3.1.57 \(\int x^4 (d+e x^2) (1+2 x^2+x^4)^5 \, dx\) [57]

Optimal. Leaf size=153 \[ \frac {d x^5}{5}+\frac {1}{7} (10 d+e) x^7+\frac {5}{9} (9 d+2 e) x^9+\frac {15}{11} (8 d+3 e) x^{11}+\frac {30}{13} (7 d+4 e) x^{13}+\frac {14}{5} (6 d+5 e) x^{15}+\frac {42}{17} (5 d+6 e) x^{17}+\frac {30}{19} (4 d+7 e) x^{19}+\frac {5}{7} (3 d+8 e) x^{21}+\frac {5}{23} (2 d+9 e) x^{23}+\frac {1}{25} (d+10 e) x^{25}+\frac {e x^{27}}{27} \]

[Out]

1/5*d*x^5+1/7*(10*d+e)*x^7+5/9*(9*d+2*e)*x^9+15/11*(8*d+3*e)*x^11+30/13*(7*d+4*e)*x^13+14/5*(6*d+5*e)*x^15+42/
17*(5*d+6*e)*x^17+30/19*(4*d+7*e)*x^19+5/7*(3*d+8*e)*x^21+5/23*(2*d+9*e)*x^23+1/25*(d+10*e)*x^25+1/27*e*x^27

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Rubi [A]
time = 0.08, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {28, 459} \begin {gather*} \frac {1}{25} x^{25} (d+10 e)+\frac {5}{23} x^{23} (2 d+9 e)+\frac {5}{7} x^{21} (3 d+8 e)+\frac {30}{19} x^{19} (4 d+7 e)+\frac {42}{17} x^{17} (5 d+6 e)+\frac {14}{5} x^{15} (6 d+5 e)+\frac {30}{13} x^{13} (7 d+4 e)+\frac {15}{11} x^{11} (8 d+3 e)+\frac {5}{9} x^9 (9 d+2 e)+\frac {1}{7} x^7 (10 d+e)+\frac {d x^5}{5}+\frac {e x^{27}}{27} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*x^5)/5 + ((10*d + e)*x^7)/7 + (5*(9*d + 2*e)*x^9)/9 + (15*(8*d + 3*e)*x^11)/11 + (30*(7*d + 4*e)*x^13)/13 +
(14*(6*d + 5*e)*x^15)/5 + (42*(5*d + 6*e)*x^17)/17 + (30*(4*d + 7*e)*x^19)/19 + (5*(3*d + 8*e)*x^21)/7 + (5*(
2*d + 9*e)*x^23)/23 + ((d + 10*e)*x^25)/25 + (e*x^27)/27

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 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin {align*} \int x^4 \left (d+e x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx &=\int x^4 \left (1+x^2\right )^{10} \left (d+e x^2\right ) \, dx\\ &=\int \left (d x^4+(10 d+e) x^6+5 (9 d+2 e) x^8+15 (8 d+3 e) x^{10}+30 (7 d+4 e) x^{12}+42 (6 d+5 e) x^{14}+42 (5 d+6 e) x^{16}+30 (4 d+7 e) x^{18}+15 (3 d+8 e) x^{20}+5 (2 d+9 e) x^{22}+(d+10 e) x^{24}+e x^{26}\right ) \, dx\\ &=\frac {d x^5}{5}+\frac {1}{7} (10 d+e) x^7+\frac {5}{9} (9 d+2 e) x^9+\frac {15}{11} (8 d+3 e) x^{11}+\frac {30}{13} (7 d+4 e) x^{13}+\frac {14}{5} (6 d+5 e) x^{15}+\frac {42}{17} (5 d+6 e) x^{17}+\frac {30}{19} (4 d+7 e) x^{19}+\frac {5}{7} (3 d+8 e) x^{21}+\frac {5}{23} (2 d+9 e) x^{23}+\frac {1}{25} (d+10 e) x^{25}+\frac {e x^{27}}{27}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 153, normalized size = 1.00 \begin {gather*} \frac {d x^5}{5}+\frac {1}{7} (10 d+e) x^7+\frac {5}{9} (9 d+2 e) x^9+\frac {15}{11} (8 d+3 e) x^{11}+\frac {30}{13} (7 d+4 e) x^{13}+\frac {14}{5} (6 d+5 e) x^{15}+\frac {42}{17} (5 d+6 e) x^{17}+\frac {30}{19} (4 d+7 e) x^{19}+\frac {5}{7} (3 d+8 e) x^{21}+\frac {5}{23} (2 d+9 e) x^{23}+\frac {1}{25} (d+10 e) x^{25}+\frac {e x^{27}}{27} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*x^5)/5 + ((10*d + e)*x^7)/7 + (5*(9*d + 2*e)*x^9)/9 + (15*(8*d + 3*e)*x^11)/11 + (30*(7*d + 4*e)*x^13)/13 +
(14*(6*d + 5*e)*x^15)/5 + (42*(5*d + 6*e)*x^17)/17 + (30*(4*d + 7*e)*x^19)/19 + (5*(3*d + 8*e)*x^21)/7 + (5*(
2*d + 9*e)*x^23)/23 + ((d + 10*e)*x^25)/25 + (e*x^27)/27

