Integrand size = 17, antiderivative size = 119 \[ \int (d+e x)^3 \left (a+c x^4\right )^2 \, dx=\frac {2}{5} a c d^3 x^5+a c d^2 e x^6+\frac {6}{7} a c d e^2 x^7+\frac {1}{4} a c e^3 x^8+\frac {1}{9} c^2 d^3 x^9+\frac {3}{10} c^2 d^2 e x^{10}+\frac {3}{11} c^2 d e^2 x^{11}+\frac {1}{12} c^2 e^3 x^{12}+\frac {a^2 (d+e x)^4}{4 e} \] Output:
2/5*a*c*d^3*x^5+a*c*d^2*e*x^6+6/7*a*c*d*e^2*x^7+1/4*a*c*e^3*x^8+1/9*c^2*d^ 3*x^9+3/10*c^2*d^2*e*x^10+3/11*c^2*d*e^2*x^11+1/12*c^2*e^3*x^12+1/4*a^2*(e *x+d)^4/e
Time = 0.01 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.24 \[ \int (d+e x)^3 \left (a+c x^4\right )^2 \, dx=a^2 d^3 x+\frac {3}{2} a^2 d^2 e x^2+a^2 d e^2 x^3+\frac {1}{4} a^2 e^3 x^4+\frac {2}{5} a c d^3 x^5+a c d^2 e x^6+\frac {6}{7} a c d e^2 x^7+\frac {1}{4} a c e^3 x^8+\frac {1}{9} c^2 d^3 x^9+\frac {3}{10} c^2 d^2 e x^{10}+\frac {3}{11} c^2 d e^2 x^{11}+\frac {1}{12} c^2 e^3 x^{12} \] Input:
Integrate[(d + e*x)^3*(a + c*x^4)^2,x]
Output:
a^2*d^3*x + (3*a^2*d^2*e*x^2)/2 + a^2*d*e^2*x^3 + (a^2*e^3*x^4)/4 + (2*a*c *d^3*x^5)/5 + a*c*d^2*e*x^6 + (6*a*c*d*e^2*x^7)/7 + (a*c*e^3*x^8)/4 + (c^2 *d^3*x^9)/9 + (3*c^2*d^2*e*x^10)/10 + (3*c^2*d*e^2*x^11)/11 + (c^2*e^3*x^1 2)/12
Time = 0.58 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2017, 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.
\(\displaystyle \int \left (a+c x^4\right )^2 (d+e x)^3 \, dx\) |
\(\Big \downarrow \) 2017 |
\(\displaystyle \int \left (c x^4+a\right )^2 \left ((d+e x)^3-e^3 x^3\right )dx+\frac {e^3 \left (a+c x^4\right )^3}{12 c}\) |
\(\Big \downarrow \) 2389 |
\(\displaystyle \int \left (3 c^2 d e^2 x^{10}+3 c^2 d^2 e x^9+c^2 d^3 x^8+6 a c d e^2 x^6+6 a c d^2 e x^5+2 a c d^3 x^4+3 a^2 d e^2 x^2+3 a^2 d^2 e x+a^2 d^3\right )dx+\frac {e^3 \left (a+c x^4\right )^3}{12 c}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^2 d^3 x+\frac {3}{2} a^2 d^2 e x^2+a^2 d e^2 x^3+\frac {2}{5} a c d^3 x^5+a c d^2 e x^6+\frac {6}{7} a c d e^2 x^7+\frac {e^3 \left (a+c x^4\right )^3}{12 c}+\frac {1}{9} c^2 d^3 x^9+\frac {3}{10} c^2 d^2 e x^{10}+\frac {3}{11} c^2 d e^2 x^{11}\) |
Input:
Int[(d + e*x)^3*(a + c*x^4)^2,x]
Output:
a^2*d^3*x + (3*a^2*d^2*e*x^2)/2 + a^2*d*e^2*x^3 + (2*a*c*d^3*x^5)/5 + a*c* d^2*e*x^6 + (6*a*c*d*e^2*x^7)/7 + (c^2*d^3*x^9)/9 + (3*c^2*d^2*e*x^10)/10 + (3*c^2*d*e^2*x^11)/11 + (e^3*(a + c*x^4)^3)/(12*c)
