Integrand size = 17, antiderivative size = 91 \[ \int (d+e x)^2 \left (a+c x^4\right )^2 \, dx=\frac {2}{5} a c d^2 x^5+\frac {2}{3} a c d e x^6+\frac {2}{7} a c e^2 x^7+\frac {1}{9} c^2 d^2 x^9+\frac {1}{5} c^2 d e x^{10}+\frac {1}{11} c^2 e^2 x^{11}+\frac {a^2 (d+e x)^3}{3 e} \] Output:
2/5*a*c*d^2*x^5+2/3*a*c*d*e*x^6+2/7*a*c*e^2*x^7+1/9*c^2*d^2*x^9+1/5*c^2*d* e*x^10+1/11*c^2*e^2*x^11+1/3*a^2*(e*x+d)^3/e
Time = 0.00 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.14 \[ \int (d+e x)^2 \left (a+c x^4\right )^2 \, dx=a^2 d^2 x+a^2 d e x^2+\frac {1}{3} a^2 e^2 x^3+\frac {2}{5} a c d^2 x^5+\frac {2}{3} a c d e x^6+\frac {2}{7} a c e^2 x^7+\frac {1}{9} c^2 d^2 x^9+\frac {1}{5} c^2 d e x^{10}+\frac {1}{11} c^2 e^2 x^{11} \] Input:
Integrate[(d + e*x)^2*(a + c*x^4)^2,x]
Output:
a^2*d^2*x + a^2*d*e*x^2 + (a^2*e^2*x^3)/3 + (2*a*c*d^2*x^5)/5 + (2*a*c*d*e *x^6)/3 + (2*a*c*e^2*x^7)/7 + (c^2*d^2*x^9)/9 + (c^2*d*e*x^10)/5 + (c^2*e^ 2*x^11)/11
Time = 0.48 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.14, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2389, 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.
\(\displaystyle \int \left (a+c x^4\right )^2 (d+e x)^2 \, dx\) |
\(\Big \downarrow \) 2389 |
\(\displaystyle \int \left (a^2 d^2+2 a^2 d e x+a^2 e^2 x^2+2 a c d^2 x^4+4 a c d e x^5+2 a c e^2 x^6+c^2 d^2 x^8+2 c^2 d e x^9+c^2 e^2 x^{10}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^2 d^2 x+a^2 d e x^2+\frac {1}{3} a^2 e^2 x^3+\frac {2}{5} a c d^2 x^5+\frac {2}{3} a c d e x^6+\frac {2}{7} a c e^2 x^7+\frac {1}{9} c^2 d^2 x^9+\frac {1}{5} c^2 d e x^{10}+\frac {1}{11} c^2 e^2 x^{11}\) |
Input:
Int[(d + e*x)^2*(a + c*x^4)^2,x]
Output:
a^2*d^2*x + a^2*d*e*x^2 + (a^2*e^2*x^3)/3 + (2*a*c*d^2*x^5)/5 + (2*a*c*d*e *x^6)/3 + (2*a*c*e^2*x^7)/7 + (c^2*d^2*x^9)/9 + (c^2*d*e*x^10)/5 + (c^2*e^ 2*x^11)/11
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Time = 0.15 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00
method | result | size |
gosper | \(\frac {1}{11} c^{2} e^{2} x^{11}+\frac {1}{5} c^{2} d e \,x^{10}+\frac {1}{9} c^{2} d^{2} x^{9}+\frac {2}{7} a c \,e^{2} x^{7}+\frac {2}{3} a c d e \,x^{6}+\frac {2}{5} a c \,d^{2} x^{5}+\frac {1}{3} x^{3} a^{2} e^{2}+a^{2} d e \,x^{2}+x \,a^{2} d^{2}\) | \(91\) |
default | \(\frac {1}{11} c^{2} e^{2} x^{11}+\frac {1}{5} c^{2} d e \,x^{10}+\frac {1}{9} c^{2} d^{2} x^{9}+\frac {2}{7} a c \,e^{2} x^{7}+\frac {2}{3} a c d e \,x^{6}+\frac {2}{5} a c \,d^{2} x^{5}+\frac {1}{3} x^{3} a^{2} e^{2}+a^{2} d e \,x^{2}+x \,a^{2} d^{2}\) | \(91\) |