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Maple [A]
time = 0.08, size = 130, normalized size = 0.85

method	result	size
norman	\(\frac {e \,x^{27}}{27}+\left (\frac {210 d}{17}+\frac {252 e}{17}\right ) x^{17}+\left (\frac {120 d}{19}+\frac {210 e}{19}\right ) x^{19}+\left (\frac {15 d}{7}+\frac {40 e}{7}\right ) x^{21}+\left (\frac {10 d}{23}+\frac {45 e}{23}\right ) x^{23}+\left (\frac {d}{25}+\frac {2 e}{5}\right ) x^{25}+\frac {d \,x^{5}}{5}+\left (\frac {10 d}{7}+\frac {e}{7}\right ) x^{7}+\left (5 d +\frac {10 e}{9}\right ) x^{9}+\left (\frac {120 d}{11}+\frac {45 e}{11}\right ) x^{11}+\left (\frac {210 d}{13}+\frac {120 e}{13}\right ) x^{13}+\left (\frac {84 d}{5}+14 e \right ) x^{15}\)	\(124\)
default	\(\frac {e \,x^{27}}{27}+\frac {\left (d +10 e \right ) x^{25}}{25}+\frac {\left (10 d +45 e \right ) x^{23}}{23}+\frac {\left (45 d +120 e \right ) x^{21}}{21}+\frac {\left (120 d +210 e \right ) x^{19}}{19}+\frac {\left (210 d +252 e \right ) x^{17}}{17}+\frac {\left (252 d +210 e \right ) x^{15}}{15}+\frac {\left (210 d +120 e \right ) x^{13}}{13}+\frac {\left (120 d +45 e \right ) x^{11}}{11}+\frac {\left (45 d +10 e \right ) x^{9}}{9}+\frac {\left (10 d +e \right ) x^{7}}{7}+\frac {d \,x^{5}}{5}\)	\(130\)
risch	\(\frac {1}{27} e \,x^{27}+\frac {1}{25} x^{25} d +\frac {2}{5} e \,x^{25}+\frac {10}{23} x^{23} d +\frac {45}{23} e \,x^{23}+\frac {15}{7} x^{21} d +\frac {40}{7} e \,x^{21}+\frac {120}{19} x^{19} d +\frac {210}{19} x^{19} e +\frac {210}{17} x^{17} d +\frac {252}{17} x^{17} e +\frac {84}{5} x^{15} d +14 x^{15} e +\frac {210}{13} x^{13} d +\frac {120}{13} x^{13} e +\frac {120}{11} x^{11} d +\frac {45}{11} e \,x^{11}+5 d \,x^{9}+\frac {10}{9} e \,x^{9}+\frac {10}{7} x^{7} d +\frac {1}{7} x^{7} e +\frac {1}{5} d \,x^{5}\)	\(134\)
gosper	\(\frac {x^{5} \left (185910725 e \,x^{22}[phone] d \,x^{20}[phone] e \,x^{20}[phone] d \,x^{18}[phone] e \,x^{18}[phone] d \,x^{16}[phone] e \,x^{16}[phone] d \,x^{14}[phone] e \,x^{14}[phone] d \,x^{12}[phone] e \,x^{12}[phone] d \,x^{10}[phone] e \,x^{10}[phone] d \,x^{8}[phone] e \,x^{8}[phone] x^{6} d [phone] e \,x^{6}[phone] d \,x^{4}[phone] e \,x^{4}[phone] d \,x^{2}[phone] e \,x^{2}[phone] d \right )}{5019589575}\)	\(136\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*e*x^27+1/25*(d+10*e)*x^25+1/23*(10*d+45*e)*x^23+1/21*(45*d+120*e)*x^21+1/19*(120*d+210*e)*x^19+1/17*(210*
d+252*e)*x^17+1/15*(252*d+210*e)*x^15+1/13*(210*d+120*e)*x^13+1/11*(120*d+45*e)*x^11+1/9*(45*d+10*e)*x^9+1/7*(
10*d+e)*x^7+1/5*d*x^5