Int[(Px_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[Coeff[Px, x, n - 1]*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] + Int[(Px - Coeff[Px, x, n - 1] *x^(n - 1))*(a + b*x^n)^p, x] /; FreeQ[{a, b}, x] && PolyQ[Px, x] && IGtQ[p , 1] && IGtQ[n, 1] && NeQ[Coeff[Px, x, n - 1], 0] && NeQ[Px, Coeff[Px, x, n - 1]*x^(n - 1)] && !MatchQ[Px, (Qx_.)*((c_) + (d_.)*x^(m_))^(q_) /; FreeQ [{c, d}, x] && PolyQ[Qx, x] && IGtQ[q, 1] && IGtQ[m, 1] && NeQ[Coeff[Qx*(a + b*x^n)^p, x, m - 1], 0] && GtQ[m*q, n*p]]
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 = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.10
method | result | size |
gosper | \(\frac {1}{12} c^{2} e^{3} x^{12}+\frac {3}{11} c^{2} d \,e^{2} x^{11}+\frac {3}{10} c^{2} d^{2} e \,x^{10}+\frac {1}{9} c^{2} d^{3} x^{9}+\frac {1}{4} a c \,e^{3} x^{8}+\frac {6}{7} a c d \,e^{2} x^{7}+a c \,d^{2} e \,x^{6}+\frac {2}{5} a c \,d^{3} x^{5}+\frac {1}{4} a^{2} e^{3} x^{4}+a^{2} d \,e^{2} x^{3}+\frac {3}{2} a^{2} d^{2} e \,x^{2}+a^{2} d^{3} x\) | \(131\) |
default | \(\frac {1}{12} c^{2} e^{3} x^{12}+\frac {3}{11} c^{2} d \,e^{2} x^{11}+\frac {3}{10} c^{2} d^{2} e \,x^{10}+\frac {1}{9} c^{2} d^{3} x^{9}+\frac {1}{4} a c \,e^{3} x^{8}+\frac {6}{7} a c d \,e^{2} x^{7}+a c \,d^{2} e \,x^{6}+\frac {2}{5} a c \,d^{3} x^{5}+\frac {1}{4} a^{2} e^{3} x^{4}+a^{2} d \,e^{2} x^{3}+\frac {3}{2} a^{2} d^{2} e \,x^{2}+a^{2} d^{3} x\) | \(131\) |
norman | \(\frac {1}{12} c^{2} e^{3} x^{12}+\frac {3}{11} c^{2} d \,e^{2} x^{11}+\frac {3}{10} c^{2} d^{2} e \,x^{10}+\frac {1}{9} c^{2} d^{3} x^{9}+\frac {1}{4} a c \,e^{3} x^{8}+\frac {6}{7} a c d \,e^{2} x^{7}+a c \,d^{2} e \,x^{6}+\frac {2}{5} a c \,d^{3} x^{5}+\frac {1}{4} a^{2} e^{3} x^{4}+a^{2} d \,e^{2} x^{3}+\frac {3}{2} a^{2} d^{2} e \,x^{2}+a^{2} d^{3} x\) | \(131\) |
risch | \(\frac {1}{12} c^{2} e^{3} x^{12}+\frac {3}{11} c^{2} d \,e^{2} x^{11}+\frac {3}{10} c^{2} d^{2} e \,x^{10}+\frac {1}{9} c^{2} d^{3} x^{9}+\frac {1}{4} a c \,e^{3} x^{8}+\frac {6}{7} a c d \,e^{2} x^{7}+a c \,d^{2} e \,x^{6}+\frac {2}{5} a c \,d^{3} x^{5}+\frac {1}{4} a^{2} e^{3} x^{4}+a^{2} d \,e^{2} x^{3}+\frac {3}{2} a^{2} d^{2} e \,x^{2}+a^{2} d^{3} x\) | \(131\) |
parallelrisch | \(\frac {1}{12} c^{2} e^{3} x^{12}+\frac {3}{11} c^{2} d \,e^{2} x^{11}+\frac {3}{10} c^{2} d^{2} e \,x^{10}+\frac {1}{9} c^{2} d^{3} x^{9}+\frac {1}{4} a c \,e^{3} x^{8}+\frac {6}{7} a c d \,e^{2} x^{7}+a c \,d^{2} e \,x^{6}+\frac {2}{5} a c \,d^{3} x^{5}+\frac {1}{4} a^{2} e^{3} x^{4}+a^{2} d \,e^{2} x^{3}+\frac {3}{2} a^{2} d^{2} e \,x^{2}+a^{2} d^{3} x\) | \(131\) |
orering | \(\frac {x \left (1155 e^{3} c^{2} x^{11}+3780 c^{2} d \,e^{2} x^{10}+4158 d^{2} e \,c^{2} x^{9}+1540 c^{2} d^{3} x^{8}+3465 a c \,e^{3} x^{7}+11880 a c d \,e^{2} x^{6}+13860 a c \,d^{2} e \,x^{5}+5544 a c \,d^{3} x^{4}+3465 a^{2} e^{3} x^{3}+13860 a^{2} d \,e^{2} x^{2}+20790 d^{2} e \,a^{2} x +13860 d^{3} a^{2}\right )}{13860}\) | \(134\) |
