\(\int x^7 (a^2+2 a b x^2+b^2 x^4)^3 \, dx\) [445]

Optimal result

Integrand size = 24, antiderivative size = 72 \[ \int x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=-\frac {a^3 \left (a+b x^2\right )^7}{14 b^4}+\frac {3 a^2 \left (a+b x^2\right )^8}{16 b^4}-\frac {a \left (a+b x^2\right )^9}{6 b^4}+\frac {\left (a+b x^2\right )^{10}}{20 b^4} \]

[Out]

-1/14*a^3*(b*x^2+a)^7/b^4+3/16*a^2*(b*x^2+a)^8/b^4-1/6*a*(b*x^2+a)^9/b^4+1/20*(b*x^2+a)^10/b^4

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 272, 45} \[ \int x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=-\frac {a^3 \left (a+b x^2\right )^7}{14 b^4}+\frac {3 a^2 \left (a+b x^2\right )^8}{16 b^4}+\frac {\left (a+b x^2\right )^{10}}{20 b^4}-\frac {a \left (a+b x^2\right )^9}{6 b^4} \]

[In]

Int[x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-1/14*(a^3*(a + b*x^2)^7)/b^4 + (3*a^2*(a + b*x^2)^8)/(16*b^4) - (a*(a + b*x^2)^9)/(6*b^4) + (a + b*x^2)^10/(2
0*b^4)

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 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {\int x^7 \left (a b+b^2 x^2\right )^6 \, dx}{b^6} \\ & = \frac {\text {Subst}\left (\int x^3 \left (a b+b^2 x\right )^6 \, dx,x,x^2\right )}{2 b^6} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {a^3 \left (a b+b^2 x\right )^6}{b^3}+\frac {3 a^2 \left (a b+b^2 x\right )^7}{b^4}-\frac {3 a \left (a b+b^2 x\right )^8}{b^5}+\frac {\left (a b+b^2 x\right )^9}{b^6}\right ) \, dx,x,x^2\right )}{2 b^6} \\ & = -\frac {a^3 \left (a+b x^2\right )^7}{14 b^4}+\frac {3 a^2 \left (a+b x^2\right )^8}{16 b^4}-\frac {a \left (a+b x^2\right )^9}{6 b^4}+\frac {\left (a+b x^2\right )^{10}}{20 b^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.14 \[ \int x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {a^6 x^8}{8}+\frac {3}{5} a^5 b x^{10}+\frac {5}{4} a^4 b^2 x^{12}+\frac {10}{7} a^3 b^3 x^{14}+\frac {15}{16} a^2 b^4 x^{16}+\frac {1}{3} a b^5 x^{18}+\frac {b^6 x^{20}}{20} \]

[In]

Integrate[x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(a^6*x^8)/8 + (3*a^5*b*x^10)/5 + (5*a^4*b^2*x^12)/4 + (10*a^3*b^3*x^14)/7 + (15*a^2*b^4*x^16)/16 + (a*b^5*x^18
)/3 + (b^6*x^20)/20

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.96

method	result	size
default	\(\frac {1}{8} a^{6} x^{8}+\frac {3}{5} a^{5} b \,x^{10}+\frac {5}{4} a^{4} b^{2} x^{12}+\frac {10}{7} a^{3} b^{3} x^{14}+\frac {15}{16} a^{2} b^{4} x^{16}+\frac {1}{3} b^{5} a \,x^{18}+\frac {1}{20} b^{6} x^{20}\)	\(69\)
norman	\(\frac {1}{8} a^{6} x^{8}+\frac {3}{5} a^{5} b \,x^{10}+\frac {5}{4} a^{4} b^{2} x^{12}+\frac {10}{7} a^{3} b^{3} x^{14}+\frac {15}{16} a^{2} b^{4} x^{16}+\frac {1}{3} b^{5} a \,x^{18}+\frac {1}{20} b^{6} x^{20}\)	\(69\)
risch	\(\frac {1}{8} a^{6} x^{8}+\frac {3}{5} a^{5} b \,x^{10}+\frac {5}{4} a^{4} b^{2} x^{12}+\frac {10}{7} a^{3} b^{3} x^{14}+\frac {15}{16} a^{2} b^{4} x^{16}+\frac {1}{3} b^{5} a \,x^{18}+\frac {1}{20} b^{6} x^{20}\)	\(69\)
parallelrisch	\(\frac {1}{8} a^{6} x^{8}+\frac {3}{5} a^{5} b \,x^{10}+\frac {5}{4} a^{4} b^{2} x^{12}+\frac {10}{7} a^{3} b^{3} x^{14}+\frac {15}{16} a^{2} b^{4} x^{16}+\frac {1}{3} b^{5} a \,x^{18}+\frac {1}{20} b^{6} x^{20}\)	\(69\)
gosper	\(\frac {x^{8} \left (84 b^{6} x^{12}+560 a \,b^{5} x^{10}+1575 a^{2} b^{4} x^{8}+2400 a^{3} b^{3} x^{6}+2100 a^{4} b^{2} x^{4}+1008 a^{5} b \,x^{2}+210 a^{6}\right )}{1680}\)	\(71\)