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Maxima [A]
time = 0.30, size = 140, normalized size = 0.92 \begin {gather*} \frac {1}{27} \, x^{27} e + \frac {1}{25} \, {\left (d + 10 \, e\right )} x^{25} + \frac {5}{23} \, {\left (2 \, d + 9 \, e\right )} x^{23} + \frac {5}{7} \, {\left (3 \, d + 8 \, e\right )} x^{21} + \frac {30}{19} \, {\left (4 \, d + 7 \, e\right )} x^{19} + \frac {42}{17} \, {\left (5 \, d + 6 \, e\right )} x^{17} + \frac {14}{5} \, {\left (6 \, d + 5 \, e\right )} x^{15} + \frac {30}{13} \, {\left (7 \, d + 4 \, e\right )} x^{13} + \frac {15}{11} \, {\left (8 \, d + 3 \, e\right )} x^{11} + \frac {5}{9} \, {\left (9 \, d + 2 \, e\right )} x^{9} + \frac {1}{7} \, {\left (10 \, d + e\right )} x^{7} + \frac {1}{5} \, d x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*x^27*e + 1/25*(d + 10*e)*x^25 + 5/23*(2*d + 9*e)*x^23 + 5/7*(3*d + 8*e)*x^21 + 30/19*(4*d + 7*e)*x^19 + 4
2/17*(5*d + 6*e)*x^17 + 14/5*(6*d + 5*e)*x^15 + 30/13*(7*d + 4*e)*x^13 + 15/11*(8*d + 3*e)*x^11 + 5/9*(9*d + 2
*e)*x^9 + 1/7*(10*d + e)*x^7 + 1/5*d*x^5

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Fricas [A]
time = 0.37, size = 127, normalized size = 0.83 \begin {gather*} \frac {1}{25} \, d x^{25} + \frac {10}{23} \, d x^{23} + \frac {15}{7} \, d x^{21} + \frac {120}{19} \, d x^{19} + \frac {210}{17} \, d x^{17} + \frac {84}{5} \, d x^{15} + \frac {210}{13} \, d x^{13} + \frac {120}{11} \, d x^{11} + 5 \, d x^{9} + \frac {10}{7} \, d x^{7} + \frac {1}{5} \, d x^{5} + \frac {1}{1003917915} \, {\left (37182145 \, x^{27} + 401567166 \, x^{25} + 1964187225 \, x^{23} + 5736673800 \, x^{21} + 11095934850 \, x^{19} + 14881606740 \, x^{17} + 14054850810 \, x^{15} + 9266934600 \, x^{13} + 4106936925 \, x^{11} + 1115464350 \, x^{9} + 143416845 \, x^{7}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*d*x^25 + 10/23*d*x^23 + 15/7*d*x^21 + 120/19*d*x^19 + 210/17*d*x^17 + 84/5*d*x^15 + 210/13*d*x^13 + 120/1
1*d*x^11 + 5*d*x^9 + 10/7*d*x^7 + 1/5*d*x^5 + 1/1003917915*(37182145*x^27 + 401567166*x^25 + 1964187225*x^23 +
5736673800*x^21 + 11095934850*x^19 + 14881606740*x^17 + 14054850810*x^15 + 9266934600*x^13 + 4106936925*x^11
+ 1115464350*x^9 + 143416845*x^7)*e