Input:
int((e*x+d)^3*(c*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
1/12*c^2*e^3*x^12+3/11*c^2*d*e^2*x^11+3/10*c^2*d^2*e*x^10+1/9*c^2*d^3*x^9+ 1/4*a*c*e^3*x^8+6/7*a*c*d*e^2*x^7+a*c*d^2*e*x^6+2/5*a*c*d^3*x^5+1/4*a^2*e^ 3*x^4+a^2*d*e^2*x^3+3/2*a^2*d^2*e*x^2+a^2*d^3*x
Time = 0.07 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.09 \[ \int (d+e x)^3 \left (a+c x^4\right )^2 \, dx=\frac {1}{12} \, c^{2} e^{3} x^{12} + \frac {3}{11} \, c^{2} d e^{2} x^{11} + \frac {3}{10} \, c^{2} d^{2} e x^{10} + \frac {1}{9} \, c^{2} d^{3} x^{9} + \frac {1}{4} \, a c e^{3} x^{8} + \frac {6}{7} \, a c d e^{2} x^{7} + a c d^{2} e x^{6} + \frac {2}{5} \, a c d^{3} x^{5} + \frac {1}{4} \, a^{2} e^{3} x^{4} + a^{2} d e^{2} x^{3} + \frac {3}{2} \, a^{2} d^{2} e x^{2} + a^{2} d^{3} x \] Input:
integrate((e*x+d)^3*(c*x^4+a)^2,x, algorithm="fricas")
Output:
1/12*c^2*e^3*x^12 + 3/11*c^2*d*e^2*x^11 + 3/10*c^2*d^2*e*x^10 + 1/9*c^2*d^ 3*x^9 + 1/4*a*c*e^3*x^8 + 6/7*a*c*d*e^2*x^7 + a*c*d^2*e*x^6 + 2/5*a*c*d^3* x^5 + 1/4*a^2*e^3*x^4 + a^2*d*e^2*x^3 + 3/2*a^2*d^2*e*x^2 + a^2*d^3*x
Time = 0.03 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.29 \[ \int (d+e x)^3 \left (a+c x^4\right )^2 \, dx=a^{2} d^{3} x + \frac {3 a^{2} d^{2} e x^{2}}{2} + a^{2} d e^{2} x^{3} + \frac {a^{2} e^{3} x^{4}}{4} + \frac {2 a c d^{3} x^{5}}{5} + a c d^{2} e x^{6} + \frac {6 a c d e^{2} x^{7}}{7} + \frac {a c e^{3} x^{8}}{4} + \frac {c^{2} d^{3} x^{9}}{9} + \frac {3 c^{2} d^{2} e x^{10}}{10} + \frac {3 c^{2} d e^{2} x^{11}}{11} + \frac {c^{2} e^{3} x^{12}}{12} \] Input:
integrate((e*x+d)**3*(c*x**4+a)**2,x)
Output:
a**2*d**3*x + 3*a**2*d**2*e*x**2/2 + a**2*d*e**2*x**3 + a**2*e**3*x**4/4 + 2*a*c*d**3*x**5/5 + a*c*d**2*e*x**6 + 6*a*c*d*e**2*x**7/7 + a*c*e**3*x**8 /4 + c**2*d**3*x**9/9 + 3*c**2*d**2*e*x**10/10 + 3*c**2*d*e**2*x**11/11 + c**2*e**3*x**12/12
Time = 0.03 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.09 \[ \int (d+e x)^3 \left (a+c x^4\right )^2 \, dx=\frac {1}{12} \, c^{2} e^{3} x^{12} + \frac {3}{11} \, c^{2} d e^{2} x^{11} + \frac {3}{10} \, c^{2} d^{2} e x^{10} + \frac {1}{9} \, c^{2} d^{3} x^{9} + \frac {1}{4} \, a c e^{3} x^{8} + \frac {6}{7} \, a c d e^{2} x^{7} + a c d^{2} e x^{6} + \frac {2}{5} \, a c d^{3} x^{5} + \frac {1}{4} \, a^{2} e^{3} x^{4} + a^{2} d e^{2} x^{3} + \frac {3}{2} \, a^{2} d^{2} e x^{2} + a^{2} d^{3} x \] Input:
