3.53 \(\int x^{11} (a+b x^2)^5 \, dx\)

Optimal. Leaf size=69 \[ \frac{5}{9} a^2 b^3 x^{18}+\frac{5}{8} a^3 b^2 x^{16}+\frac{5}{14} a^4 b x^{14}+\frac{a^5 x^{12}}{12}+\frac{1}{4} a b^4 x^{20}+\frac{b^5 x^{22}}{22} \]

[Out]

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

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Rubi [A]  time = 0.044137, antiderivative size = 69, normalized size of antiderivative = 1., 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} \[ \frac{5}{9} a^2 b^3 x^{18}+\frac{5}{8} a^3 b^2 x^{16}+\frac{5}{14} a^4 b x^{14}+\frac{a^5 x^{12}}{12}+\frac{1}{4} a b^4 x^{20}+\frac{b^5 x^{22}}{22} \]

Antiderivative was successfully verified.

[In]

Int[x^11*(a + b*x^2)^5,x]

[Out]

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

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]]

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])

Rubi steps

\begin{align*} \int x^{11} \left (a+b x^2\right )^5 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^5 (a+b x)^5 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a^5 x^5+5 a^4 b x^6+10 a^3 b^2 x^7+10 a^2 b^3 x^8+5 a b^4 x^9+b^5 x^{10}\right ) \, dx,x,x^2\right )\\ &=\frac{a^5 x^{12}}{12}+\frac{5}{14} a^4 b x^{14}+\frac{5}{8} a^3 b^2 x^{16}+\frac{5}{9} a^2 b^3 x^{18}+\frac{1}{4} a b^4 x^{20}+\frac{b^5 x^{22}}{22}\\ \end{align*}

Mathematica [A]  time = 0.002101, size = 69, normalized size = 1. \[ \frac{5}{9} a^2 b^3 x^{18}+\frac{5}{8} a^3 b^2 x^{16}+\frac{5}{14} a^4 b x^{14}+\frac{a^5 x^{12}}{12}+\frac{1}{4} a b^4 x^{20}+\frac{b^5 x^{22}}{22} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11*(a + b*x^2)^5,x]

[Out]

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

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Maple [A]  time = 0., size = 58, normalized size = 0.8 \begin{align*}{\frac{{a}^{5}{x}^{12}}{12}}+{\frac{5\,{a}^{4}b{x}^{14}}{14}}+{\frac{5\,{a}^{3}{b}^{2}{x}^{16}}{8}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{18}}{9}}+{\frac{a{b}^{4}{x}^{20}}{4}}+{\frac{{b}^{5}{x}^{22}}{22}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(b*x^2+a)^5,x)

[Out]

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

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Maxima [A]  time = 1.3269, size = 77, normalized size = 1.12 \begin{align*} \frac{1}{22} \, b^{5} x^{22} + \frac{1}{4} \, a b^{4} x^{20} + \frac{5}{9} \, a^{2} b^{3} x^{18} + \frac{5}{8} \, a^{3} b^{2} x^{16} + \frac{5}{14} \, a^{4} b x^{14} + \frac{1}{12} \, a^{5} x^{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^2+a)^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.07144, size = 140, normalized size = 2.03 \begin{align*} \frac{1}{22} x^{22} b^{5} + \frac{1}{4} x^{20} b^{4} a + \frac{5}{9} x^{18} b^{3} a^{2} + \frac{5}{8} x^{16} b^{2} a^{3} + \frac{5}{14} x^{14} b a^{4} + \frac{1}{12} x^{12} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^2+a)^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.070696, size = 65, normalized size = 0.94 \begin{align*} \frac{a^{5} x^{12}}{12} + \frac{5 a^{4} b x^{14}}{14} + \frac{5 a^{3} b^{2} x^{16}}{8} + \frac{5 a^{2} b^{3} x^{18}}{9} + \frac{a b^{4} x^{20}}{4} + \frac{b^{5} x^{22}}{22} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11*(b*x**2+a)**5,x)

[Out]

a**5*x**12/12 + 5*a**4*b*x**14/14 + 5*a**3*b**2*x**16/8 + 5*a**2*b**3*x**18/9 + a*b**4*x**20/4 + b**5*x**22/22

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Giac [A]  time = 2.51753, size = 77, normalized size = 1.12 \begin{align*} \frac{1}{22} \, b^{5} x^{22} + \frac{1}{4} \, a b^{4} x^{20} + \frac{5}{9} \, a^{2} b^{3} x^{18} + \frac{5}{8} \, a^{3} b^{2} x^{16} + \frac{5}{14} \, a^{4} b x^{14} + \frac{1}{12} \, a^{5} x^{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^2+a)^5,x, algorithm="giac")

[Out]

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