norman | \(\frac {1}{11} c^{2} e^{2} x^{11}+\frac {1}{5} c^{2} d e \,x^{10}+\frac {1}{9} c^{2} d^{2} x^{9}+\frac {2}{7} a c \,e^{2} x^{7}+\frac {2}{3} a c d e \,x^{6}+\frac {2}{5} a c \,d^{2} x^{5}+\frac {1}{3} x^{3} a^{2} e^{2}+a^{2} d e \,x^{2}+x \,a^{2} d^{2}\) | \(91\) |
risch | \(\frac {1}{11} c^{2} e^{2} x^{11}+\frac {1}{5} c^{2} d e \,x^{10}+\frac {1}{9} c^{2} d^{2} x^{9}+\frac {2}{7} a c \,e^{2} x^{7}+\frac {2}{3} a c d e \,x^{6}+\frac {2}{5} a c \,d^{2} x^{5}+\frac {1}{3} x^{3} a^{2} e^{2}+a^{2} d e \,x^{2}+x \,a^{2} d^{2}\) | \(91\) |
parallelrisch | \(\frac {1}{11} c^{2} e^{2} x^{11}+\frac {1}{5} c^{2} d e \,x^{10}+\frac {1}{9} c^{2} d^{2} x^{9}+\frac {2}{7} a c \,e^{2} x^{7}+\frac {2}{3} a c d e \,x^{6}+\frac {2}{5} a c \,d^{2} x^{5}+\frac {1}{3} x^{3} a^{2} e^{2}+a^{2} d e \,x^{2}+x \,a^{2} d^{2}\) | \(91\) |
orering | \(\frac {x \left (315 c^{2} e^{2} x^{10}+693 c^{2} d e \,x^{9}+385 c^{2} d^{2} x^{8}+990 a c \,e^{2} x^{6}+2310 a c d e \,x^{5}+1386 d^{2} x^{4} a c +1155 a^{2} e^{2} x^{2}+3465 a^{2} d e x +3465 a^{2} d^{2}\right )}{3465}\) | \(93\) |
Input:
int((e*x+d)^2*(c*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
1/11*c^2*e^2*x^11+1/5*c^2*d*e*x^10+1/9*c^2*d^2*x^9+2/7*a*c*e^2*x^7+2/3*a*c *d*e*x^6+2/5*a*c*d^2*x^5+1/3*x^3*a^2*e^2+a^2*d*e*x^2+x*a^2*d^2
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.99 \[ \int (d+e x)^2 \left (a+c x^4\right )^2 \, dx=\frac {1}{11} \, c^{2} e^{2} x^{11} + \frac {1}{5} \, c^{2} d e x^{10} + \frac {1}{9} \, c^{2} d^{2} x^{9} + \frac {2}{7} \, a c e^{2} x^{7} + \frac {2}{3} \, a c d e x^{6} + \frac {2}{5} \, a c d^{2} x^{5} + \frac {1}{3} \, a^{2} e^{2} x^{3} + a^{2} d e x^{2} + a^{2} d^{2} x \] Input:
integrate((e*x+d)^2*(c*x^4+a)^2,x, algorithm="fricas")
Output:
1/11*c^2*e^2*x^11 + 1/5*c^2*d*e*x^10 + 1/9*c^2*d^2*x^9 + 2/7*a*c*e^2*x^7 + 2/3*a*c*d*e*x^6 + 2/5*a*c*d^2*x^5 + 1/3*a^2*e^2*x^3 + a^2*d*e*x^2 + a^2*d ^2*x
Time = 0.03 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.15 \[ \int (d+e x)^2 \left (a+c x^4\right )^2 \, dx=a^{2} d^{2} x + a^{2} d e x^{2} + \frac {a^{2} e^{2} x^{3}}{3} + \frac {2 a c d^{2} x^{5}}{5} + \frac {2 a c d e x^{6}}{3} + \frac {2 a c e^{2} x^{7}}{7} + \frac {c^{2} d^{2} x^{9}}{9} + \frac {c^{2} d e x^{10}}{5} + \frac {c^{2} e^{2} x^{11}}{11} \] Input:
integrate((e*x+d)**2*(c*x**4+a)**2,x)
Output:
