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

Optimal. Leaf size=69 \[ \frac {a^5 x^9}{9}+\frac {5}{11} a^4 b x^{11}+\frac {10}{13} a^3 b^2 x^{13}+\frac {2}{3} a^2 b^3 x^{15}+\frac {5}{17} a b^4 x^{17}+\frac {b^5 x^{19}}{19} \]

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Rubi [A]  time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \begin {gather*} \frac {2}{3} a^2 b^3 x^{15}+\frac {10}{13} a^3 b^2 x^{13}+\frac {5}{11} a^4 b x^{11}+\frac {a^5 x^9}{9}+\frac {5}{17} a b^4 x^{17}+\frac {b^5 x^{19}}{19} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^8*(a + b*x^2)^5,x]

[Out]

(a^5*x^9)/9 + (5*a^4*b*x^11)/11 + (10*a^3*b^2*x^13)/13 + (2*a^2*b^3*x^15)/3 + (5*a*b^4*x^17)/17 + (b^5*x^19)/1
9

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int x^8 \left (a+b x^2\right )^5 \, dx &=\int \left (a^5 x^8+5 a^4 b x^{10}+10 a^3 b^2 x^{12}+10 a^2 b^3 x^{14}+5 a b^4 x^{16}+b^5 x^{18}\right ) \, dx\\ &=\frac {a^5 x^9}{9}+\frac {5}{11} a^4 b x^{11}+\frac {10}{13} a^3 b^2 x^{13}+\frac {2}{3} a^2 b^3 x^{15}+\frac {5}{17} a b^4 x^{17}+\frac {b^5 x^{19}}{19}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 69, normalized size = 1.00 \begin {gather*} \frac {a^5 x^9}{9}+\frac {5}{11} a^4 b x^{11}+\frac {10}{13} a^3 b^2 x^{13}+\frac {2}{3} a^2 b^3 x^{15}+\frac {5}{17} a b^4 x^{17}+\frac {b^5 x^{19}}{19} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^8*(a + b*x^2)^5,x]

[Out]

(a^5*x^9)/9 + (5*a^4*b*x^11)/11 + (10*a^3*b^2*x^13)/13 + (2*a^2*b^3*x^15)/3 + (5*a*b^4*x^17)/17 + (b^5*x^19)/1
9

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.50, size = 57, normalized size = 0.83 \begin {gather*} \frac {1}{19} x^{19} b^{5} + \frac {5}{17} x^{17} b^{4} a + \frac {2}{3} x^{15} b^{3} a^{2} + \frac {10}{13} x^{13} b^{2} a^{3} + \frac {5}{11} x^{11} b a^{4} + \frac {1}{9} x^{9} a^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/19*x^19*b^5 + 5/17*x^17*b^4*a + 2/3*x^15*b^3*a^2 + 10/13*x^13*b^2*a^3 + 5/11*x^11*b*a^4 + 1/9*x^9*a^5

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giac [A]  time = 1.05, size = 57, normalized size = 0.83 \begin {gather*} \frac {1}{19} \, b^{5} x^{19} + \frac {5}{17} \, a b^{4} x^{17} + \frac {2}{3} \, a^{2} b^{3} x^{15} + \frac {10}{13} \, a^{3} b^{2} x^{13} + \frac {5}{11} \, a^{4} b x^{11} + \frac {1}{9} \, a^{5} x^{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/19*b^5*x^19 + 5/17*a*b^4*x^17 + 2/3*a^2*b^3*x^15 + 10/13*a^3*b^2*x^13 + 5/11*a^4*b*x^11 + 1/9*a^5*x^9

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maple [A]  time = 0.00, size = 58, normalized size = 0.84 \begin {gather*} \frac {1}{19} b^{5} x^{19}+\frac {5}{17} a \,b^{4} x^{17}+\frac {2}{3} a^{2} b^{3} x^{15}+\frac {10}{13} a^{3} b^{2} x^{13}+\frac {5}{11} a^{4} b \,x^{11}+\frac {1}{9} a^{5} x^{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*a^5*x^9+5/11*a^4*b*x^11+10/13*a^3*b^2*x^13+2/3*a^2*b^3*x^15+5/17*a*b^4*x^17+1/19*b^5*x^19

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maxima [A]  time = 1.32, size = 57, normalized size = 0.83 \begin {gather*} \frac {1}{19} \, b^{5} x^{19} + \frac {5}{17} \, a b^{4} x^{17} + \frac {2}{3} \, a^{2} b^{3} x^{15} + \frac {10}{13} \, a^{3} b^{2} x^{13} + \frac {5}{11} \, a^{4} b x^{11} + \frac {1}{9} \, a^{5} x^{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/19*b^5*x^19 + 5/17*a*b^4*x^17 + 2/3*a^2*b^3*x^15 + 10/13*a^3*b^2*x^13 + 5/11*a^4*b*x^11 + 1/9*a^5*x^9

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mupad [B]  time = 0.02, size = 57, normalized size = 0.83 \begin {gather*} \frac {a^5\,x^9}{9}+\frac {5\,a^4\,b\,x^{11}}{11}+\frac {10\,a^3\,b^2\,x^{13}}{13}+\frac {2\,a^2\,b^3\,x^{15}}{3}+\frac {5\,a\,b^4\,x^{17}}{17}+\frac {b^5\,x^{19}}{19} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(a + b*x^2)^5,x)

[Out]

(a^5*x^9)/9 + (b^5*x^19)/19 + (5*a^4*b*x^11)/11 + (5*a*b^4*x^17)/17 + (10*a^3*b^2*x^13)/13 + (2*a^2*b^3*x^15)/
3

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sympy [A]  time = 0.08, size = 66, normalized size = 0.96 \begin {gather*} \frac {a^{5} x^{9}}{9} + \frac {5 a^{4} b x^{11}}{11} + \frac {10 a^{3} b^{2} x^{13}}{13} + \frac {2 a^{2} b^{3} x^{15}}{3} + \frac {5 a b^{4} x^{17}}{17} + \frac {b^{5} x^{19}}{19} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*x**9/9 + 5*a**4*b*x**11/11 + 10*a**3*b**2*x**13/13 + 2*a**2*b**3*x**15/3 + 5*a*b**4*x**17/17 + b**5*x**19
/19

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