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

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

[Out]

a^2*x + (a*b*x^4)/2 + (b^2*x^7)/7

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Rubi [A]  time = 0.006953, antiderivative size = 25, normalized size of antiderivative = 1., 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 = {194} \[ a^2 x+\frac{1}{2} a b x^4+\frac{b^2 x^7}{7} \]

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 194

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*}

Mathematica [A]  time = 0.0007463, size = 25, normalized size = 1. \[ a^2 x+\frac{1}{2} a b x^4+\frac{b^2 x^7}{7} \]

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., size = 22, normalized size = 0.9 \begin{align*}{a}^{2}x+{\frac{ab{x}^{4}}{2}}+{\frac{{b}^{2}{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.953087, size = 28, normalized size = 1.12 \begin{align*} \frac{1}{7} \, b^{2} x^{7} + \frac{1}{2} \, a b x^{4} + a^{2} x \end{align*}

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 = 1.42111, size = 47, normalized size = 1.88 \begin{align*} \frac{1}{7} x^{7} b^{2} + \frac{1}{2} x^{4} b a + x a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*b^2 + 1/2*x^4*b*a + x*a^2

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Sympy [A]  time = 0.072196, size = 20, normalized size = 0.8 \begin{align*} a^{2} x + \frac{a b x^{4}}{2} + \frac{b^{2} x^{7}}{7} \end{align*}

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 = 1.10828, size = 28, normalized size = 1.12 \begin{align*} \frac{1}{7} \, b^{2} x^{7} + \frac{1}{2} \, a b x^{4} + a^{2} x \end{align*}

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