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

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

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Rubi [A]  time = 0.01, antiderivative size = 43, 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} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 43, normalized size = 1.00 \begin {gather*} \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 {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^4 \left (a+b x^2\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^4*(a + b*x^2)^3,x]

[Out]

IntegrateAlgebraic[x^4*(a + b*x^2)^3, x]

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fricas [A]  time = 0.96, size = 35, normalized size = 0.81 \begin {gather*} \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 {gather*}

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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giac [A]  time = 1.03, size = 35, normalized size = 0.81 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.00, size = 36, normalized size = 0.84 \begin {gather*} \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 {gather*}

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 = 1.38, size = 35, normalized size = 0.81 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.04, size = 35, normalized size = 0.81 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*x^2)^3,x)

[Out]

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

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sympy [A]  time = 0.07, size = 37, normalized size = 0.86 \begin {gather*} \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 {gather*}

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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