3.2.2 \(\int x (b+2 c x^2) (b x^2+c x^4)^{13} \, dx\) [102]

Optimal. Leaf size=16 \[ \frac {1}{28} x^{28} \left (b+c x^2\right )^{14} \]

[Out]

1/28*x^28*(c*x^2+b)^14

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Rubi [A]
time = 0.04, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1598, 457, 75} \begin {gather*} \frac {1}{28} x^{28} \left (b+c x^2\right )^{14} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(b + 2*c*x^2)*(b*x^2 + c*x^4)^13,x]

[Out]

(x^28*(b + c*x^2)^14)/28

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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]

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[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]]

Rule 1598

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {align*} \int x \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{13} \, dx &=\int x^{27} \left (b+c x^2\right )^{13} \left (b+2 c x^2\right ) \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int x^{13} (b+c x)^{13} (b+2 c x) \, dx,x,x^2\right )\\ &=\frac {1}{28} x^{28} \left (b+c x^2\right )^{14}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(182\) vs. \(2(16)=32\).
time = 0.00, size = 182, normalized size = 11.38 \begin {gather*} \frac {b^{14} x^{28}}{28}+\frac {1}{2} b^{13} c x^{30}+\frac {13}{4} b^{12} c^2 x^{32}+13 b^{11} c^3 x^{34}+\frac {143}{4} b^{10} c^4 x^{36}+\frac {143}{2} b^9 c^5 x^{38}+\frac {429}{4} b^8 c^6 x^{40}+\frac {858}{7} b^7 c^7 x^{42}+\frac {429}{4} b^6 c^8 x^{44}+\frac {143}{2} b^5 c^9 x^{46}+\frac {143}{4} b^4 c^{10} x^{48}+13 b^3 c^{11} x^{50}+\frac {13}{4} b^2 c^{12} x^{52}+\frac {1}{2} b c^{13} x^{54}+\frac {c^{14} x^{56}}{28} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(b + 2*c*x^2)*(b*x^2 + c*x^4)^13,x]

[Out]

(b^14*x^28)/28 + (b^13*c*x^30)/2 + (13*b^12*c^2*x^32)/4 + 13*b^11*c^3*x^34 + (143*b^10*c^4*x^36)/4 + (143*b^9*
c^5*x^38)/2 + (429*b^8*c^6*x^40)/4 + (858*b^7*c^7*x^42)/7 + (429*b^6*c^8*x^44)/4 + (143*b^5*c^9*x^46)/2 + (143
*b^4*c^10*x^48)/4 + 13*b^3*c^11*x^50 + (13*b^2*c^12*x^52)/4 + (b*c^13*x^54)/2 + (c^14*x^56)/28

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Maple [A]
time = 0.23, size = 24, normalized size = 1.50

method	result	size
gosper	\(\frac {x^{28} \left (c \,x^{2}+b \right )^{14}}{28}\)	\(15\)
default	\(\frac {\left (b^{2}-\left (2 c \,x^{2}+b \right )^{2}\right )^{14}}{7516192768 c^{14}}\)	\(24\)
risch	\(\frac {1}{28} c^{14} x^{56}+\frac {1}{2} b \,c^{13} x^{54}+\frac {13}{4} x^{52} b^{2} c^{12}+\frac {143}{2} x^{38} b^{9} c^{5}+\frac {429}{4} x^{40} b^{8} c^{6}+\frac {858}{7} x^{42} b^{7} c^{7}+\frac {429}{4} x^{44} b^{6} c^{8}+\frac {143}{2} x^{46} b^{5} c^{9}+\frac {143}{4} x^{48} b^{4} c^{10}+13 x^{50} b^{3} c^{11}+\frac {1}{2} x^{30} b^{13} c +\frac {13}{4} x^{32} b^{12} c^{2}+13 x^{34} b^{11} c^{3}+\frac {143}{4} x^{36} b^{10} c^{4}+\frac {1}{28} x^{28} b^{14}\)	\(157\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(2*c*x^2+b)*(c*x^4+b*x^2)^13,x,method=_RETURNVERBOSE)

[Out]

1/7516192768*(b^2-(2*c*x^2+b)^2)^14/c^14

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (14) = 28\).
time = 0.29, size = 156, normalized size = 9.75 \begin {gather*} \frac {1}{28} \, c^{14} x^{56} + \frac {1}{2} \, b c^{13} x^{54} + \frac {13}{4} \, b^{2} c^{12} x^{52} + 13 \, b^{3} c^{11} x^{50} + \frac {143}{4} \, b^{4} c^{10} x^{48} + \frac {143}{2} \, b^{5} c^{9} x^{46} + \frac {429}{4} \, b^{6} c^{8} x^{44} + \frac {858}{7} \, b^{7} c^{7} x^{42} + \frac {429}{4} \, b^{8} c^{6} x^{40} + \frac {143}{2} \, b^{9} c^{5} x^{38} + \frac {143}{4} \, b^{10} c^{4} x^{36} + 13 \, b^{11} c^{3} x^{34} + \frac {13}{4} \, b^{12} c^{2} x^{32} + \frac {1}{2} \, b^{13} c x^{30} + \frac {1}{28} \, b^{14} x^{28} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*c*x^2+b)*(c*x^4+b*x^2)^13,x, algorithm="maxima")

[Out]

1/28*c^14*x^56 + 1/2*b*c^13*x^54 + 13/4*b^2*c^12*x^52 + 13*b^3*c^11*x^50 + 143/4*b^4*c^10*x^48 + 143/2*b^5*c^9
*x^46 + 429/4*b^6*c^8*x^44 + 858/7*b^7*c^7*x^42 + 429/4*b^8*c^6*x^40 + 143/2*b^9*c^5*x^38 + 143/4*b^10*c^4*x^3
6 + 13*b^11*c^3*x^34 + 13/4*b^12*c^2*x^32 + 1/2*b^13*c*x^30 + 1/28*b^14*x^28

