Integrand size = 20, antiderivative size = 149 \[ \int x^3 \left (d+e x^2\right ) \left (a+c x^4\right )^5 \, dx=\frac {1}{4} a^5 d x^4+\frac {1}{6} a^5 e x^6+\frac {5}{8} a^4 c d x^8+\frac {1}{2} a^4 c e x^{10}+\frac {5}{6} a^3 c^2 d x^{12}+\frac {5}{7} a^3 c^2 e x^{14}+\frac {5}{8} a^2 c^3 d x^{16}+\frac {5}{9} a^2 c^3 e x^{18}+\frac {1}{4} a c^4 d x^{20}+\frac {5}{22} a c^4 e x^{22}+\frac {1}{24} c^5 d x^{24}+\frac {1}{26} c^5 e x^{26} \]
1/4*a^5*d*x^4+1/6*a^5*e*x^6+5/8*a^4*c*d*x^8+1/2*a^4*c*e*x^10+5/6*a^3*c^2*d *x^12+5/7*a^3*c^2*e*x^14+5/8*a^2*c^3*d*x^16+5/9*a^2*c^3*e*x^18+1/4*a*c^4*d *x^20+5/22*a*c^4*e*x^22+1/24*c^5*d*x^24+1/26*c^5*e*x^26
Time = 0.01 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00 \[ \int x^3 \left (d+e x^2\right ) \left (a+c x^4\right )^5 \, dx=\frac {1}{4} a^5 d x^4+\frac {1}{6} a^5 e x^6+\frac {5}{8} a^4 c d x^8+\frac {1}{2} a^4 c e x^{10}+\frac {5}{6} a^3 c^2 d x^{12}+\frac {5}{7} a^3 c^2 e x^{14}+\frac {5}{8} a^2 c^3 d x^{16}+\frac {5}{9} a^2 c^3 e x^{18}+\frac {1}{4} a c^4 d x^{20}+\frac {5}{22} a c^4 e x^{22}+\frac {1}{24} c^5 d x^{24}+\frac {1}{26} c^5 e x^{26} \]
(a^5*d*x^4)/4 + (a^5*e*x^6)/6 + (5*a^4*c*d*x^8)/8 + (a^4*c*e*x^10)/2 + (5* a^3*c^2*d*x^12)/6 + (5*a^3*c^2*e*x^14)/7 + (5*a^2*c^3*d*x^16)/8 + (5*a^2*c ^3*e*x^18)/9 + (a*c^4*d*x^20)/4 + (5*a*c^4*e*x^22)/22 + (c^5*d*x^24)/24 + (c^5*e*x^26)/26
Time = 0.35 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1579, 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.
\(\displaystyle \int x^3 \left (a+c x^4\right )^5 \left (d+e x^2\right ) \, dx\) |
\(\Big \downarrow \) 1579 |
\(\displaystyle \frac {1}{2} \int x^2 \left (e x^2+d\right ) \left (c x^4+a\right )^5dx^2\) |
\(\Big \downarrow \) 522 |
\(\displaystyle \frac {1}{2} \int \left (c^5 e x^{24}+c^5 d x^{22}+5 a c^4 e x^{20}+5 a c^4 d x^{18}+10 a^2 c^3 e x^{16}+10 a^2 c^3 d x^{14}+10 a^3 c^2 e x^{12}+10 a^3 c^2 d x^{10}+5 a^4 c e x^8+5 a^4 c d x^6+a^5 e x^4+a^5 d x^2\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} a^5 d x^4+\frac {1}{3} a^5 e x^6+\frac {5}{4} a^4 c d x^8+a^4 c e x^{10}+\frac {5}{3} a^3 c^2 d x^{12}+\frac {10}{7} a^3 c^2 e x^{14}+\frac {5}{4} a^2 c^3 d x^{16}+\frac {10}{9} a^2 c^3 e x^{18}+\frac {1}{2} a c^4 d x^{20}+\frac {5}{11} a c^4 e x^{22}+\frac {1}{12} c^5 d x^{24}+\frac {1}{13} c^5 e x^{26}\right )\) |
