \int (d+e x)^3 \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{8} \, c e^{3} x^{8} + \frac {3}{7} \, c d e^{2} x^{7} + \frac {1}{6} \, {\left (3 \, c d^{2} e + b e^{3}\right )} x^{6} + \frac {3}{2} \, a d^{2} e x^{2} + \frac {1}{5} \, {\left (c d^{3} + 3 \, b d e^{2}\right )} x^{5} + a d^{3} x + \frac {1}{4} \, {\left (3 \, b d^{2} e + a e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b d^{3} + 3 \, a d e^{2}\right )} x^{3} \] Input:
integrate((e*x+d)^3*(c*x^4+b*x^2+a),x, algorithm="maxima")
Output:
1/8*c*e^3*x^8 + 3/7*c*d*e^2*x^7 + 1/6*(3*c*d^2*e + b*e^3)*x^6 + 3/2*a*d^2* e*x^2 + 1/5*(c*d^3 + 3*b*d*e^2)*x^5 + a*d^3*x + 1/4*(3*b*d^2*e + a*e^3)*x^ 4 + 1/3*(b*d^3 + 3*a*d*e^2)*x^3
Time = 0.11 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.09 \[ \int (d+e x)^3 \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{8} \, c e^{3} x^{8} + \frac {3}{7} \, c d e^{2} x^{7} + \frac {1}{2} \, c d^{2} e x^{6} + \frac {1}{6} \, b e^{3} x^{6} + \frac {1}{5} \, c d^{3} x^{5} + \frac {3}{5} \, b d e^{2} x^{5} + \frac {3}{4} \, b d^{2} e x^{4} + \frac {1}{4} \, a e^{3} x^{4} + \frac {1}{3} \, b d^{3} x^{3} + a d e^{2} x^{3} + \frac {3}{2} \, a d^{2} e x^{2} + a d^{3} x \] Input:
integrate((e*x+d)^3*(c*x^4+b*x^2+a),x, algorithm="giac")
Output:
1/8*c*e^3*x^8 + 3/7*c*d*e^2*x^7 + 1/2*c*d^2*e*x^6 + 1/6*b*e^3*x^6 + 1/5*c* d^3*x^5 + 3/5*b*d*e^2*x^5 + 3/4*b*d^2*e*x^4 + 1/4*a*e^3*x^4 + 1/3*b*d^3*x^ 3 + a*d*e^2*x^3 + 3/2*a*d^2*e*x^2 + a*d^3*x
Time = 0.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.05 \[ \int (d+e x)^3 \left (a+b x^2+c x^4\right ) \, dx=x^3\,\left (\frac {b\,d^3}{3}+a\,d\,e^2\right )+x^4\,\left (\frac {3\,b\,d^2\,e}{4}+\frac {a\,e^3}{4}\right )+x^5\,\left (\frac {c\,d^3}{5}+\frac {3\,b\,d\,e^2}{5}\right )+x^6\,\left (\frac {c\,d^2\,e}{2}+\frac {b\,e^3}{6}\right )+\frac {c\,e^3\,x^8}{8}+a\,d^3\,x+\frac {3\,a\,d^2\,e\,x^2}{2}+\frac {3\,c\,d\,e^2\,x^7}{7} \] Input:
int((d + e*x)^3*(a + b*x^2 + c*x^4),x)
Output:
x^3*((b*d^3)/3 + a*d*e^2) + x^4*((a*e^3)/4 + (3*b*d^2*e)/4) + x^5*((c*d^3) /5 + (3*b*d*e^2)/5) + x^6*((b*e^3)/6 + (c*d^2*e)/2) + (c*e^3*x^8)/8 + a*d^ 3*x + (3*a*d^2*e*x^2)/2 + (3*c*d*e^2*x^7)/7
Time = 0.16 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.11 \[ \int (d+e x)^3 \left (a+b x^2+c x^4\right ) \, dx=\frac {x \left (105 c \,e^{3} x^{7}+360 c d \,e^{2} x^{6}+140 b \,e^{3} x^{5}+420 c \,d^{2} e \,x^{5}+504 b d \,e^{2} x^{4}+168 c \,d^{3} x^{4}+210 a \,e^{3} x^{3}+630 b \,d^{2} e \,x^{3}+840 a d \,e^{2} x^{2}+280 b \,d^{3} x^{2}+1260 a \,d^{2} e x +840 a \,d^{3}\right )}{840} \] Input:
int((e*x+d)^3*(c*x^4+b*x^2+a),x)
Output:
(x*(840*a*d**3 + 1260*a*d**2*e*x + 840*a*d*e**2*x**2 + 210*a*e**3*x**3 + 2 80*b*d**3*x**2 + 630*b*d**2*e*x**3 + 504*b*d*e**2*x**4 + 140*b*e**3*x**5 + 168*c*d**3*x**4 + 420*c*d**2*e*x**5 + 360*c*d*e**2*x**6 + 105*c*e**3*x**7 ))/840