3.1.56 \(\int x^5 (d+e x^2) (1+2 x^2+x^4)^5 \, dx\) [56]

Optimal. Leaf size=63 \[ \frac {1}{22} (d-e) \left (1+x^2\right )^{11}-\frac {1}{24} (2 d-3 e) \left (1+x^2\right )^{12}+\frac {1}{26} (d-3 e) \left (1+x^2\right )^{13}+\frac {1}{28} e \left (1+x^2\right )^{14} \]

[Out]

1/22*(d-e)*(x^2+1)^11-1/24*(2*d-3*e)*(x^2+1)^12+1/26*(d-3*e)*(x^2+1)^13+1/28*e*(x^2+1)^14

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Rubi [A]
time = 0.13, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {28, 457, 77} \begin {gather*} \frac {1}{26} \left (x^2+1\right )^{13} (d-3 e)-\frac {1}{24} \left (x^2+1\right )^{12} (2 d-3 e)+\frac {1}{22} \left (x^2+1\right )^{11} (d-e)+\frac {1}{28} e \left (x^2+1\right )^{14} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*(d + e*x^2)*(1 + 2*x^2 + x^4)^5,x]

[Out]

((d - e)*(1 + x^2)^11)/22 - ((2*d - 3*e)*(1 + x^2)^12)/24 + ((d - 3*e)*(1 + x^2)^13)/26 + (e*(1 + x^2)^14)/28

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 77

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

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]]

Rubi steps

\begin {align*} \int x^5 \left (d+e x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx &=\int x^5 \left (1+x^2\right )^{10} \left (d+e x^2\right ) \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int x^2 (1+x)^{10} (d+e x) \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left ((d-e) (1+x)^{10}+(-2 d+3 e) (1+x)^{11}+(d-3 e) (1+x)^{12}+e (1+x)^{13}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{22} (d-e) \left (1+x^2\right )^{11}-\frac {1}{24} (2 d-3 e) \left (1+x^2\right )^{12}+\frac {1}{26} (d-3 e) \left (1+x^2\right )^{13}+\frac {1}{28} e \left (1+x^2\right )^{14}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(153\) vs. \(2(63)=126\).
time = 0.02, size = 153, normalized size = 2.43 \begin {gather*} \frac {d x^6}{6}+\frac {1}{8} (10 d+e) x^8+\frac {1}{2} (9 d+2 e) x^{10}+\frac {5}{4} (8 d+3 e) x^{12}+\frac {15}{7} (7 d+4 e) x^{14}+\frac {21}{8} (6 d+5 e) x^{16}+\frac {7}{3} (5 d+6 e) x^{18}+\frac {3}{2} (4 d+7 e) x^{20}+\frac {15}{22} (3 d+8 e) x^{22}+\frac {5}{24} (2 d+9 e) x^{24}+\frac {1}{26} (d+10 e) x^{26}+\frac {e x^{28}}{28} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*(d + e*x^2)*(1 + 2*x^2 + x^4)^5,x]

[Out]

(d*x^6)/6 + ((10*d + e)*x^8)/8 + ((9*d + 2*e)*x^10)/2 + (5*(8*d + 3*e)*x^12)/4 + (15*(7*d + 4*e)*x^14)/7 + (21
*(6*d + 5*e)*x^16)/8 + (7*(5*d + 6*e)*x^18)/3 + (3*(4*d + 7*e)*x^20)/2 + (15*(3*d + 8*e)*x^22)/22 + (5*(2*d +
9*e)*x^24)/24 + ((d + 10*e)*x^26)/26 + (e*x^28)/28