((a^5*d*x^4)/2 + (a^5*e*x^6)/3 + (5*a^4*c*d*x^8)/4 + a^4*c*e*x^10 + (5*a^3 *c^2*d*x^12)/3 + (10*a^3*c^2*e*x^14)/7 + (5*a^2*c^3*d*x^16)/4 + (10*a^2*c^ 3*e*x^18)/9 + (a*c^4*d*x^20)/2 + (5*a*c^4*e*x^22)/11 + (c^5*d*x^24)/12 + ( c^5*e*x^26)/13)/2
3.1.1.3.1 Defintions of rubi rules used
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]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]
Time = 0.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.85
method | result | size |
gosper | \(\frac {1}{4} a^{5} d \,x^{4}+\frac {1}{6} a^{5} e \,x^{6}+\frac {5}{8} a^{4} c d \,x^{8}+\frac {1}{2} a^{4} c e \,x^{10}+\frac {5}{6} a^{3} c^{2} d \,x^{12}+\frac {5}{7} a^{3} c^{2} e \,x^{14}+\frac {5}{8} a^{2} c^{3} d \,x^{16}+\frac {5}{9} a^{2} c^{3} e \,x^{18}+\frac {1}{4} a \,c^{4} d \,x^{20}+\frac {5}{22} a \,c^{4} e \,x^{22}+\frac {1}{24} c^{5} d \,x^{24}+\frac {1}{26} c^{5} e \,x^{26}\) | \(126\) |
default | \(\frac {1}{4} a^{5} d \,x^{4}+\frac {1}{6} a^{5} e \,x^{6}+\frac {5}{8} a^{4} c d \,x^{8}+\frac {1}{2} a^{4} c e \,x^{10}+\frac {5}{6} a^{3} c^{2} d \,x^{12}+\frac {5}{7} a^{3} c^{2} e \,x^{14}+\frac {5}{8} a^{2} c^{3} d \,x^{16}+\frac {5}{9} a^{2} c^{3} e \,x^{18}+\frac {1}{4} a \,c^{4} d \,x^{20}+\frac {5}{22} a \,c^{4} e \,x^{22}+\frac {1}{24} c^{5} d \,x^{24}+\frac {1}{26} c^{5} e \,x^{26}\) | \(126\) |
norman | \(\frac {1}{4} a^{5} d \,x^{4}+\frac {1}{6} a^{5} e \,x^{6}+\frac {5}{8} a^{4} c d \,x^{8}+\frac {1}{2} a^{4} c e \,x^{10}+\frac {5}{6} a^{3} c^{2} d \,x^{12}+\frac {5}{7} a^{3} c^{2} e \,x^{14}+\frac {5}{8} a^{2} c^{3} d \,x^{16}+\frac {5}{9} a^{2} c^{3} e \,x^{18}+\frac {1}{4} a \,c^{4} d \,x^{20}+\frac {5}{22} a \,c^{4} e \,x^{22}+\frac {1}{24} c^{5} d \,x^{24}+\frac {1}{26} c^{5} e \,x^{26}\) | \(126\) |
risch | \(\frac {1}{4} a^{5} d \,x^{4}+\frac {1}{6} a^{5} e \,x^{6}+\frac {5}{8} a^{4} c d \,x^{8}+\frac {1}{2} a^{4} c e \,x^{10}+\frac {5}{6} a^{3} c^{2} d \,x^{12}+\frac {5}{7} a^{3} c^{2} e \,x^{14}+\frac {5}{8} a^{2} c^{3} d \,x^{16}+\frac {5}{9} a^{2} c^{3} e \,x^{18}+\frac {1}{4} a \,c^{4} d \,x^{20}+\frac {5}{22} a \,c^{4} e \,x^{22}+\frac {1}{24} c^{5} d \,x^{24}+\frac {1}{26} c^{5} e \,x^{26}\) | \(126\) |
