3.3.22 \(\int (a+b x^3)^2 \, dx\) [222]

Optimal. Leaf size=25 \[ a^2 x+\frac {1}{2} a b x^4+\frac {b^2 x^7}{7} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {200} \begin {gather*} a^2 x+\frac {1}{2} a b x^4+\frac {b^2 x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2,x]

[Out]

a^2*x + (a*b*x^4)/2 + (b^2*x^7)/7

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 25, normalized size = 1.00 \begin {gather*} a^2 x+\frac {1}{2} a b x^4+\frac {b^2 x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2,x]

[Out]

a^2*x + (a*b*x^4)/2 + (b^2*x^7)/7

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Maple [A]
time = 0.11, size = 22, normalized size = 0.88

method	result	size
gosper	\(a^{2} x +\frac {1}{2} a b \,x^{4}+\frac {1}{7} b^{2} x^{7}\)	\(22\)
default	\(a^{2} x +\frac {1}{2} a b \,x^{4}+\frac {1}{7} b^{2} x^{7}\)	\(22\)
norman	\(a^{2} x +\frac {1}{2} a b \,x^{4}+\frac {1}{7} b^{2} x^{7}\)	\(22\)
risch	\(a^{2} x +\frac {1}{2} a b \,x^{4}+\frac {1}{7} b^{2} x^{7}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 21, normalized size = 0.84 \begin {gather*} \frac {1}{7} \, b^{2} x^{7} + \frac {1}{2} \, a b x^{4} + a^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^2*x^7 + 1/2*a*b*x^4 + a^2*x

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Fricas [A]
time = 0.33, size = 21, normalized size = 0.84 \begin {gather*} \frac {1}{7} \, b^{2} x^{7} + \frac {1}{2} \, a b x^{4} + a^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^2*x^7 + 1/2*a*b*x^4 + a^2*x

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Sympy [A]
time = 0.01, size = 20, normalized size = 0.80 \begin {gather*} a^{2} x + \frac {a b x^{4}}{2} + \frac {b^{2} x^{7}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x + a*b*x**4/2 + b**2*x**7/7

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Giac [A]
time = 2.54, size = 21, normalized size = 0.84 \begin {gather*} \frac {1}{7} \, b^{2} x^{7} + \frac {1}{2} \, a b x^{4} + a^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^2*x^7 + 1/2*a*b*x^4 + a^2*x

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Mupad [B]
time = 0.03, size = 21, normalized size = 0.84 \begin {gather*} a^2\,x+\frac {a\,b\,x^4}{2}+\frac {b^2\,x^7}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^2,x)

[Out]

a^2*x + (b^2*x^7)/7 + (a*b*x^4)/2

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