Integrand size = 23, antiderivative size = 16 \[ \int x^{28} \left (b+2 c x^3\right ) \left (b x+c x^4\right )^{13} \, dx=\frac {1}{42} x^{42} \left (b+c x^3\right )^{14} \] Output:
1/42*x^42*(c*x^3+b)^14
Leaf count is larger than twice the leaf count of optimal. \(186\) vs. \(2(16)=32\).
Time = 0.01 (sec) , antiderivative size = 186, normalized size of antiderivative = 11.62 \[ \int x^{28} \left (b+2 c x^3\right ) \left (b x+c x^4\right )^{13} \, dx=\frac {b^{14} x^{42}}{42}+\frac {1}{3} b^{13} c x^{45}+\frac {13}{6} b^{12} c^2 x^{48}+\frac {26}{3} b^{11} c^3 x^{51}+\frac {143}{6} b^{10} c^4 x^{54}+\frac {143}{3} b^9 c^5 x^{57}+\frac {143}{2} b^8 c^6 x^{60}+\frac {572}{7} b^7 c^7 x^{63}+\frac {143}{2} b^6 c^8 x^{66}+\frac {143}{3} b^5 c^9 x^{69}+\frac {143}{6} b^4 c^{10} x^{72}+\frac {26}{3} b^3 c^{11} x^{75}+\frac {13}{6} b^2 c^{12} x^{78}+\frac {1}{3} b c^{13} x^{81}+\frac {c^{14} x^{84}}{42} \] Input:
Integrate[x^28*(b + 2*c*x^3)*(b*x + c*x^4)^13,x]
Output:
(b^14*x^42)/42 + (b^13*c*x^45)/3 + (13*b^12*c^2*x^48)/6 + (26*b^11*c^3*x^5 1)/3 + (143*b^10*c^4*x^54)/6 + (143*b^9*c^5*x^57)/3 + (143*b^8*c^6*x^60)/2 + (572*b^7*c^7*x^63)/7 + (143*b^6*c^8*x^66)/2 + (143*b^5*c^9*x^69)/3 + (1 43*b^4*c^10*x^72)/6 + (26*b^3*c^11*x^75)/3 + (13*b^2*c^12*x^78)/6 + (b*c^1 3*x^81)/3 + (c^14*x^84)/42
Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {9, 948, 83}
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.
\(\displaystyle \int x^{28} \left (b+2 c x^3\right ) \left (b x+c x^4\right )^{13} \, dx\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \int x^{41} \left (b+c x^3\right )^{13} \left (b+2 c x^3\right )dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{3} \int x^{39} \left (c x^3+b\right )^{13} \left (2 c x^3+b\right )dx^3\) |
\(\Big \downarrow \) 83 |
\(\displaystyle \frac {1}{42} x^{42} \left (b+c x^3\right )^{14}\) |
Input:
Int[x^28*(b + 2*c*x^3)*(b*x + c*x^4)^13,x]
Output:
(x^42*(b + c*x^3)^14)/42
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f *(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.33 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94
method | result | size |
gosper | \(\frac {x^{42} \left (c \,x^{3}+b \right )^{14}}{42}\) | \(15\) |
orering | \(\frac {\left (c \,x^{3}+b \right ) x^{29} \left (c \,x^{4}+b x \right )^{13}}{42}\) | \(24\) |
default | \(\frac {143}{3} x^{69} b^{5} c^{9}+\frac {1}{42} x^{42} b^{14}+\frac {13}{6} x^{48} b^{12} c^{2}+\frac {143}{3} x^{57} b^{9} c^{5}+\frac {26}{3} x^{75} b^{3} c^{11}+\frac {26}{3} x^{51} b^{11} c^{3}+\frac {1}{3} x^{45} b^{13} c +\frac {1}{3} b \,c^{13} x^{81}+\frac {143}{2} x^{66} b^{6} c^{8}+\frac {143}{6} x^{54} b^{10} c^{4}+\frac {1}{42} c^{14} x^{84}+\frac {143}{6} x^{72} b^{4} c^{10}+\frac {143}{2} x^{60} b^{8} c^{6}+\frac {13}{6} x^{78} b^{2} c^{12}+\frac {572}{7} x^{63} b^{7} c^{7}\) | \(157\) |