parallelrisch | \(\frac {1}{4} a^{5} d \,x^{4}+\frac {1}{6} a^{5} e \,x^{6}+\frac {5}{8} a^{4} c d \,x^{8}+\frac {1}{2} a^{4} c e \,x^{10}+\frac {5}{6} a^{3} c^{2} d \,x^{12}+\frac {5}{7} a^{3} c^{2} e \,x^{14}+\frac {5}{8} a^{2} c^{3} d \,x^{16}+\frac {5}{9} a^{2} c^{3} e \,x^{18}+\frac {1}{4} a \,c^{4} d \,x^{20}+\frac {5}{22} a \,c^{4} e \,x^{22}+\frac {1}{24} c^{5} d \,x^{24}+\frac {1}{26} c^{5} e \,x^{26}\) | \(126\) |
1/4*a^5*d*x^4+1/6*a^5*e*x^6+5/8*a^4*c*d*x^8+1/2*a^4*c*e*x^10+5/6*a^3*c^2*d *x^12+5/7*a^3*c^2*e*x^14+5/8*a^2*c^3*d*x^16+5/9*a^2*c^3*e*x^18+1/4*a*c^4*d *x^20+5/22*a*c^4*e*x^22+1/24*c^5*d*x^24+1/26*c^5*e*x^26
Time = 0.25 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.84 \[ \int x^3 \left (d+e x^2\right ) \left (a+c x^4\right )^5 \, dx=\frac {1}{26} \, c^{5} e x^{26} + \frac {1}{24} \, c^{5} d x^{24} + \frac {5}{22} \, a c^{4} e x^{22} + \frac {1}{4} \, a c^{4} d x^{20} + \frac {5}{9} \, a^{2} c^{3} e x^{18} + \frac {5}{8} \, a^{2} c^{3} d x^{16} + \frac {5}{7} \, a^{3} c^{2} e x^{14} + \frac {5}{6} \, a^{3} c^{2} d x^{12} + \frac {1}{2} \, a^{4} c e x^{10} + \frac {5}{8} \, a^{4} c d x^{8} + \frac {1}{6} \, a^{5} e x^{6} + \frac {1}{4} \, a^{5} d x^{4} \]
1/26*c^5*e*x^26 + 1/24*c^5*d*x^24 + 5/22*a*c^4*e*x^22 + 1/4*a*c^4*d*x^20 + 5/9*a^2*c^3*e*x^18 + 5/8*a^2*c^3*d*x^16 + 5/7*a^3*c^2*e*x^14 + 5/6*a^3*c^ 2*d*x^12 + 1/2*a^4*c*e*x^10 + 5/8*a^4*c*d*x^8 + 1/6*a^5*e*x^6 + 1/4*a^5*d* x^4
Time = 0.02 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.01 \[ \int x^3 \left (d+e x^2\right ) \left (a+c x^4\right )^5 \, dx=\frac {a^{5} d x^{4}}{4} + \frac {a^{5} e x^{6}}{6} + \frac {5 a^{4} c d x^{8}}{8} + \frac {a^{4} c e x^{10}}{2} + \frac {5 a^{3} c^{2} d x^{12}}{6} + \frac {5 a^{3} c^{2} e x^{14}}{7} + \frac {5 a^{2} c^{3} d x^{16}}{8} + \frac {5 a^{2} c^{3} e x^{18}}{9} + \frac {a c^{4} d x^{20}}{4} + \frac {5 a c^{4} e x^{22}}{22} + \frac {c^{5} d x^{24}}{24} + \frac {c^{5} e x^{26}}{26} \]
a**5*d*x**4/4 + a**5*e*x**6/6 + 5*a**4*c*d*x**8/8 + a**4*c*e*x**10/2 + 5*a **3*c**2*d*x**12/6 + 5*a**3*c**2*e*x**14/7 + 5*a**2*c**3*d*x**16/8 + 5*a** 2*c**3*e*x**18/9 + a*c**4*d*x**20/4 + 5*a*c**4*e*x**22/22 + c**5*d*x**24/2 4 + c**5*e*x**26/26
