3.7.24 \(\int x^2 (a+b x^4)^2 \, dx\) [624]

Optimal. Leaf size=30 \[ \frac {a^2 x^3}{3}+\frac {2}{7} a b x^7+\frac {b^2 x^{11}}{11} \]

[Out]

1/3*a^2*x^3+2/7*a*b*x^7+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 = {276} \begin {gather*} \frac {a^2 x^3}{3}+\frac {2}{7} a b x^7+\frac {b^2 x^{11}}{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^4)^2,x]

[Out]

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

Rule 276

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

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Mathematica [A]
time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} \frac {a^2 x^3}{3}+\frac {2}{7} a b x^7+\frac {b^2 x^{11}}{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^4)^2,x]

[Out]

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

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Maple [A]
time = 0.15, size = 25, normalized size = 0.83

method	result	size
gosper	\(\frac {1}{3} a^{2} x^{3}+\frac {2}{7} a b \,x^{7}+\frac {1}{11} b^{2} x^{11}\)	\(25\)
default	\(\frac {1}{3} a^{2} x^{3}+\frac {2}{7} a b \,x^{7}+\frac {1}{11} b^{2} x^{11}\)	\(25\)
norman	\(\frac {1}{3} a^{2} x^{3}+\frac {2}{7} a b \,x^{7}+\frac {1}{11} b^{2} x^{11}\)	\(25\)
risch	\(\frac {1}{3} a^{2} x^{3}+\frac {2}{7} a b \,x^{7}+\frac {1}{11} b^{2} x^{11}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x^4+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{11} \, b^{2} x^{11} + \frac {2}{7} \, a b x^{7} + \frac {1}{3} \, a^{2} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.35, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{11} \, b^{2} x^{11} + \frac {2}{7} \, a b x^{7} + \frac {1}{3} \, a^{2} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.01, size = 26, normalized size = 0.87 \begin {gather*} \frac {a^{2} x^{3}}{3} + \frac {2 a b x^{7}}{7} + \frac {b^{2} x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.67, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{11} \, b^{2} x^{11} + \frac {2}{7} \, a b x^{7} + \frac {1}{3} \, a^{2} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.03, size = 24, normalized size = 0.80 \begin {gather*} \frac {a^2\,x^3}{3}+\frac {2\,a\,b\,x^7}{7}+\frac {b^2\,x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x^4)^2,x)

[Out]

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

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