a**2*d**2*x + a**2*d*e*x**2 + a**2*e**2*x**3/3 + 2*a*c*d**2*x**5/5 + 2*a*c *d*e*x**6/3 + 2*a*c*e**2*x**7/7 + c**2*d**2*x**9/9 + c**2*d*e*x**10/5 + c* *2*e**2*x**11/11
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.99 \[ \int (d+e x)^2 \left (a+c x^4\right )^2 \, dx=\frac {1}{11} \, c^{2} e^{2} x^{11} + \frac {1}{5} \, c^{2} d e x^{10} + \frac {1}{9} \, c^{2} d^{2} x^{9} + \frac {2}{7} \, a c e^{2} x^{7} + \frac {2}{3} \, a c d e x^{6} + \frac {2}{5} \, a c d^{2} x^{5} + \frac {1}{3} \, a^{2} e^{2} x^{3} + a^{2} d e x^{2} + a^{2} d^{2} x \] Input:
integrate((e*x+d)^2*(c*x^4+a)^2,x, algorithm="maxima")
Output:
1/11*c^2*e^2*x^11 + 1/5*c^2*d*e*x^10 + 1/9*c^2*d^2*x^9 + 2/7*a*c*e^2*x^7 + 2/3*a*c*d*e*x^6 + 2/5*a*c*d^2*x^5 + 1/3*a^2*e^2*x^3 + a^2*d*e*x^2 + a^2*d ^2*x
Time = 0.13 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.99 \[ \int (d+e x)^2 \left (a+c x^4\right )^2 \, dx=\frac {1}{11} \, c^{2} e^{2} x^{11} + \frac {1}{5} \, c^{2} d e x^{10} + \frac {1}{9} \, c^{2} d^{2} x^{9} + \frac {2}{7} \, a c e^{2} x^{7} + \frac {2}{3} \, a c d e x^{6} + \frac {2}{5} \, a c d^{2} x^{5} + \frac {1}{3} \, a^{2} e^{2} x^{3} + a^{2} d e x^{2} + a^{2} d^{2} x \] Input:
integrate((e*x+d)^2*(c*x^4+a)^2,x, algorithm="giac")
Output:
1/11*c^2*e^2*x^11 + 1/5*c^2*d*e*x^10 + 1/9*c^2*d^2*x^9 + 2/7*a*c*e^2*x^7 + 2/3*a*c*d*e*x^6 + 2/5*a*c*d^2*x^5 + 1/3*a^2*e^2*x^3 + a^2*d*e*x^2 + a^2*d ^2*x
Time = 0.05 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.99 \[ \int (d+e x)^2 \left (a+c x^4\right )^2 \, dx=a^2\,d^2\,x+a^2\,d\,e\,x^2+\frac {a^2\,e^2\,x^3}{3}+\frac {2\,a\,c\,d^2\,x^5}{5}+\frac {2\,a\,c\,d\,e\,x^6}{3}+\frac {2\,a\,c\,e^2\,x^7}{7}+\frac {c^2\,d^2\,x^9}{9}+\frac {c^2\,d\,e\,x^{10}}{5}+\frac {c^2\,e^2\,x^{11}}{11} \] Input:
int((a + c*x^4)^2*(d + e*x)^2,x)
Output:
a^2*d^2*x + (a^2*e^2*x^3)/3 + (c^2*d^2*x^9)/9 + (c^2*e^2*x^11)/11 + (2*a*c *d^2*x^5)/5 + (2*a*c*e^2*x^7)/7 + a^2*d*e*x^2 + (c^2*d*e*x^10)/5 + (2*a*c* d*e*x^6)/3
Time = 0.15 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.01 \[ \int (d+e x)^2 \left (a+c x^4\right )^2 \, dx=\frac {x \left (315 c^{2} e^{2} x^{10}+693 c^{2} d e \,x^{9}+385 c^{2} d^{2} x^{8}+990 a c \,e^{2} x^{6}+2310 a c d e \,x^{5}+1386 a c \,d^{2} x^{4}+1155 a^{2} e^{2} x^{2}+3465 a^{2} d e x +3465 a^{2} d^{2}\right )}{3465} \] Input:
int((e*x+d)^2*(c*x^4+a)^2,x)
Output:
(x*(3465*a**2*d**2 + 3465*a**2*d*e*x + 1155*a**2*e**2*x**2 + 1386*a*c*d**2 *x**4 + 2310*a*c*d*e*x**5 + 990*a*c*e**2*x**6 + 385*c**2*d**2*x**8 + 693*c **2*d*e*x**9 + 315*c**2*e**2*x**10))/3465