Time = 0.19 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.84 \[ \int x^3 \left (d+e x^2\right ) \left (a+c x^4\right )^5 \, dx=\frac {1}{26} \, c^{5} e x^{26} + \frac {1}{24} \, c^{5} d x^{24} + \frac {5}{22} \, a c^{4} e x^{22} + \frac {1}{4} \, a c^{4} d x^{20} + \frac {5}{9} \, a^{2} c^{3} e x^{18} + \frac {5}{8} \, a^{2} c^{3} d x^{16} + \frac {5}{7} \, a^{3} c^{2} e x^{14} + \frac {5}{6} \, a^{3} c^{2} d x^{12} + \frac {1}{2} \, a^{4} c e x^{10} + \frac {5}{8} \, a^{4} c d x^{8} + \frac {1}{6} \, a^{5} e x^{6} + \frac {1}{4} \, a^{5} d x^{4} \]
1/26*c^5*e*x^26 + 1/24*c^5*d*x^24 + 5/22*a*c^4*e*x^22 + 1/4*a*c^4*d*x^20 + 5/9*a^2*c^3*e*x^18 + 5/8*a^2*c^3*d*x^16 + 5/7*a^3*c^2*e*x^14 + 5/6*a^3*c^ 2*d*x^12 + 1/2*a^4*c*e*x^10 + 5/8*a^4*c*d*x^8 + 1/6*a^5*e*x^6 + 1/4*a^5*d* x^4
Time = 0.26 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.84 \[ \int x^3 \left (d+e x^2\right ) \left (a+c x^4\right )^5 \, dx=\frac {1}{26} \, c^{5} e x^{26} + \frac {1}{24} \, c^{5} d x^{24} + \frac {5}{22} \, a c^{4} e x^{22} + \frac {1}{4} \, a c^{4} d x^{20} + \frac {5}{9} \, a^{2} c^{3} e x^{18} + \frac {5}{8} \, a^{2} c^{3} d x^{16} + \frac {5}{7} \, a^{3} c^{2} e x^{14} + \frac {5}{6} \, a^{3} c^{2} d x^{12} + \frac {1}{2} \, a^{4} c e x^{10} + \frac {5}{8} \, a^{4} c d x^{8} + \frac {1}{6} \, a^{5} e x^{6} + \frac {1}{4} \, a^{5} d x^{4} \]
1/26*c^5*e*x^26 + 1/24*c^5*d*x^24 + 5/22*a*c^4*e*x^22 + 1/4*a*c^4*d*x^20 + 5/9*a^2*c^3*e*x^18 + 5/8*a^2*c^3*d*x^16 + 5/7*a^3*c^2*e*x^14 + 5/6*a^3*c^ 2*d*x^12 + 1/2*a^4*c*e*x^10 + 5/8*a^4*c*d*x^8 + 1/6*a^5*e*x^6 + 1/4*a^5*d* x^4
Time = 0.06 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.84 \[ \int x^3 \left (d+e x^2\right ) \left (a+c x^4\right )^5 \, dx=\frac {e\,a^5\,x^6}{6}+\frac {d\,a^5\,x^4}{4}+\frac {e\,a^4\,c\,x^{10}}{2}+\frac {5\,d\,a^4\,c\,x^8}{8}+\frac {5\,e\,a^3\,c^2\,x^{14}}{7}+\frac {5\,d\,a^3\,c^2\,x^{12}}{6}+\frac {5\,e\,a^2\,c^3\,x^{18}}{9}+\frac {5\,d\,a^2\,c^3\,x^{16}}{8}+\frac {5\,e\,a\,c^4\,x^{22}}{22}+\frac {d\,a\,c^4\,x^{20}}{4}+\frac {e\,c^5\,x^{26}}{26}+\frac {d\,c^5\,x^{24}}{24} \]