3.1.89 \(\int x^5 (a+b x^2)^8 \, dx\)

Optimal. Leaf size=53 \[ \frac {a^2 \left (a+b x^2\right )^9}{18 b^3}+\frac {\left (a+b x^2\right )^{11}}{22 b^3}-\frac {a \left (a+b x^2\right )^{10}}{10 b^3} \]

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Rubi [A]  time = 0.08, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \begin {gather*} \frac {a^2 \left (a+b x^2\right )^9}{18 b^3}+\frac {\left (a+b x^2\right )^{11}}{22 b^3}-\frac {a \left (a+b x^2\right )^{10}}{10 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x^2)^8,x]

[Out]

(a^2*(a + b*x^2)^9)/(18*b^3) - (a*(a + b*x^2)^10)/(10*b^3) + (a + b*x^2)^11/(22*b^3)

Rule 43

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 266

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*} \int x^5 \left (a+b x^2\right )^8 \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^2 (a+b x)^8 \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^2 (a+b x)^8}{b^2}-\frac {2 a (a+b x)^9}{b^2}+\frac {(a+b x)^{10}}{b^2}\right ) \, dx,x,x^2\right )\\ &=\frac {a^2 \left (a+b x^2\right )^9}{18 b^3}-\frac {a \left (a+b x^2\right )^{10}}{10 b^3}+\frac {\left (a+b x^2\right )^{11}}{22 b^3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 103, normalized size = 1.94 \begin {gather*} \frac {a^8 x^6}{6}+a^7 b x^8+\frac {14}{5} a^6 b^2 x^{10}+\frac {14}{3} a^5 b^3 x^{12}+5 a^4 b^4 x^{14}+\frac {7}{2} a^3 b^5 x^{16}+\frac {14}{9} a^2 b^6 x^{18}+\frac {2}{5} a b^7 x^{20}+\frac {b^8 x^{22}}{22} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x^2)^8,x]

[Out]

(a^8*x^6)/6 + a^7*b*x^8 + (14*a^6*b^2*x^10)/5 + (14*a^5*b^3*x^12)/3 + 5*a^4*b^4*x^14 + (7*a^3*b^5*x^16)/2 + (1
4*a^2*b^6*x^18)/9 + (2*a*b^7*x^20)/5 + (b^8*x^22)/22

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^5 \left (a+b x^2\right )^8 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^5*(a + b*x^2)^8,x]

[Out]

IntegrateAlgebraic[x^5*(a + b*x^2)^8, x]

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fricas [A]  time = 0.98, size = 89, normalized size = 1.68 \begin {gather*} \frac {1}{22} x^{22} b^{8} + \frac {2}{5} x^{20} b^{7} a + \frac {14}{9} x^{18} b^{6} a^{2} + \frac {7}{2} x^{16} b^{5} a^{3} + 5 x^{14} b^{4} a^{4} + \frac {14}{3} x^{12} b^{3} a^{5} + \frac {14}{5} x^{10} b^{2} a^{6} + x^{8} b a^{7} + \frac {1}{6} x^{6} a^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^2+a)^8,x, algorithm="fricas")

[Out]

1/22*x^22*b^8 + 2/5*x^20*b^7*a + 14/9*x^18*b^6*a^2 + 7/2*x^16*b^5*a^3 + 5*x^14*b^4*a^4 + 14/3*x^12*b^3*a^5 + 1
4/5*x^10*b^2*a^6 + x^8*b*a^7 + 1/6*x^6*a^8

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giac [A]  time = 1.13, size = 89, normalized size = 1.68 \begin {gather*} \frac {1}{22} \, b^{8} x^{22} + \frac {2}{5} \, a b^{7} x^{20} + \frac {14}{9} \, a^{2} b^{6} x^{18} + \frac {7}{2} \, a^{3} b^{5} x^{16} + 5 \, a^{4} b^{4} x^{14} + \frac {14}{3} \, a^{5} b^{3} x^{12} + \frac {14}{5} \, a^{6} b^{2} x^{10} + a^{7} b x^{8} + \frac {1}{6} \, a^{8} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^2+a)^8,x, algorithm="giac")

[Out]