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (14) = 28\).
time = 0.33, size = 156, normalized size = 9.75 \begin {gather*} \frac {1}{28} \, c^{14} x^{56} + \frac {1}{2} \, b c^{13} x^{54} + \frac {13}{4} \, b^{2} c^{12} x^{52} + 13 \, b^{3} c^{11} x^{50} + \frac {143}{4} \, b^{4} c^{10} x^{48} + \frac {143}{2} \, b^{5} c^{9} x^{46} + \frac {429}{4} \, b^{6} c^{8} x^{44} + \frac {858}{7} \, b^{7} c^{7} x^{42} + \frac {429}{4} \, b^{8} c^{6} x^{40} + \frac {143}{2} \, b^{9} c^{5} x^{38} + \frac {143}{4} \, b^{10} c^{4} x^{36} + 13 \, b^{11} c^{3} x^{34} + \frac {13}{4} \, b^{12} c^{2} x^{32} + \frac {1}{2} \, b^{13} c x^{30} + \frac {1}{28} \, b^{14} x^{28} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*c*x^2+b)*(c*x^4+b*x^2)^13,x, algorithm="fricas")

[Out]

1/28*c^14*x^56 + 1/2*b*c^13*x^54 + 13/4*b^2*c^12*x^52 + 13*b^3*c^11*x^50 + 143/4*b^4*c^10*x^48 + 143/2*b^5*c^9
*x^46 + 429/4*b^6*c^8*x^44 + 858/7*b^7*c^7*x^42 + 429/4*b^8*c^6*x^40 + 143/2*b^9*c^5*x^38 + 143/4*b^10*c^4*x^3
6 + 13*b^11*c^3*x^34 + 13/4*b^12*c^2*x^32 + 1/2*b^13*c*x^30 + 1/28*b^14*x^28

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (12) = 24\).
time = 0.05, size = 182, normalized size = 11.38 \begin {gather*} \frac {b^{14} x^{28}}{28} + \frac {b^{13} c x^{30}}{2} + \frac {13 b^{12} c^{2} x^{32}}{4} + 13 b^{11} c^{3} x^{34} + \frac {143 b^{10} c^{4} x^{36}}{4} + \frac {143 b^{9} c^{5} x^{38}}{2} + \frac {429 b^{8} c^{6} x^{40}}{4} + \frac {858 b^{7} c^{7} x^{42}}{7} + \frac {429 b^{6} c^{8} x^{44}}{4} + \frac {143 b^{5} c^{9} x^{46}}{2} + \frac {143 b^{4} c^{10} x^{48}}{4} + 13 b^{3} c^{11} x^{50} + \frac {13 b^{2} c^{12} x^{52}}{4} + \frac {b c^{13} x^{54}}{2} + \frac {c^{14} x^{56}}{28} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*c*x**2+b)*(c*x**4+b*x**2)**13,x)

[Out]

b**14*x**28/28 + b**13*c*x**30/2 + 13*b**12*c**2*x**32/4 + 13*b**11*c**3*x**34 + 143*b**10*c**4*x**36/4 + 143*
b**9*c**5*x**38/2 + 429*b**8*c**6*x**40/4 + 858*b**7*c**7*x**42/7 + 429*b**6*c**8*x**44/4 + 143*b**5*c**9*x**4
6/2 + 143*b**4*c**10*x**48/4 + 13*b**3*c**11*x**50 + 13*b**2*c**12*x**52/4 + b*c**13*x**54/2 + c**14*x**56/28

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Giac [A]
time = 4.07, size = 15, normalized size = 0.94 \begin {gather*} \frac {1}{28} \, {\left (c x^{4} + b x^{2}\right )}^{14} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*c*x^2+b)*(c*x^4+b*x^2)^13,x, algorithm="giac")

[Out]

1/28*(c*x^4 + b*x^2)^14

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Mupad [B]
time = 2.08, size = 156, normalized size = 9.75 \begin {gather*} \frac {b^{14}\,x^{28}}{28}+\frac {b^{13}\,c\,x^{30}}{2}+\frac {13\,b^{12}\,c^2\,x^{32}}{4}+13\,b^{11}\,c^3\,x^{34}+\frac {143\,b^{10}\,c^4\,x^{36}}{4}+\frac {143\,b^9\,c^5\,x^{38}}{2}+\frac {429\,b^8\,c^6\,x^{40}}{4}+\frac {858\,b^7\,c^7\,x^{42}}{7}+\frac {429\,b^6\,c^8\,x^{44}}{4}+\frac {143\,b^5\,c^9\,x^{46}}{2}+\frac {143\,b^4\,c^{10}\,x^{48}}{4}+13\,b^3\,c^{11}\,x^{50}+\frac {13\,b^2\,c^{12}\,x^{52}}{4}+\frac {b\,c^{13}\,x^{54}}{2}+\frac {c^{14}\,x^{56}}{28} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b + 2*c*x^2)*(b*x^2 + c*x^4)^13,x)

[Out]

(b^14*x^28)/28 + (c^14*x^56)/28 + (b^13*c*x^30)/2 + (b*c^13*x^54)/2 + (13*b^12*c^2*x^32)/4 + 13*b^11*c^3*x^34
+ (143*b^10*c^4*x^36)/4 + (143*b^9*c^5*x^38)/2 + (429*b^8*c^6*x^40)/4 + (858*b^7*c^7*x^42)/7 + (429*b^6*c^8*x^
44)/4 + (143*b^5*c^9*x^46)/2 + (143*b^4*c^10*x^48)/4 + 13*b^3*c^11*x^50 + (13*b^2*c^12*x^52)/4

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