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

Optimal. Leaf size=30 \[ \frac {a^2 x^5}{5}+\frac {1}{4} a b x^8+\frac {b^2 x^{11}}{11} \]

[Out]

1/5*a^2*x^5+1/4*a*b*x^8+1/11*b^2*x^11

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Rubi [A]  time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, 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} \[ \frac {a^2 x^5}{5}+\frac {1}{4} a b x^8+\frac {b^2 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x^3)^2,x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 30, normalized size = 1.00 \[ \frac {a^2 x^5}{5}+\frac {1}{4} a b x^8+\frac {b^2 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x^3)^2,x]

[Out]

(a^2*x^5)/5 + (a*b*x^8)/4 + (b^2*x^11)/11

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fricas [A]  time = 0.46, size = 24, normalized size = 0.80 \[ \frac {1}{11} x^{11} b^{2} + \frac {1}{4} x^{8} b a + \frac {1}{5} x^{5} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*x^11*b^2 + 1/4*x^8*b*a + 1/5*x^5*a^2

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giac [A]  time = 0.15, size = 24, normalized size = 0.80 \[ \frac {1}{11} \, b^{2} x^{11} + \frac {1}{4} \, a b x^{8} + \frac {1}{5} \, a^{2} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b^2*x^11 + 1/4*a*b*x^8 + 1/5*a^2*x^5

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maple [A]  time = 0.00, size = 25, normalized size = 0.83 \[ \frac {1}{11} b^{2} x^{11}+\frac {1}{4} a b \,x^{8}+\frac {1}{5} a^{2} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a^2*x^5+1/4*a*b*x^8+1/11*b^2*x^11

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maxima [A]  time = 1.37, size = 24, normalized size = 0.80 \[ \frac {1}{11} \, b^{2} x^{11} + \frac {1}{4} \, a b x^{8} + \frac {1}{5} \, a^{2} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b^2*x^11 + 1/4*a*b*x^8 + 1/5*a^2*x^5

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mupad [B]  time = 0.04, size = 24, normalized size = 0.80 \[ \frac {a^2\,x^5}{5}+\frac {a\,b\,x^8}{4}+\frac {b^2\,x^{11}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*x^3)^2,x)

[Out]

(a^2*x^5)/5 + (b^2*x^11)/11 + (a*b*x^8)/4

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sympy [A]  time = 0.07, size = 24, normalized size = 0.80 \[ \frac {a^{2} x^{5}}{5} + \frac {a b x^{8}}{4} + \frac {b^{2} x^{11}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**5/5 + a*b*x**8/4 + b**2*x**11/11

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