risch | \(\frac {143}{3} x^{69} b^{5} c^{9}+\frac {1}{42} x^{42} b^{14}+\frac {13}{6} x^{48} b^{12} c^{2}+\frac {143}{3} x^{57} b^{9} c^{5}+\frac {26}{3} x^{75} b^{3} c^{11}+\frac {26}{3} x^{51} b^{11} c^{3}+\frac {1}{3} x^{45} b^{13} c +\frac {1}{3} b \,c^{13} x^{81}+\frac {143}{2} x^{66} b^{6} c^{8}+\frac {143}{6} x^{54} b^{10} c^{4}+\frac {1}{42} c^{14} x^{84}+\frac {143}{6} x^{72} b^{4} c^{10}+\frac {143}{2} x^{60} b^{8} c^{6}+\frac {13}{6} x^{78} b^{2} c^{12}+\frac {572}{7} x^{63} b^{7} c^{7}\) | \(157\) |
parallelrisch | \(\frac {143}{3} x^{69} b^{5} c^{9}+\frac {1}{42} x^{42} b^{14}+\frac {13}{6} x^{48} b^{12} c^{2}+\frac {143}{3} x^{57} b^{9} c^{5}+\frac {26}{3} x^{75} b^{3} c^{11}+\frac {26}{3} x^{51} b^{11} c^{3}+\frac {1}{3} x^{45} b^{13} c +\frac {1}{3} b \,c^{13} x^{81}+\frac {143}{2} x^{66} b^{6} c^{8}+\frac {143}{6} x^{54} b^{10} c^{4}+\frac {1}{42} c^{14} x^{84}+\frac {143}{6} x^{72} b^{4} c^{10}+\frac {143}{2} x^{60} b^{8} c^{6}+\frac {13}{6} x^{78} b^{2} c^{12}+\frac {572}{7} x^{63} b^{7} c^{7}\) | \(157\) |
Input:
int(x^28*(2*c*x^3+b)*(c*x^4+b*x)^13,x,method=_RETURNVERBOSE)
Output:
1/42*x^42*(c*x^3+b)^14
Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (14) = 28\).
Time = 0.07 (sec) , antiderivative size = 156, normalized size of antiderivative = 9.75 \[ \int x^{28} \left (b+2 c x^3\right ) \left (b x+c x^4\right )^{13} \, dx=\frac {1}{42} \, c^{14} x^{84} + \frac {1}{3} \, b c^{13} x^{81} + \frac {13}{6} \, b^{2} c^{12} x^{78} + \frac {26}{3} \, b^{3} c^{11} x^{75} + \frac {143}{6} \, b^{4} c^{10} x^{72} + \frac {143}{3} \, b^{5} c^{9} x^{69} + \frac {143}{2} \, b^{6} c^{8} x^{66} + \frac {572}{7} \, b^{7} c^{7} x^{63} + \frac {143}{2} \, b^{8} c^{6} x^{60} + \frac {143}{3} \, b^{9} c^{5} x^{57} + \frac {143}{6} \, b^{10} c^{4} x^{54} + \frac {26}{3} \, b^{11} c^{3} x^{51} + \frac {13}{6} \, b^{12} c^{2} x^{48} + \frac {1}{3} \, b^{13} c x^{45} + \frac {1}{42} \, b^{14} x^{42} \] Input:
integrate(x^28*(2*c*x^3+b)*(c*x^4+b*x)^13,x, algorithm="fricas")
Output:
1/42*c^14*x^84 + 1/3*b*c^13*x^81 + 13/6*b^2*c^12*x^78 + 26/3*b^3*c^11*x^75 + 143/6*b^4*c^10*x^72 + 143/3*b^5*c^9*x^69 + 143/2*b^6*c^8*x^66 + 572/7*b ^7*c^7*x^63 + 143/2*b^8*c^6*x^60 + 143/3*b^9*c^5*x^57 + 143/6*b^10*c^4*x^5 4 + 26/3*b^11*c^3*x^51 + 13/6*b^12*c^2*x^48 + 1/3*b^13*c*x^45 + 1/42*b^14* x^42
Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (12) = 24\).