integrate((e*x+d)^3*(c*x^4+a)^2,x, algorithm="maxima")
Output:
1/12*c^2*e^3*x^12 + 3/11*c^2*d*e^2*x^11 + 3/10*c^2*d^2*e*x^10 + 1/9*c^2*d^ 3*x^9 + 1/4*a*c*e^3*x^8 + 6/7*a*c*d*e^2*x^7 + a*c*d^2*e*x^6 + 2/5*a*c*d^3* x^5 + 1/4*a^2*e^3*x^4 + a^2*d*e^2*x^3 + 3/2*a^2*d^2*e*x^2 + a^2*d^3*x
Time = 0.11 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.09 \[ \int (d+e x)^3 \left (a+c x^4\right )^2 \, dx=\frac {1}{12} \, c^{2} e^{3} x^{12} + \frac {3}{11} \, c^{2} d e^{2} x^{11} + \frac {3}{10} \, c^{2} d^{2} e x^{10} + \frac {1}{9} \, c^{2} d^{3} x^{9} + \frac {1}{4} \, a c e^{3} x^{8} + \frac {6}{7} \, a c d e^{2} x^{7} + a c d^{2} e x^{6} + \frac {2}{5} \, a c d^{3} x^{5} + \frac {1}{4} \, a^{2} e^{3} x^{4} + a^{2} d e^{2} x^{3} + \frac {3}{2} \, a^{2} d^{2} e x^{2} + a^{2} d^{3} x \] Input:
integrate((e*x+d)^3*(c*x^4+a)^2,x, algorithm="giac")
Output:
1/12*c^2*e^3*x^12 + 3/11*c^2*d*e^2*x^11 + 3/10*c^2*d^2*e*x^10 + 1/9*c^2*d^ 3*x^9 + 1/4*a*c*e^3*x^8 + 6/7*a*c*d*e^2*x^7 + a*c*d^2*e*x^6 + 2/5*a*c*d^3* x^5 + 1/4*a^2*e^3*x^4 + a^2*d*e^2*x^3 + 3/2*a^2*d^2*e*x^2 + a^2*d^3*x
Time = 21.50 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.09 \[ \int (d+e x)^3 \left (a+c x^4\right )^2 \, dx=a^2\,d^3\,x+\frac {3\,a^2\,d^2\,e\,x^2}{2}+a^2\,d\,e^2\,x^3+\frac {a^2\,e^3\,x^4}{4}+\frac {2\,a\,c\,d^3\,x^5}{5}+a\,c\,d^2\,e\,x^6+\frac {6\,a\,c\,d\,e^2\,x^7}{7}+\frac {a\,c\,e^3\,x^8}{4}+\frac {c^2\,d^3\,x^9}{9}+\frac {3\,c^2\,d^2\,e\,x^{10}}{10}+\frac {3\,c^2\,d\,e^2\,x^{11}}{11}+\frac {c^2\,e^3\,x^{12}}{12} \] Input:
int((a + c*x^4)^2*(d + e*x)^3,x)
Output:
a^2*d^3*x + (a^2*e^3*x^4)/4 + (c^2*d^3*x^9)/9 + (c^2*e^3*x^12)/12 + (3*a^2 *d^2*e*x^2)/2 + a^2*d*e^2*x^3 + (3*c^2*d^2*e*x^10)/10 + (3*c^2*d*e^2*x^11) /11 + (2*a*c*d^3*x^5)/5 + (a*c*e^3*x^8)/4 + a*c*d^2*e*x^6 + (6*a*c*d*e^2*x ^7)/7
Time = 0.14 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.12 \[ \int (d+e x)^3 \left (a+c x^4\right )^2 \, dx=\frac {x \left (1155 c^{2} e^{3} x^{11}+3780 c^{2} d \,e^{2} x^{10}+4158 c^{2} d^{2} e \,x^{9}+1540 c^{2} d^{3} x^{8}+3465 a c \,e^{3} x^{7}+11880 a c d \,e^{2} x^{6}+13860 a c \,d^{2} e \,x^{5}+5544 a c \,d^{3} x^{4}+3465 a^{2} e^{3} x^{3}+13860 a^{2} d \,e^{2} x^{2}+20790 a^{2} d^{2} e x +13860 a^{2} d^{3}\right )}{13860} \] Input:
int((e*x+d)^3*(c*x^4+a)^2,x)
Output:
(x*(13860*a**2*d**3 + 20790*a**2*d**2*e*x + 13860*a**2*d*e**2*x**2 + 3465* a**2*e**3*x**3 + 5544*a*c*d**3*x**4 + 13860*a*c*d**2*e*x**5 + 11880*a*c*d* e**2*x**6 + 3465*a*c*e**3*x**7 + 1540*c**2*d**3*x**8 + 4158*c**2*d**2*e*x* *9 + 3780*c**2*d*e**2*x**10 + 1155*c**2*e**3*x**11))/13860