3.52 \(\int x^{13} (a+b x^2)^5 \, dx\)

Optimal. Leaf size=69 \[ \frac{1}{2} a^2 b^3 x^{20}+\frac{5}{9} a^3 b^2 x^{18}+\frac{5}{16} a^4 b x^{16}+\frac{a^5 x^{14}}{14}+\frac{5}{22} a b^4 x^{22}+\frac{b^5 x^{24}}{24} \]

[Out]

(a^5*x^14)/14 + (5*a^4*b*x^16)/16 + (5*a^3*b^2*x^18)/9 + (a^2*b^3*x^20)/2 + (5*a*b^4*x^22)/22 + (b^5*x^24)/24

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Rubi [A]  time = 0.0488556, 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} \[ \frac{1}{2} a^2 b^3 x^{20}+\frac{5}{9} a^3 b^2 x^{18}+\frac{5}{16} a^4 b x^{16}+\frac{a^5 x^{14}}{14}+\frac{5}{22} a b^4 x^{22}+\frac{b^5 x^{24}}{24} \]

Antiderivative was successfully verified.

[In]

Int[x^13*(a + b*x^2)^5,x]

[Out]

(a^5*x^14)/14 + (5*a^4*b*x^16)/16 + (5*a^3*b^2*x^18)/9 + (a^2*b^3*x^20)/2 + (5*a*b^4*x^22)/22 + (b^5*x^24)/24

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^{13} \left (a+b x^2\right )^5 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^6 (a+b x)^5 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a^5 x^6+5 a^4 b x^7+10 a^3 b^2 x^8+10 a^2 b^3 x^9+5 a b^4 x^{10}+b^5 x^{11}\right ) \, dx,x,x^2\right )\\ &=\frac{a^5 x^{14}}{14}+\frac{5}{16} a^4 b x^{16}+\frac{5}{9} a^3 b^2 x^{18}+\frac{1}{2} a^2 b^3 x^{20}+\frac{5}{22} a b^4 x^{22}+\frac{b^5 x^{24}}{24}\\ \end{align*}

Mathematica [A]  time = 0.002849, size = 69, normalized size = 1. \[ \frac{1}{2} a^2 b^3 x^{20}+\frac{5}{9} a^3 b^2 x^{18}+\frac{5}{16} a^4 b x^{16}+\frac{a^5 x^{14}}{14}+\frac{5}{22} a b^4 x^{22}+\frac{b^5 x^{24}}{24} \]

Antiderivative was successfully verified.

[In]

Integrate[x^13*(a + b*x^2)^5,x]

[Out]

(a^5*x^14)/14 + (5*a^4*b*x^16)/16 + (5*a^3*b^2*x^18)/9 + (a^2*b^3*x^20)/2 + (5*a*b^4*x^22)/22 + (b^5*x^24)/24

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Maple [A]  time = 0.002, size = 58, normalized size = 0.8 \begin{align*}{\frac{{a}^{5}{x}^{14}}{14}}+{\frac{5\,{a}^{4}b{x}^{16}}{16}}+{\frac{5\,{a}^{3}{b}^{2}{x}^{18}}{9}}+{\frac{{a}^{2}{b}^{3}{x}^{20}}{2}}+{\frac{5\,a{b}^{4}{x}^{22}}{22}}+{\frac{{b}^{5}{x}^{24}}{24}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^13*(b*x^2+a)^5,x)

[Out]

1/14*a^5*x^14+5/16*a^4*b*x^16+5/9*a^3*b^2*x^18+1/2*a^2*b^3*x^20+5/22*a*b^4*x^22+1/24*b^5*x^24

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Maxima [A]  time = 2.1205, size = 77, normalized size = 1.12 \begin{align*} \frac{1}{24} \, b^{5} x^{24} + \frac{5}{22} \, a b^{4} x^{22} + \frac{1}{2} \, a^{2} b^{3} x^{20} + \frac{5}{9} \, a^{3} b^{2} x^{18} + \frac{5}{16} \, a^{4} b x^{16} + \frac{1}{14} \, a^{5} x^{14} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13*(b*x^2+a)^5,x, algorithm="maxima")

[Out]

1/24*b^5*x^24 + 5/22*a*b^4*x^22 + 1/2*a^2*b^3*x^20 + 5/9*a^3*b^2*x^18 + 5/16*a^4*b*x^16 + 1/14*a^5*x^14

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Fricas [A]  time = 1.11462, size = 142, normalized size = 2.06 \begin{align*} \frac{1}{24} x^{24} b^{5} + \frac{5}{22} x^{22} b^{4} a + \frac{1}{2} x^{20} b^{3} a^{2} + \frac{5}{9} x^{18} b^{2} a^{3} + \frac{5}{16} x^{16} b a^{4} + \frac{1}{14} x^{14} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13*(b*x^2+a)^5,x, algorithm="fricas")

[Out]

1/24*x^24*b^5 + 5/22*x^22*b^4*a + 1/2*x^20*b^3*a^2 + 5/9*x^18*b^2*a^3 + 5/16*x^16*b*a^4 + 1/14*x^14*a^5

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Sympy [A]  time = 0.072784, size = 65, normalized size = 0.94 \begin{align*} \frac{a^{5} x^{14}}{14} + \frac{5 a^{4} b x^{16}}{16} + \frac{5 a^{3} b^{2} x^{18}}{9} + \frac{a^{2} b^{3} x^{20}}{2} + \frac{5 a b^{4} x^{22}}{22} + \frac{b^{5} x^{24}}{24} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**13*(b*x**2+a)**5,x)

[Out]

a**5*x**14/14 + 5*a**4*b*x**16/16 + 5*a**3*b**2*x**18/9 + a**2*b**3*x**20/2 + 5*a*b**4*x**22/22 + b**5*x**24/2
4

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Giac [A]  time = 2.58715, size = 77, normalized size = 1.12 \begin{align*} \frac{1}{24} \, b^{5} x^{24} + \frac{5}{22} \, a b^{4} x^{22} + \frac{1}{2} \, a^{2} b^{3} x^{20} + \frac{5}{9} \, a^{3} b^{2} x^{18} + \frac{5}{16} \, a^{4} b x^{16} + \frac{1}{14} \, a^{5} x^{14} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13*(b*x^2+a)^5,x, algorithm="giac")

[Out]

1/24*b^5*x^24 + 5/22*a*b^4*x^22 + 1/2*a^2*b^3*x^20 + 5/9*a^3*b^2*x^18 + 5/16*a^4*b*x^16 + 1/14*a^5*x^14