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Sympy [A]
time = 0.02, size = 141, normalized size = 0.92 \begin {gather*} \frac {d x^{5}}{5} + \frac {e x^{27}}{27} + x^{25} \left (\frac {d}{25} + \frac {2 e}{5}\right ) + x^{23} \cdot \left (\frac {10 d}{23} + \frac {45 e}{23}\right ) + x^{21} \cdot \left (\frac {15 d}{7} + \frac {40 e}{7}\right ) + x^{19} \cdot \left (\frac {120 d}{19} + \frac {210 e}{19}\right ) + x^{17} \cdot \left (\frac {210 d}{17} + \frac {252 e}{17}\right ) + x^{15} \cdot \left (\frac {84 d}{5} + 14 e\right ) + x^{13} \cdot \left (\frac {210 d}{13} + \frac {120 e}{13}\right ) + x^{11} \cdot \left (\frac {120 d}{11} + \frac {45 e}{11}\right ) + x^{9} \cdot \left (5 d + \frac {10 e}{9}\right ) + x^{7} \cdot \left (\frac {10 d}{7} + \frac {e}{7}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*x**5/5 + e*x**27/27 + x**25*(d/25 + 2*e/5) + x**23*(10*d/23 + 45*e/23) + x**21*(15*d/7 + 40*e/7) + x**19*(12
0*d/19 + 210*e/19) + x**17*(210*d/17 + 252*e/17) + x**15*(84*d/5 + 14*e) + x**13*(210*d/13 + 120*e/13) + x**11
*(120*d/11 + 45*e/11) + x**9*(5*d + 10*e/9) + x**7*(10*d/7 + e/7)

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Giac [A]
time = 4.40, size = 144, normalized size = 0.94 \begin {gather*} \frac {1}{27} \, x^{27} e + \frac {1}{25} \, d x^{25} + \frac {2}{5} \, x^{25} e + \frac {10}{23} \, d x^{23} + \frac {45}{23} \, x^{23} e + \frac {15}{7} \, d x^{21} + \frac {40}{7} \, x^{21} e + \frac {120}{19} \, d x^{19} + \frac {210}{19} \, x^{19} e + \frac {210}{17} \, d x^{17} + \frac {252}{17} \, x^{17} e + \frac {84}{5} \, d x^{15} + 14 \, x^{15} e + \frac {210}{13} \, d x^{13} + \frac {120}{13} \, x^{13} e + \frac {120}{11} \, d x^{11} + \frac {45}{11} \, x^{11} e + 5 \, d x^{9} + \frac {10}{9} \, x^{9} e + \frac {10}{7} \, d x^{7} + \frac {1}{7} \, x^{7} e + \frac {1}{5} \, d x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*x^27*e + 1/25*d*x^25 + 2/5*x^25*e + 10/23*d*x^23 + 45/23*x^23*e + 15/7*d*x^21 + 40/7*x^21*e + 120/19*d*x^
19 + 210/19*x^19*e + 210/17*d*x^17 + 252/17*x^17*e + 84/5*d*x^15 + 14*x^15*e + 210/13*d*x^13 + 120/13*x^13*e +
120/11*d*x^11 + 45/11*x^11*e + 5*d*x^9 + 10/9*x^9*e + 10/7*d*x^7 + 1/7*x^7*e + 1/5*d*x^5

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Mupad [B]
time = 0.12, size = 123, normalized size = 0.80 \begin {gather*} \frac {e\,x^{27}}{27}+\left (\frac {d}{25}+\frac {2\,e}{5}\right )\,x^{25}+\left (\frac {10\,d}{23}+\frac {45\,e}{23}\right )\,x^{23}+\left (\frac {15\,d}{7}+\frac {40\,e}{7}\right )\,x^{21}+\left (\frac {120\,d}{19}+\frac {210\,e}{19}\right )\,x^{19}+\left (\frac {210\,d}{17}+\frac {252\,e}{17}\right )\,x^{17}+\left (\frac {84\,d}{5}+14\,e\right )\,x^{15}+\left (\frac {210\,d}{13}+\frac {120\,e}{13}\right )\,x^{13}+\left (\frac {120\,d}{11}+\frac {45\,e}{11}\right )\,x^{11}+\left (5\,d+\frac {10\,e}{9}\right )\,x^9+\left (\frac {10\,d}{7}+\frac {e}{7}\right )\,x^7+\frac {d\,x^5}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^7*((10*d)/7 + e/7) + x^9*(5*d + (10*e)/9) + x^25*(d/25 + (2*e)/5) + x^21*((15*d)/7 + (40*e)/7) + x^15*((84*d
)/5 + 14*e) + x^23*((10*d)/23 + (45*e)/23) + x^11*((120*d)/11 + (45*e)/11) + x^13*((210*d)/13 + (120*e)/13) +
x^19*((120*d)/19 + (210*e)/19) + x^17*((210*d)/17 + (252*e)/17) + (d*x^5)/5 + (e*x^27)/27

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