Time = 0.05 (sec) , antiderivative size = 185, normalized size of antiderivative = 11.56 \[ \int x^{28} \left (b+2 c x^3\right ) \left (b x+c x^4\right )^{13} \, dx=\frac {b^{14} x^{42}}{42} + \frac {b^{13} c x^{45}}{3} + \frac {13 b^{12} c^{2} x^{48}}{6} + \frac {26 b^{11} c^{3} x^{51}}{3} + \frac {143 b^{10} c^{4} x^{54}}{6} + \frac {143 b^{9} c^{5} x^{57}}{3} + \frac {143 b^{8} c^{6} x^{60}}{2} + \frac {572 b^{7} c^{7} x^{63}}{7} + \frac {143 b^{6} c^{8} x^{66}}{2} + \frac {143 b^{5} c^{9} x^{69}}{3} + \frac {143 b^{4} c^{10} x^{72}}{6} + \frac {26 b^{3} c^{11} x^{75}}{3} + \frac {13 b^{2} c^{12} x^{78}}{6} + \frac {b c^{13} x^{81}}{3} + \frac {c^{14} x^{84}}{42} \] Input:
integrate(x**28*(2*c*x**3+b)*(c*x**4+b*x)**13,x)
Output:
b**14*x**42/42 + b**13*c*x**45/3 + 13*b**12*c**2*x**48/6 + 26*b**11*c**3*x **51/3 + 143*b**10*c**4*x**54/6 + 143*b**9*c**5*x**57/3 + 143*b**8*c**6*x* *60/2 + 572*b**7*c**7*x**63/7 + 143*b**6*c**8*x**66/2 + 143*b**5*c**9*x**6 9/3 + 143*b**4*c**10*x**72/6 + 26*b**3*c**11*x**75/3 + 13*b**2*c**12*x**78 /6 + b*c**13*x**81/3 + c**14*x**84/42
Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (14) = 28\).
Time = 0.03 (sec) , antiderivative size = 156, normalized size of antiderivative = 9.75 \[ \int x^{28} \left (b+2 c x^3\right ) \left (b x+c x^4\right )^{13} \, dx=\frac {1}{42} \, c^{14} x^{84} + \frac {1}{3} \, b c^{13} x^{81} + \frac {13}{6} \, b^{2} c^{12} x^{78} + \frac {26}{3} \, b^{3} c^{11} x^{75} + \frac {143}{6} \, b^{4} c^{10} x^{72} + \frac {143}{3} \, b^{5} c^{9} x^{69} + \frac {143}{2} \, b^{6} c^{8} x^{66} + \frac {572}{7} \, b^{7} c^{7} x^{63} + \frac {143}{2} \, b^{8} c^{6} x^{60} + \frac {143}{3} \, b^{9} c^{5} x^{57} + \frac {143}{6} \, b^{10} c^{4} x^{54} + \frac {26}{3} \, b^{11} c^{3} x^{51} + \frac {13}{6} \, b^{12} c^{2} x^{48} + \frac {1}{3} \, b^{13} c x^{45} + \frac {1}{42} \, b^{14} x^{42} \] Input:
integrate(x^28*(2*c*x^3+b)*(c*x^4+b*x)^13,x, algorithm="maxima")
Output:
1/42*c^14*x^84 + 1/3*b*c^13*x^81 + 13/6*b^2*c^12*x^78 + 26/3*b^3*c^11*x^75 + 143/6*b^4*c^10*x^72 + 143/3*b^5*c^9*x^69 + 143/2*b^6*c^8*x^66 + 572/7*b ^7*c^7*x^63 + 143/2*b^8*c^6*x^60 + 143/3*b^9*c^5*x^57 + 143/6*b^10*c^4*x^5 4 + 26/3*b^11*c^3*x^51 + 13/6*b^12*c^2*x^48 + 1/3*b^13*c*x^45 + 1/42*b^14* x^42
Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (14) = 28\).
