Integrand size = 25, antiderivative size = 109 \[ \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^2 \, dx=a^2 c x+\frac {1}{2} a^2 d x^2+\frac {1}{3} a^2 e x^3+\frac {2}{5} a b c x^5+\frac {1}{3} a b d x^6+\frac {2}{7} a b e x^7+\frac {1}{9} b^2 c x^9+\frac {1}{10} b^2 d x^{10}+\frac {1}{11} b^2 e x^{11}+\frac {f \left (a+b x^4\right )^3}{12 b} \]
a^2*c*x+1/2*a^2*d*x^2+1/3*a^2*e*x^3+2/5*a*b*c*x^5+1/3*a*b*d*x^6+2/7*a*b*e* x^7+1/9*b^2*c*x^9+1/10*b^2*d*x^10+1/11*b^2*e*x^11+1/12*f*(b*x^4+a)^3/b
Time = 0.00 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.14 \[ \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^2 \, dx=a^2 c x+\frac {1}{2} a^2 d x^2+\frac {1}{3} a^2 e x^3+\frac {1}{4} a^2 f x^4+\frac {2}{5} a b c x^5+\frac {1}{3} a b d x^6+\frac {2}{7} a b e x^7+\frac {1}{4} a b f x^8+\frac {1}{9} b^2 c x^9+\frac {1}{10} b^2 d x^{10}+\frac {1}{11} b^2 e x^{11}+\frac {1}{12} b^2 f x^{12} \]
a^2*c*x + (a^2*d*x^2)/2 + (a^2*e*x^3)/3 + (a^2*f*x^4)/4 + (2*a*b*c*x^5)/5 + (a*b*d*x^6)/3 + (2*a*b*e*x^7)/7 + (a*b*f*x^8)/4 + (b^2*c*x^9)/9 + (b^2*d *x^10)/10 + (b^2*e*x^11)/11 + (b^2*f*x^12)/12
Time = 0.30 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2017, 2188, 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.
\(\displaystyle \int \left (a+b x^4\right )^2 \left (c+d x+e x^2+f x^3\right ) \, dx\) |
\(\Big \downarrow \) 2017 |
\(\displaystyle \int \left (e x^2+d x+c\right ) \left (b x^4+a\right )^2dx+\frac {f \left (a+b x^4\right )^3}{12 b}\) |
\(\Big \downarrow \) 2188 |
\(\displaystyle \int \left (b^2 e x^{10}+b^2 d x^9+b^2 c x^8+2 a b e x^6+2 a b d x^5+2 a b c x^4+a^2 e x^2+a^2 d x+a^2 c\right )dx+\frac {f \left (a+b x^4\right )^3}{12 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^2 c x+\frac {1}{2} a^2 d x^2+\frac {1}{3} a^2 e x^3+\frac {2}{5} a b c x^5+\frac {1}{3} a b d x^6+\frac {2}{7} a b e x^7+\frac {f \left (a+b x^4\right )^3}{12 b}+\frac {1}{9} b^2 c x^9+\frac {1}{10} b^2 d x^{10}+\frac {1}{11} b^2 e x^{11}\) |
a^2*c*x + (a^2*d*x^2)/2 + (a^2*e*x^3)/3 + (2*a*b*c*x^5)/5 + (a*b*d*x^6)/3 + (2*a*b*e*x^7)/7 + (b^2*c*x^9)/9 + (b^2*d*x^10)/10 + (b^2*e*x^11)/11 + (f *(a + b*x^4)^3)/(12*b)
3.5.79.3.1 Defintions of rubi rules used
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_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq , x] && IGtQ[p, -2]
Time = 1.66 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.94
method | result | size |
