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

Optimal. Leaf size=66 \[ \frac{10}{9} a^2 b^3 x^9+\frac{10}{7} a^3 b^2 x^7+a^4 b x^5+\frac{a^5 x^3}{3}+\frac{5}{11} a b^4 x^{11}+\frac{b^5 x^{13}}{13} \]

[Out]

(a^5*x^3)/3 + a^4*b*x^5 + (10*a^3*b^2*x^7)/7 + (10*a^2*b^3*x^9)/9 + (5*a*b^4*x^11)/11 + (b^5*x^13)/13

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Rubi [A]  time = 0.0218526, antiderivative size = 66, normalized size of antiderivative = 1., 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} \[ \frac{10}{9} a^2 b^3 x^9+\frac{10}{7} a^3 b^2 x^7+a^4 b x^5+\frac{a^5 x^3}{3}+\frac{5}{11} a b^4 x^{11}+\frac{b^5 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^2)^5,x]

[Out]

(a^5*x^3)/3 + a^4*b*x^5 + (10*a^3*b^2*x^7)/7 + (10*a^2*b^3*x^9)/9 + (5*a*b^4*x^11)/11 + (b^5*x^13)/13

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^2 \left (a+b x^2\right )^5 \, dx &=\int \left (a^5 x^2+5 a^4 b x^4+10 a^3 b^2 x^6+10 a^2 b^3 x^8+5 a b^4 x^{10}+b^5 x^{12}\right ) \, dx\\ &=\frac{a^5 x^3}{3}+a^4 b x^5+\frac{10}{7} a^3 b^2 x^7+\frac{10}{9} a^2 b^3 x^9+\frac{5}{11} a b^4 x^{11}+\frac{b^5 x^{13}}{13}\\ \end{align*}

Mathematica [A]  time = 0.0020046, size = 66, normalized size = 1. \[ \frac{10}{9} a^2 b^3 x^9+\frac{10}{7} a^3 b^2 x^7+a^4 b x^5+\frac{a^5 x^3}{3}+\frac{5}{11} a b^4 x^{11}+\frac{b^5 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^2)^5,x]

[Out]

(a^5*x^3)/3 + a^4*b*x^5 + (10*a^3*b^2*x^7)/7 + (10*a^2*b^3*x^9)/9 + (5*a*b^4*x^11)/11 + (b^5*x^13)/13

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Maple [A]  time = 0., size = 57, normalized size = 0.9 \begin{align*}{\frac{{a}^{5}{x}^{3}}{3}}+{a}^{4}b{x}^{5}+{\frac{10\,{a}^{3}{b}^{2}{x}^{7}}{7}}+{\frac{10\,{a}^{2}{b}^{3}{x}^{9}}{9}}+{\frac{5\,a{b}^{4}{x}^{11}}{11}}+{\frac{{b}^{5}{x}^{13}}{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^5*x^3+a^4*b*x^5+10/7*a^3*b^2*x^7+10/9*a^2*b^3*x^9+5/11*a*b^4*x^11+1/13*b^5*x^13

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Maxima [A]  time = 1.58331, size = 76, normalized size = 1.15 \begin{align*} \frac{1}{13} \, b^{5} x^{13} + \frac{5}{11} \, a b^{4} x^{11} + \frac{10}{9} \, a^{2} b^{3} x^{9} + \frac{10}{7} \, a^{3} b^{2} x^{7} + a^{4} b x^{5} + \frac{1}{3} \, a^{5} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*b^5*x^13 + 5/11*a*b^4*x^11 + 10/9*a^2*b^3*x^9 + 10/7*a^3*b^2*x^7 + a^4*b*x^5 + 1/3*a^5*x^3

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Fricas [A]  time = 1.13154, size = 131, normalized size = 1.98 \begin{align*} \frac{1}{13} x^{13} b^{5} + \frac{5}{11} x^{11} b^{4} a + \frac{10}{9} x^{9} b^{3} a^{2} + \frac{10}{7} x^{7} b^{2} a^{3} + x^{5} b a^{4} + \frac{1}{3} x^{3} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*x^13*b^5 + 5/11*x^11*b^4*a + 10/9*x^9*b^3*a^2 + 10/7*x^7*b^2*a^3 + x^5*b*a^4 + 1/3*x^3*a^5

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Sympy [A]  time = 0.0711, size = 63, normalized size = 0.95 \begin{align*} \frac{a^{5} x^{3}}{3} + a^{4} b x^{5} + \frac{10 a^{3} b^{2} x^{7}}{7} + \frac{10 a^{2} b^{3} x^{9}}{9} + \frac{5 a b^{4} x^{11}}{11} + \frac{b^{5} x^{13}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*x**3/3 + a**4*b*x**5 + 10*a**3*b**2*x**7/7 + 10*a**2*b**3*x**9/9 + 5*a*b**4*x**11/11 + b**5*x**13/13

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Giac [A]  time = 1.39158, size = 76, normalized size = 1.15 \begin{align*} \frac{1}{13} \, b^{5} x^{13} + \frac{5}{11} \, a b^{4} x^{11} + \frac{10}{9} \, a^{2} b^{3} x^{9} + \frac{10}{7} \, a^{3} b^{2} x^{7} + a^{4} b x^{5} + \frac{1}{3} \, a^{5} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*b^5*x^13 + 5/11*a*b^4*x^11 + 10/9*a^2*b^3*x^9 + 10/7*a^3*b^2*x^7 + a^4*b*x^5 + 1/3*a^5*x^3