3.3.19 \(\int x^3 (a+b x^3)^2 \, dx\) [219]

Optimal. Leaf size=30 \[ \frac {a^2 x^4}{4}+\frac {2}{7} a b x^7+\frac {b^2 x^{10}}{10} \]

[Out]

1/4*a^2*x^4+2/7*a*b*x^7+1/10*b^2*x^10

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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^4}{4}+\frac {2}{7} a b x^7+\frac {b^2 x^{10}}{10} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^3)^2,x]

[Out]

(a^2*x^4)/4 + (2*a*b*x^7)/7 + (b^2*x^10)/10

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

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Mathematica [A]
time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} \frac {a^2 x^4}{4}+\frac {2}{7} a b x^7+\frac {b^2 x^{10}}{10} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x^3)^2,x]

[Out]

(a^2*x^4)/4 + (2*a*b*x^7)/7 + (b^2*x^10)/10

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Maple [A]
time = 0.14, size = 25, normalized size = 0.83

method	result	size
gosper	\(\frac {1}{4} a^{2} x^{4}+\frac {2}{7} a b \,x^{7}+\frac {1}{10} b^{2} x^{10}\)	\(25\)
default	\(\frac {1}{4} a^{2} x^{4}+\frac {2}{7} a b \,x^{7}+\frac {1}{10} b^{2} x^{10}\)	\(25\)
norman	\(\frac {1}{4} a^{2} x^{4}+\frac {2}{7} a b \,x^{7}+\frac {1}{10} b^{2} x^{10}\)	\(25\)
risch	\(\frac {1}{4} a^{2} x^{4}+\frac {2}{7} a b \,x^{7}+\frac {1}{10} b^{2} x^{10}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x^3+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/4*a^2*x^4+2/7*a*b*x^7+1/10*b^2*x^10

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Maxima [A]
time = 0.29, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{10} \, b^{2} x^{10} + \frac {2}{7} \, a b x^{7} + \frac {1}{4} \, a^{2} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^2*x^10 + 2/7*a*b*x^7 + 1/4*a^2*x^4

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Fricas [A]
time = 0.34, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{10} \, b^{2} x^{10} + \frac {2}{7} \, a b x^{7} + \frac {1}{4} \, a^{2} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^2*x^10 + 2/7*a*b*x^7 + 1/4*a^2*x^4

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Sympy [A]
time = 0.01, size = 26, normalized size = 0.87 \begin {gather*} \frac {a^{2} x^{4}}{4} + \frac {2 a b x^{7}}{7} + \frac {b^{2} x^{10}}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**4/4 + 2*a*b*x**7/7 + b**2*x**10/10

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Giac [A]
time = 3.16, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{10} \, b^{2} x^{10} + \frac {2}{7} \, a b x^{7} + \frac {1}{4} \, a^{2} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^2*x^10 + 2/7*a*b*x^7 + 1/4*a^2*x^4

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Mupad [B]
time = 0.03, size = 24, normalized size = 0.80 \begin {gather*} \frac {a^2\,x^4}{4}+\frac {2\,a\,b\,x^7}{7}+\frac {b^2\,x^{10}}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x^3)^2,x)

[Out]

(a^2*x^4)/4 + (b^2*x^10)/10 + (2*a*b*x^7)/7

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