[In]

int(x^7*(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/8*a^6*x^8+3/5*a^5*b*x^10+5/4*a^4*b^2*x^12+10/7*a^3*b^3*x^14+15/16*a^2*b^4*x^16+1/3*b^5*a*x^18+1/20*b^6*x^20

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.94 \[ \int x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {1}{20} \, b^{6} x^{20} + \frac {1}{3} \, a b^{5} x^{18} + \frac {15}{16} \, a^{2} b^{4} x^{16} + \frac {10}{7} \, a^{3} b^{3} x^{14} + \frac {5}{4} \, a^{4} b^{2} x^{12} + \frac {3}{5} \, a^{5} b x^{10} + \frac {1}{8} \, a^{6} x^{8} \]

[In]

integrate(x^7*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

1/20*b^6*x^20 + 1/3*a*b^5*x^18 + 15/16*a^2*b^4*x^16 + 10/7*a^3*b^3*x^14 + 5/4*a^4*b^2*x^12 + 3/5*a^5*b*x^10 +
1/8*a^6*x^8

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.08 \[ \int x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {a^{6} x^{8}}{8} + \frac {3 a^{5} b x^{10}}{5} + \frac {5 a^{4} b^{2} x^{12}}{4} + \frac {10 a^{3} b^{3} x^{14}}{7} + \frac {15 a^{2} b^{4} x^{16}}{16} + \frac {a b^{5} x^{18}}{3} + \frac {b^{6} x^{20}}{20} \]

[In]

integrate(x**7*(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

a**6*x**8/8 + 3*a**5*b*x**10/5 + 5*a**4*b**2*x**12/4 + 10*a**3*b**3*x**14/7 + 15*a**2*b**4*x**16/16 + a*b**5*x
**18/3 + b**6*x**20/20

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.94 \[ \int x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {1}{20} \, b^{6} x^{20} + \frac {1}{3} \, a b^{5} x^{18} + \frac {15}{16} \, a^{2} b^{4} x^{16} + \frac {10}{7} \, a^{3} b^{3} x^{14} + \frac {5}{4} \, a^{4} b^{2} x^{12} + \frac {3}{5} \, a^{5} b x^{10} + \frac {1}{8} \, a^{6} x^{8} \]

[In]

integrate(x^7*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

1/20*b^6*x^20 + 1/3*a*b^5*x^18 + 15/16*a^2*b^4*x^16 + 10/7*a^3*b^3*x^14 + 5/4*a^4*b^2*x^12 + 3/5*a^5*b*x^10 +
1/8*a^6*x^8

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.94 \[ \int x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {1}{20} \, b^{6} x^{20} + \frac {1}{3} \, a b^{5} x^{18} + \frac {15}{16} \, a^{2} b^{4} x^{16} + \frac {10}{7} \, a^{3} b^{3} x^{14} + \frac {5}{4} \, a^{4} b^{2} x^{12} + \frac {3}{5} \, a^{5} b x^{10} + \frac {1}{8} \, a^{6} x^{8} \]

[In]

integrate(x^7*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

1/20*b^6*x^20 + 1/3*a*b^5*x^18 + 15/16*a^2*b^4*x^16 + 10/7*a^3*b^3*x^14 + 5/4*a^4*b^2*x^12 + 3/5*a^5*b*x^10 +
1/8*a^6*x^8

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.94 \[ \int x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {a^6\,x^8}{8}+\frac {3\,a^5\,b\,x^{10}}{5}+\frac {5\,a^4\,b^2\,x^{12}}{4}+\frac {10\,a^3\,b^3\,x^{14}}{7}+\frac {15\,a^2\,b^4\,x^{16}}{16}+\frac {a\,b^5\,x^{18}}{3}+\frac {b^6\,x^{20}}{20} \]

[In]

int(x^7*(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)

[Out]

(a^6*x^8)/8 + (b^6*x^20)/20 + (3*a^5*b*x^10)/5 + (a*b^5*x^18)/3 + (5*a^4*b^2*x^12)/4 + (10*a^3*b^3*x^14)/7 + (
15*a^2*b^4*x^16)/16