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(129\) vs. \(2(55)=110\).
time = 0.09, size = 130, normalized size = 2.06

method	result	size
norman	\(\left (10 d +\frac {15 e}{4}\right ) x^{12}+\left (15 d +\frac {60 e}{7}\right ) x^{14}+\left (\frac {63 d}{4}+\frac {105 e}{8}\right ) x^{16}+\left (\frac {35 d}{3}+14 e \right ) x^{18}+\left (6 d +\frac {21 e}{2}\right ) x^{20}+\left (\frac {45 d}{22}+\frac {60 e}{11}\right ) x^{22}+\left (\frac {5 d}{12}+\frac {15 e}{8}\right ) x^{24}+\left (\frac {d}{26}+\frac {5 e}{13}\right ) x^{26}+\frac {e \,x^{28}}{28}+\left (\frac {5 d}{4}+\frac {e}{8}\right ) x^{8}+\left (\frac {9 d}{2}+e \right ) x^{10}+\frac {x^{6} d}{6}\)	\(122\)
default	\(\frac {e \,x^{28}}{28}+\frac {\left (d +10 e \right ) x^{26}}{26}+\frac {\left (10 d +45 e \right ) x^{24}}{24}+\frac {\left (45 d +120 e \right ) x^{22}}{22}+\frac {\left (120 d +210 e \right ) x^{20}}{20}+\frac {\left (210 d +252 e \right ) x^{18}}{18}+\frac {\left (252 d +210 e \right ) x^{16}}{16}+\frac {\left (210 d +120 e \right ) x^{14}}{14}+\frac {\left (120 d +45 e \right ) x^{12}}{12}+\frac {\left (45 d +10 e \right ) x^{10}}{10}+\frac {\left (10 d +e \right ) x^{8}}{8}+\frac {x^{6} d}{6}\)	\(130\)
risch	\(\frac {1}{28} e \,x^{28}+\frac {1}{26} x^{26} d +\frac {5}{13} x^{26} e +\frac {5}{12} x^{24} d +\frac {15}{8} x^{24} e +\frac {45}{22} x^{22} d +\frac {60}{11} e \,x^{22}+6 d \,x^{20}+\frac {21}{2} e \,x^{20}+\frac {35}{3} d \,x^{18}+14 e \,x^{18}+\frac {63}{4} d \,x^{16}+\frac {105}{8} e \,x^{16}+15 d \,x^{14}+\frac {60}{7} e \,x^{14}+10 d \,x^{12}+\frac {15}{4} e \,x^{12}+\frac {9}{2} d \,x^{10}+e \,x^{10}+\frac {5}{4} d \,x^{8}+\frac {1}{8} e \,x^{8}+\frac {1}{6} x^{6} d\)	\(133\)
gosper	\(\frac {x^{6} \left (858 e \,x^{22}+924 d \,x^{20}+9240 e \,x^{20}+10010 d \,x^{18}+45045 e \,x^{18}+49140 d \,x^{16}+131040 e \,x^{16}+144144 d \,x^{14}+252252 e \,x^{14}+280280 d \,x^{12}+336336 e \,x^{12}+378378 d \,x^{10}+315315 e \,x^{10}+360360 d \,x^{8}+205920 e \,x^{8}+240240 x^{6} d +90090 e \,x^{6}+108108 d \,x^{4}+24024 e \,x^{4}+30030 d \,x^{2}+3003 e \,x^{2}+4004 d \right )}{24024}\)	\(136\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(e*x^2+d)*(x^4+2*x^2+1)^5,x,method=_RETURNVERBOSE)

[Out]

1/28*e*x^28+1/26*(d+10*e)*x^26+1/24*(10*d+45*e)*x^24+1/22*(45*d+120*e)*x^22+1/20*(120*d+210*e)*x^20+1/18*(210*
d+252*e)*x^18+1/16*(252*d+210*e)*x^16+1/14*(210*d+120*e)*x^14+1/12*(120*d+45*e)*x^12+1/10*(45*d+10*e)*x^10+1/8
*(10*d+e)*x^8+1/6*x^6*d