1/22*b^8*x^22 + 2/5*a*b^7*x^20 + 14/9*a^2*b^6*x^18 + 7/2*a^3*b^5*x^16 + 5*a^4*b^4*x^14 + 14/3*a^5*b^3*x^12 + 1
4/5*a^6*b^2*x^10 + a^7*b*x^8 + 1/6*a^8*x^6

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maple [A]  time = 0.00, size = 90, normalized size = 1.70 \begin {gather*} \frac {1}{22} b^{8} x^{22}+\frac {2}{5} a \,b^{7} x^{20}+\frac {14}{9} a^{2} b^{6} x^{18}+\frac {7}{2} a^{3} b^{5} x^{16}+5 a^{4} b^{4} x^{14}+\frac {14}{3} a^{5} b^{3} x^{12}+\frac {14}{5} a^{6} b^{2} x^{10}+a^{7} b \,x^{8}+\frac {1}{6} a^{8} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(b*x^2+a)^8,x)

[Out]

1/22*b^8*x^22+2/5*a*b^7*x^20+14/9*a^2*b^6*x^18+7/2*a^3*b^5*x^16+5*a^4*b^4*x^14+14/3*a^5*b^3*x^12+14/5*a^6*b^2*
x^10+a^7*b*x^8+1/6*a^8*x^6

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maxima [A]  time = 1.35, size = 89, normalized size = 1.68 \begin {gather*} \frac {1}{22} \, b^{8} x^{22} + \frac {2}{5} \, a b^{7} x^{20} + \frac {14}{9} \, a^{2} b^{6} x^{18} + \frac {7}{2} \, a^{3} b^{5} x^{16} + 5 \, a^{4} b^{4} x^{14} + \frac {14}{3} \, a^{5} b^{3} x^{12} + \frac {14}{5} \, a^{6} b^{2} x^{10} + a^{7} b x^{8} + \frac {1}{6} \, a^{8} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^2+a)^8,x, algorithm="maxima")

[Out]

1/22*b^8*x^22 + 2/5*a*b^7*x^20 + 14/9*a^2*b^6*x^18 + 7/2*a^3*b^5*x^16 + 5*a^4*b^4*x^14 + 14/3*a^5*b^3*x^12 + 1
4/5*a^6*b^2*x^10 + a^7*b*x^8 + 1/6*a^8*x^6

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mupad [B]  time = 0.09, size = 89, normalized size = 1.68 \begin {gather*} \frac {a^8\,x^6}{6}+a^7\,b\,x^8+\frac {14\,a^6\,b^2\,x^{10}}{5}+\frac {14\,a^5\,b^3\,x^{12}}{3}+5\,a^4\,b^4\,x^{14}+\frac {7\,a^3\,b^5\,x^{16}}{2}+\frac {14\,a^2\,b^6\,x^{18}}{9}+\frac {2\,a\,b^7\,x^{20}}{5}+\frac {b^8\,x^{22}}{22} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b*x^2)^8,x)

[Out]

(a^8*x^6)/6 + (b^8*x^22)/22 + a^7*b*x^8 + (2*a*b^7*x^20)/5 + (14*a^6*b^2*x^10)/5 + (14*a^5*b^3*x^12)/3 + 5*a^4
*b^4*x^14 + (7*a^3*b^5*x^16)/2 + (14*a^2*b^6*x^18)/9

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sympy [B]  time = 0.09, size = 102, normalized size = 1.92 \begin {gather*} \frac {a^{8} x^{6}}{6} + a^{7} b x^{8} + \frac {14 a^{6} b^{2} x^{10}}{5} + \frac {14 a^{5} b^{3} x^{12}}{3} + 5 a^{4} b^{4} x^{14} + \frac {7 a^{3} b^{5} x^{16}}{2} + \frac {14 a^{2} b^{6} x^{18}}{9} + \frac {2 a b^{7} x^{20}}{5} + \frac {b^{8} x^{22}}{22} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(b*x**2+a)**8,x)

[Out]

a**8*x**6/6 + a**7*b*x**8 + 14*a**6*b**2*x**10/5 + 14*a**5*b**3*x**12/3 + 5*a**4*b**4*x**14 + 7*a**3*b**5*x**1
6/2 + 14*a**2*b**6*x**18/9 + 2*a*b**7*x**20/5 + b**8*x**22/22

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