gosper | \(\frac {1}{12} b^{2} f \,x^{12}+\frac {1}{11} b^{2} e \,x^{11}+\frac {1}{10} b^{2} d \,x^{10}+\frac {1}{9} b^{2} c \,x^{9}+\frac {1}{4} a b f \,x^{8}+\frac {2}{7} a b e \,x^{7}+\frac {1}{3} a b d \,x^{6}+\frac {2}{5} a b c \,x^{5}+\frac {1}{4} a^{2} f \,x^{4}+\frac {1}{3} a^{2} e \,x^{3}+\frac {1}{2} a^{2} d \,x^{2}+a^{2} c x\) | \(103\) |
default | \(\frac {1}{12} b^{2} f \,x^{12}+\frac {1}{11} b^{2} e \,x^{11}+\frac {1}{10} b^{2} d \,x^{10}+\frac {1}{9} b^{2} c \,x^{9}+\frac {1}{4} a b f \,x^{8}+\frac {2}{7} a b e \,x^{7}+\frac {1}{3} a b d \,x^{6}+\frac {2}{5} a b c \,x^{5}+\frac {1}{4} a^{2} f \,x^{4}+\frac {1}{3} a^{2} e \,x^{3}+\frac {1}{2} a^{2} d \,x^{2}+a^{2} c x\) | \(103\) |
norman | \(\frac {1}{12} b^{2} f \,x^{12}+\frac {1}{11} b^{2} e \,x^{11}+\frac {1}{10} b^{2} d \,x^{10}+\frac {1}{9} b^{2} c \,x^{9}+\frac {1}{4} a b f \,x^{8}+\frac {2}{7} a b e \,x^{7}+\frac {1}{3} a b d \,x^{6}+\frac {2}{5} a b c \,x^{5}+\frac {1}{4} a^{2} f \,x^{4}+\frac {1}{3} a^{2} e \,x^{3}+\frac {1}{2} a^{2} d \,x^{2}+a^{2} c x\) | \(103\) |
risch | \(\frac {1}{12} b^{2} f \,x^{12}+\frac {1}{11} b^{2} e \,x^{11}+\frac {1}{10} b^{2} d \,x^{10}+\frac {1}{9} b^{2} c \,x^{9}+\frac {1}{4} a b f \,x^{8}+\frac {2}{7} a b e \,x^{7}+\frac {1}{3} a b d \,x^{6}+\frac {2}{5} a b c \,x^{5}+\frac {1}{4} a^{2} f \,x^{4}+\frac {1}{3} a^{2} e \,x^{3}+\frac {1}{2} a^{2} d \,x^{2}+a^{2} c x\) | \(103\) |
parallelrisch | \(\frac {1}{12} b^{2} f \,x^{12}+\frac {1}{11} b^{2} e \,x^{11}+\frac {1}{10} b^{2} d \,x^{10}+\frac {1}{9} b^{2} c \,x^{9}+\frac {1}{4} a b f \,x^{8}+\frac {2}{7} a b e \,x^{7}+\frac {1}{3} a b d \,x^{6}+\frac {2}{5} a b c \,x^{5}+\frac {1}{4} a^{2} f \,x^{4}+\frac {1}{3} a^{2} e \,x^{3}+\frac {1}{2} a^{2} d \,x^{2}+a^{2} c x\) | \(103\) |
1/12*b^2*f*x^12+1/11*b^2*e*x^11+1/10*b^2*d*x^10+1/9*b^2*c*x^9+1/4*a*b*f*x^ 8+2/7*a*b*e*x^7+1/3*a*b*d*x^6+2/5*a*b*c*x^5+1/4*a^2*f*x^4+1/3*a^2*e*x^3+1/ 2*a^2*d*x^2+a^2*c*x
Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.94 \[ \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^2 \, dx=\frac {1}{12} \, b^{2} f x^{12} + \frac {1}{11} \, b^{2} e x^{11} + \frac {1}{10} \, b^{2} d x^{10} + \frac {1}{9} \, b^{2} c x^{9} + \frac {1}{4} \, a b f x^{8} + \frac {2}{7} \, a b e x^{7} + \frac {1}{3} \, a b d x^{6} + \frac {2}{5} \, a b c x^{5} + \frac {1}{4} \, a^{2} f x^{4} + \frac {1}{3} \, a^{2} e x^{3} + \frac {1}{2} \, a^{2} d x^{2} + a^{2} c x \]
1/12*b^2*f*x^12 + 1/11*b^2*e*x^11 + 1/10*b^2*d*x^10 + 1/9*b^2*c*x^9 + 1/4* a*b*f*x^8 + 2/7*a*b*e*x^7 + 1/3*a*b*d*x^6 + 2/5*a*b*c*x^5 + 1/4*a^2*f*x^4 + 1/3*a^2*e*x^3 + 1/2*a^2*d*x^2 + a^2*c*x