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (59) = 118\).
time = 0.27, size = 140, normalized size = 2.22 \begin {gather*} \frac {1}{28} \, x^{28} e + \frac {1}{26} \, {\left (d + 10 \, e\right )} x^{26} + \frac {5}{24} \, {\left (2 \, d + 9 \, e\right )} x^{24} + \frac {15}{22} \, {\left (3 \, d + 8 \, e\right )} x^{22} + \frac {3}{2} \, {\left (4 \, d + 7 \, e\right )} x^{20} + \frac {7}{3} \, {\left (5 \, d + 6 \, e\right )} x^{18} + \frac {21}{8} \, {\left (6 \, d + 5 \, e\right )} x^{16} + \frac {15}{7} \, {\left (7 \, d + 4 \, e\right )} x^{14} + \frac {5}{4} \, {\left (8 \, d + 3 \, e\right )} x^{12} + \frac {1}{2} \, {\left (9 \, d + 2 \, e\right )} x^{10} + \frac {1}{8} \, {\left (10 \, d + e\right )} x^{8} + \frac {1}{6} \, d x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x^2+d)*(x^4+2*x^2+1)^5,x, algorithm="maxima")

[Out]

1/28*x^28*e + 1/26*(d + 10*e)*x^26 + 5/24*(2*d + 9*e)*x^24 + 15/22*(3*d + 8*e)*x^22 + 3/2*(4*d + 7*e)*x^20 + 7
/3*(5*d + 6*e)*x^18 + 21/8*(6*d + 5*e)*x^16 + 15/7*(7*d + 4*e)*x^14 + 5/4*(8*d + 3*e)*x^12 + 1/2*(9*d + 2*e)*x
^10 + 1/8*(10*d + e)*x^8 + 1/6*d*x^6

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (59) = 118\).
time = 0.34, size = 127, normalized size = 2.02 \begin {gather*} \frac {1}{26} \, d x^{26} + \frac {5}{12} \, d x^{24} + \frac {45}{22} \, d x^{22} + 6 \, d x^{20} + \frac {35}{3} \, d x^{18} + \frac {63}{4} \, d x^{16} + 15 \, d x^{14} + 10 \, d x^{12} + \frac {9}{2} \, d x^{10} + \frac {5}{4} \, d x^{8} + \frac {1}{6} \, d x^{6} + \frac {1}{8008} \, {\left (286 \, x^{28} + 3080 \, x^{26} + 15015 \, x^{24} + 43680 \, x^{22} + 84084 \, x^{20} + 112112 \, x^{18} + 105105 \, x^{16} + 68640 \, x^{14} + 30030 \, x^{12} + 8008 \, x^{10} + 1001 \, x^{8}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x^2+d)*(x^4+2*x^2+1)^5,x, algorithm="fricas")

[Out]

1/26*d*x^26 + 5/12*d*x^24 + 45/22*d*x^22 + 6*d*x^20 + 35/3*d*x^18 + 63/4*d*x^16 + 15*d*x^14 + 10*d*x^12 + 9/2*
d*x^10 + 5/4*d*x^8 + 1/6*d*x^6 + 1/8008*(286*x^28 + 3080*x^26 + 15015*x^24 + 43680*x^22 + 84084*x^20 + 112112*
x^18 + 105105*x^16 + 68640*x^14 + 30030*x^12 + 8008*x^10 + 1001*x^8)*e

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (51) = 102\).
time = 0.03, size = 134, normalized size = 2.13 \begin {gather*} \frac {d x^{6}}{6} + \frac {e x^{28}}{28} + x^{26} \left (\frac {d}{26} + \frac {5 e}{13}\right ) + x^{24} \cdot \left (\frac {5 d}{12} + \frac {15 e}{8}\right ) + x^{22} \cdot \left (\frac {45 d}{22} + \frac {60 e}{11}\right ) + x^{20} \cdot \left (6 d + \frac {21 e}{2}\right ) + x^{18} \cdot \left (\frac {35 d}{3} + 14 e\right ) + x^{16} \cdot \left (\frac {63 d}{4} + \frac {105 e}{8}\right ) + x^{14} \cdot \left (15 d + \frac {60 e}{7}\right ) + x^{12} \cdot \left (10 d + \frac {15 e}{4}\right ) + x^{10} \cdot \left (\frac {9 d}{2} + e\right ) + x^{8} \cdot \left (\frac {5 d}{4} + \frac {e}{8}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(e*x**2+d)*(x**4+2*x**2+1)**5,x)

