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

Optimal. Leaf size=53 \[ \frac{a^2 \left (a+b x^2\right )^6}{12 b^3}+\frac{\left (a+b x^2\right )^8}{16 b^3}-\frac{a \left (a+b x^2\right )^7}{7 b^3} \]

[Out]

(a^2*(a + b*x^2)^6)/(12*b^3) - (a*(a + b*x^2)^7)/(7*b^3) + (a + b*x^2)^8/(16*b^3)

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Rubi [A]  time = 0.0647074, antiderivative size = 53, 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} \[ \frac{a^2 \left (a+b x^2\right )^6}{12 b^3}+\frac{\left (a+b x^2\right )^8}{16 b^3}-\frac{a \left (a+b x^2\right )^7}{7 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x^2)^5,x]

[Out]

(a^2*(a + b*x^2)^6)/(12*b^3) - (a*(a + b*x^2)^7)/(7*b^3) + (a + b*x^2)^8/(16*b^3)

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^5 \left (a+b x^2\right )^5 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 (a+b x)^5 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^2 (a+b x)^5}{b^2}-\frac{2 a (a+b x)^6}{b^2}+\frac{(a+b x)^7}{b^2}\right ) \, dx,x,x^2\right )\\ &=\frac{a^2 \left (a+b x^2\right )^6}{12 b^3}-\frac{a \left (a+b x^2\right )^7}{7 b^3}+\frac{\left (a+b x^2\right )^8}{16 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0020264, size = 66, normalized size = 1.25 \[ \frac{5}{6} a^2 b^3 x^{12}+a^3 b^2 x^{10}+\frac{5}{8} a^4 b x^8+\frac{a^5 x^6}{6}+\frac{5}{14} a b^4 x^{14}+\frac{b^5 x^{16}}{16} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x^2)^5,x]

[Out]

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

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Maple [A]  time = 0.001, size = 57, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}{x}^{16}}{16}}+{\frac{5\,a{b}^{4}{x}^{14}}{14}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{12}}{6}}+{a}^{3}{b}^{2}{x}^{10}+{\frac{5\,{a}^{4}b{x}^{8}}{8}}+{\frac{{a}^{5}{x}^{6}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 2.87543, size = 76, normalized size = 1.43 \begin{align*} \frac{1}{16} \, b^{5} x^{16} + \frac{5}{14} \, a b^{4} x^{14} + \frac{5}{6} \, a^{2} b^{3} x^{12} + a^{3} b^{2} x^{10} + \frac{5}{8} \, a^{4} b x^{8} + \frac{1}{6} \, a^{5} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.984943, size = 131, normalized size = 2.47 \begin{align*} \frac{1}{16} x^{16} b^{5} + \frac{5}{14} x^{14} b^{4} a + \frac{5}{6} x^{12} b^{3} a^{2} + x^{10} b^{2} a^{3} + \frac{5}{8} x^{8} b a^{4} + \frac{1}{6} x^{6} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.069944, size = 63, normalized size = 1.19 \begin{align*} \frac{a^{5} x^{6}}{6} + \frac{5 a^{4} b x^{8}}{8} + a^{3} b^{2} x^{10} + \frac{5 a^{2} b^{3} x^{12}}{6} + \frac{5 a b^{4} x^{14}}{14} + \frac{b^{5} x^{16}}{16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 2.01114, size = 76, normalized size = 1.43 \begin{align*} \frac{1}{16} \, b^{5} x^{16} + \frac{5}{14} \, a b^{4} x^{14} + \frac{5}{6} \, a^{2} b^{3} x^{12} + a^{3} b^{2} x^{10} + \frac{5}{8} \, a^{4} b x^{8} + \frac{1}{6} \, a^{5} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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