3.43 \(\int x^4 (a+b x^2)^3 \, dx\)

Optimal. Leaf size=43 \[ \frac{3}{7} a^2 b x^7+\frac{a^3 x^5}{5}+\frac{1}{3} a b^2 x^9+\frac{b^3 x^{11}}{11} \]

[Out]

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

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Rubi [A]  time = 0.0134052, antiderivative size = 43, 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} \[ \frac{3}{7} a^2 b x^7+\frac{a^3 x^5}{5}+\frac{1}{3} a b^2 x^9+\frac{b^3 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x^2)^3,x]

[Out]

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

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^4 \left (a+b x^2\right )^3 \, dx &=\int \left (a^3 x^4+3 a^2 b x^6+3 a b^2 x^8+b^3 x^{10}\right ) \, dx\\ &=\frac{a^3 x^5}{5}+\frac{3}{7} a^2 b x^7+\frac{1}{3} a b^2 x^9+\frac{b^3 x^{11}}{11}\\ \end{align*}

Mathematica [A]  time = 0.0016709, size = 43, normalized size = 1. \[ \frac{3}{7} a^2 b x^7+\frac{a^3 x^5}{5}+\frac{1}{3} a b^2 x^9+\frac{b^3 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x^2)^3,x]

[Out]

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

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Maple [A]  time = 0.001, size = 36, normalized size = 0.8 \begin{align*}{\frac{{a}^{3}{x}^{5}}{5}}+{\frac{3\,{a}^{2}b{x}^{7}}{7}}+{\frac{a{b}^{2}{x}^{9}}{3}}+{\frac{{b}^{3}{x}^{11}}{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x^2+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.99563, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{11} \, b^{3} x^{11} + \frac{1}{3} \, a b^{2} x^{9} + \frac{3}{7} \, a^{2} b x^{7} + \frac{1}{5} \, a^{3} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.08659, size = 82, normalized size = 1.91 \begin{align*} \frac{1}{11} x^{11} b^{3} + \frac{1}{3} x^{9} b^{2} a + \frac{3}{7} x^{7} b a^{2} + \frac{1}{5} x^{5} a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.069345, size = 37, normalized size = 0.86 \begin{align*} \frac{a^{3} x^{5}}{5} + \frac{3 a^{2} b x^{7}}{7} + \frac{a b^{2} x^{9}}{3} + \frac{b^{3} x^{11}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x**2+a)**3,x)

[Out]

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

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Giac [A]  time = 2.54028, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{11} \, b^{3} x^{11} + \frac{1}{3} \, a b^{2} x^{9} + \frac{3}{7} \, a^{2} b x^{7} + \frac{1}{5} \, a^{3} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^3,x, algorithm="giac")

[Out]

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