[Out]

d*x**6/6 + e*x**28/28 + x**26*(d/26 + 5*e/13) + x**24*(5*d/12 + 15*e/8) + x**22*(45*d/22 + 60*e/11) + x**20*(6
*d + 21*e/2) + x**18*(35*d/3 + 14*e) + x**16*(63*d/4 + 105*e/8) + x**14*(15*d + 60*e/7) + x**12*(10*d + 15*e/4
) + x**10*(9*d/2 + e) + x**8*(5*d/4 + e/8)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (59) = 118\).
time = 4.03, size = 143, normalized size = 2.27 \begin {gather*} \frac {1}{28} \, x^{28} e + \frac {1}{26} \, d x^{26} + \frac {5}{13} \, x^{26} e + \frac {5}{12} \, d x^{24} + \frac {15}{8} \, x^{24} e + \frac {45}{22} \, d x^{22} + \frac {60}{11} \, x^{22} e + 6 \, d x^{20} + \frac {21}{2} \, x^{20} e + \frac {35}{3} \, d x^{18} + 14 \, x^{18} e + \frac {63}{4} \, d x^{16} + \frac {105}{8} \, x^{16} e + 15 \, d x^{14} + \frac {60}{7} \, x^{14} e + 10 \, d x^{12} + \frac {15}{4} \, x^{12} e + \frac {9}{2} \, d x^{10} + x^{10} e + \frac {5}{4} \, d x^{8} + \frac {1}{8} \, x^{8} e + \frac {1}{6} \, d x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x^2+d)*(x^4+2*x^2+1)^5,x, algorithm="giac")

[Out]

1/28*x^28*e + 1/26*d*x^26 + 5/13*x^26*e + 5/12*d*x^24 + 15/8*x^24*e + 45/22*d*x^22 + 60/11*x^22*e + 6*d*x^20 +
21/2*x^20*e + 35/3*d*x^18 + 14*x^18*e + 63/4*d*x^16 + 105/8*x^16*e + 15*d*x^14 + 60/7*x^14*e + 10*d*x^12 + 15
/4*x^12*e + 9/2*d*x^10 + x^10*e + 5/4*d*x^8 + 1/8*x^8*e + 1/6*d*x^6

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Mupad [B]
time = 0.09, size = 121, normalized size = 1.92 \begin {gather*} \frac {e\,x^{28}}{28}+\left (\frac {d}{26}+\frac {5\,e}{13}\right )\,x^{26}+\left (\frac {5\,d}{12}+\frac {15\,e}{8}\right )\,x^{24}+\left (\frac {45\,d}{22}+\frac {60\,e}{11}\right )\,x^{22}+\left (6\,d+\frac {21\,e}{2}\right )\,x^{20}+\left (\frac {35\,d}{3}+14\,e\right )\,x^{18}+\left (\frac {63\,d}{4}+\frac {105\,e}{8}\right )\,x^{16}+\left (15\,d+\frac {60\,e}{7}\right )\,x^{14}+\left (10\,d+\frac {15\,e}{4}\right )\,x^{12}+\left (\frac {9\,d}{2}+e\right )\,x^{10}+\left (\frac {5\,d}{4}+\frac {e}{8}\right )\,x^8+\frac {d\,x^6}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(d + e*x^2)*(2*x^2 + x^4 + 1)^5,x)

[Out]

x^8*((5*d)/4 + e/8) + x^12*(10*d + (15*e)/4) + x^20*(6*d + (21*e)/2) + x^24*((5*d)/12 + (15*e)/8) + x^18*((35*
d)/3 + 14*e) + x^26*(d/26 + (5*e)/13) + x^14*(15*d + (60*e)/7) + x^22*((45*d)/22 + (60*e)/11) + x^16*((63*d)/4
+ (105*e)/8) + (d*x^6)/6 + (e*x^28)/28 + x^10*((9*d)/2 + e)

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