Time = 0.11 (sec) , antiderivative size = 156, normalized size of antiderivative = 9.75 \[ \int x^{28} \left (b+2 c x^3\right ) \left (b x+c x^4\right )^{13} \, dx=\frac {1}{42} \, c^{14} x^{84} + \frac {1}{3} \, b c^{13} x^{81} + \frac {13}{6} \, b^{2} c^{12} x^{78} + \frac {26}{3} \, b^{3} c^{11} x^{75} + \frac {143}{6} \, b^{4} c^{10} x^{72} + \frac {143}{3} \, b^{5} c^{9} x^{69} + \frac {143}{2} \, b^{6} c^{8} x^{66} + \frac {572}{7} \, b^{7} c^{7} x^{63} + \frac {143}{2} \, b^{8} c^{6} x^{60} + \frac {143}{3} \, b^{9} c^{5} x^{57} + \frac {143}{6} \, b^{10} c^{4} x^{54} + \frac {26}{3} \, b^{11} c^{3} x^{51} + \frac {13}{6} \, b^{12} c^{2} x^{48} + \frac {1}{3} \, b^{13} c x^{45} + \frac {1}{42} \, b^{14} x^{42} \] Input:
integrate(x^28*(2*c*x^3+b)*(c*x^4+b*x)^13,x, algorithm="giac")
Output:
1/42*c^14*x^84 + 1/3*b*c^13*x^81 + 13/6*b^2*c^12*x^78 + 26/3*b^3*c^11*x^75 + 143/6*b^4*c^10*x^72 + 143/3*b^5*c^9*x^69 + 143/2*b^6*c^8*x^66 + 572/7*b ^7*c^7*x^63 + 143/2*b^8*c^6*x^60 + 143/3*b^9*c^5*x^57 + 143/6*b^10*c^4*x^5 4 + 26/3*b^11*c^3*x^51 + 13/6*b^12*c^2*x^48 + 1/3*b^13*c*x^45 + 1/42*b^14* x^42
Time = 22.25 (sec) , antiderivative size = 156, normalized size of antiderivative = 9.75 \[ \int x^{28} \left (b+2 c x^3\right ) \left (b x+c x^4\right )^{13} \, dx=\frac {b^{14}\,x^{42}}{42}+\frac {b^{13}\,c\,x^{45}}{3}+\frac {13\,b^{12}\,c^2\,x^{48}}{6}+\frac {26\,b^{11}\,c^3\,x^{51}}{3}+\frac {143\,b^{10}\,c^4\,x^{54}}{6}+\frac {143\,b^9\,c^5\,x^{57}}{3}+\frac {143\,b^8\,c^6\,x^{60}}{2}+\frac {572\,b^7\,c^7\,x^{63}}{7}+\frac {143\,b^6\,c^8\,x^{66}}{2}+\frac {143\,b^5\,c^9\,x^{69}}{3}+\frac {143\,b^4\,c^{10}\,x^{72}}{6}+\frac {26\,b^3\,c^{11}\,x^{75}}{3}+\frac {13\,b^2\,c^{12}\,x^{78}}{6}+\frac {b\,c^{13}\,x^{81}}{3}+\frac {c^{14}\,x^{84}}{42} \] Input:
int(x^28*(b*x + c*x^4)^13*(b + 2*c*x^3),x)
Output:
(b^14*x^42)/42 + (c^14*x^84)/42 + (b^13*c*x^45)/3 + (b*c^13*x^81)/3 + (13* b^12*c^2*x^48)/6 + (26*b^11*c^3*x^51)/3 + (143*b^10*c^4*x^54)/6 + (143*b^9 *c^5*x^57)/3 + (143*b^8*c^6*x^60)/2 + (572*b^7*c^7*x^63)/7 + (143*b^6*c^8* x^66)/2 + (143*b^5*c^9*x^69)/3 + (143*b^4*c^10*x^72)/6 + (26*b^3*c^11*x^75 )/3 + (13*b^2*c^12*x^78)/6
Time = 0.14 (sec) , antiderivative size = 155, normalized size of antiderivative = 9.69 \[ \int x^{28} \left (b+2 c x^3\right ) \left (b x+c x^4\right )^{13} \, dx=\frac {x^{42} \left (c^{14} x^{42}+14 b \,c^{13} x^{39}+91 b^{2} c^{12} x^{36}+364 b^{3} c^{11} x^{33}+1001 b^{4} c^{10} x^{30}+2002 b^{5} c^{9} x^{27}+3003 b^{6} c^{8} x^{24}+3432 b^{7} c^{7} x^{21}+3003 b^{8} c^{6} x^{18}+2002 b^{9} c^{5} x^{15}+1001 b^{10} c^{4} x^{12}+364 b^{11} c^{3} x^{9}+91 b^{12} c^{2} x^{6}+14 b^{13} c \,x^{3}+b^{14}\right )}{42} \] Input:
int(x^28*(2*c*x^3+b)*(c*x^4+b*x)^13,x)
Output:
(x**42*(b**14 + 14*b**13*c*x**3 + 91*b**12*c**2*x**6 + 364*b**11*c**3*x**9 + 1001*b**10*c**4*x**12 + 2002*b**9*c**5*x**15 + 3003*b**8*c**6*x**18 + 3 432*b**7*c**7*x**21 + 3003*b**6*c**8*x**24 + 2002*b**5*c**9*x**27 + 1001*b **4*c**10*x**30 + 364*b**3*c**11*x**33 + 91*b**2*c**12*x**36 + 14*b*c**13* x**39 + c**14*x**42))/42