Time = 0.03 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.11 \[ \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^2 \, dx=a^{2} c x + \frac {a^{2} d x^{2}}{2} + \frac {a^{2} e x^{3}}{3} + \frac {a^{2} f x^{4}}{4} + \frac {2 a b c x^{5}}{5} + \frac {a b d x^{6}}{3} + \frac {2 a b e x^{7}}{7} + \frac {a b f x^{8}}{4} + \frac {b^{2} c x^{9}}{9} + \frac {b^{2} d x^{10}}{10} + \frac {b^{2} e x^{11}}{11} + \frac {b^{2} f x^{12}}{12} \]
a**2*c*x + a**2*d*x**2/2 + a**2*e*x**3/3 + a**2*f*x**4/4 + 2*a*b*c*x**5/5 + a*b*d*x**6/3 + 2*a*b*e*x**7/7 + a*b*f*x**8/4 + b**2*c*x**9/9 + b**2*d*x* *10/10 + b**2*e*x**11/11 + b**2*f*x**12/12
Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.94 \[ \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^2 \, dx=\frac {1}{12} \, b^{2} f x^{12} + \frac {1}{11} \, b^{2} e x^{11} + \frac {1}{10} \, b^{2} d x^{10} + \frac {1}{9} \, b^{2} c x^{9} + \frac {1}{4} \, a b f x^{8} + \frac {2}{7} \, a b e x^{7} + \frac {1}{3} \, a b d x^{6} + \frac {2}{5} \, a b c x^{5} + \frac {1}{4} \, a^{2} f x^{4} + \frac {1}{3} \, a^{2} e x^{3} + \frac {1}{2} \, a^{2} d x^{2} + a^{2} c x \]
1/12*b^2*f*x^12 + 1/11*b^2*e*x^11 + 1/10*b^2*d*x^10 + 1/9*b^2*c*x^9 + 1/4* a*b*f*x^8 + 2/7*a*b*e*x^7 + 1/3*a*b*d*x^6 + 2/5*a*b*c*x^5 + 1/4*a^2*f*x^4 + 1/3*a^2*e*x^3 + 1/2*a^2*d*x^2 + a^2*c*x
Time = 0.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.94 \[ \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^2 \, dx=\frac {1}{12} \, b^{2} f x^{12} + \frac {1}{11} \, b^{2} e x^{11} + \frac {1}{10} \, b^{2} d x^{10} + \frac {1}{9} \, b^{2} c x^{9} + \frac {1}{4} \, a b f x^{8} + \frac {2}{7} \, a b e x^{7} + \frac {1}{3} \, a b d x^{6} + \frac {2}{5} \, a b c x^{5} + \frac {1}{4} \, a^{2} f x^{4} + \frac {1}{3} \, a^{2} e x^{3} + \frac {1}{2} \, a^{2} d x^{2} + a^{2} c x \]
1/12*b^2*f*x^12 + 1/11*b^2*e*x^11 + 1/10*b^2*d*x^10 + 1/9*b^2*c*x^9 + 1/4* a*b*f*x^8 + 2/7*a*b*e*x^7 + 1/3*a*b*d*x^6 + 2/5*a*b*c*x^5 + 1/4*a^2*f*x^4 + 1/3*a^2*e*x^3 + 1/2*a^2*d*x^2 + a^2*c*x
Time = 0.09 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.94 \[ \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^2 \, dx=\frac {f\,a^2\,x^4}{4}+\frac {e\,a^2\,x^3}{3}+\frac {d\,a^2\,x^2}{2}+c\,a^2\,x+\frac {f\,a\,b\,x^8}{4}+\frac {2\,e\,a\,b\,x^7}{7}+\frac {d\,a\,b\,x^6}{3}+\frac {2\,c\,a\,b\,x^5}{5}+\frac {f\,b^2\,x^{12}}{12}+\frac {e\,b^2\,x^{11}}{11}+\frac {d\,b^2\,x^{10}}{10}+\frac {c\,b^2\,x^9}{9} \]