3.10 \(\int (a+b x^3)^2 (c+d x^3) \, dx\)

Optimal. Leaf size=50 \[ a^2 c x+\frac{1}{7} b x^7 (2 a d+b c)+\frac{1}{4} a x^4 (a d+2 b c)+\frac{1}{10} b^2 d x^{10} \]

[Out]

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

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Rubi [A]  time = 0.0292321, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {373} \[ a^2 c x+\frac{1}{7} b x^7 (2 a d+b c)+\frac{1}{4} a x^4 (a d+2 b c)+\frac{1}{10} b^2 d x^{10} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2*(c + d*x^3),x]

[Out]

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

Rule 373

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

Rubi steps

\begin{align*} \int \left (a+b x^3\right )^2 \left (c+d x^3\right ) \, dx &=\int \left (a^2 c+a (2 b c+a d) x^3+b (b c+2 a d) x^6+b^2 d x^9\right ) \, dx\\ &=a^2 c x+\frac{1}{4} a (2 b c+a d) x^4+\frac{1}{7} b (b c+2 a d) x^7+\frac{1}{10} b^2 d x^{10}\\ \end{align*}

Mathematica [A]  time = 0.0070966, size = 50, normalized size = 1. \[ a^2 c x+\frac{1}{7} b x^7 (2 a d+b c)+\frac{1}{4} a x^4 (a d+2 b c)+\frac{1}{10} b^2 d x^{10} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2*(c + d*x^3),x]

[Out]

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

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Maple [A]  time = 0., size = 49, normalized size = 1. \begin{align*}{\frac{{b}^{2}d{x}^{10}}{10}}+{\frac{ \left ( 2\,abd+{b}^{2}c \right ){x}^{7}}{7}}+{\frac{ \left ({a}^{2}d+2\,abc \right ){x}^{4}}{4}}+{a}^{2}cx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^2*d*x^10+1/7*(2*a*b*d+b^2*c)*x^7+1/4*(a^2*d+2*a*b*c)*x^4+a^2*c*x

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Maxima [A]  time = 0.955875, size = 65, normalized size = 1.3 \begin{align*} \frac{1}{10} \, b^{2} d x^{10} + \frac{1}{7} \,{\left (b^{2} c + 2 \, a b d\right )} x^{7} + \frac{1}{4} \,{\left (2 \, a b c + a^{2} d\right )} x^{4} + a^{2} c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^2*d*x^10 + 1/7*(b^2*c + 2*a*b*d)*x^7 + 1/4*(2*a*b*c + a^2*d)*x^4 + a^2*c*x

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Fricas [A]  time = 1.29771, size = 123, normalized size = 2.46 \begin{align*} \frac{1}{10} x^{10} d b^{2} + \frac{1}{7} x^{7} c b^{2} + \frac{2}{7} x^{7} d b a + \frac{1}{2} x^{4} c b a + \frac{1}{4} x^{4} d a^{2} + x c a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.100682, size = 51, normalized size = 1.02 \begin{align*} a^{2} c x + \frac{b^{2} d x^{10}}{10} + x^{7} \left (\frac{2 a b d}{7} + \frac{b^{2} c}{7}\right ) + x^{4} \left (\frac{a^{2} d}{4} + \frac{a b c}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.11853, size = 68, normalized size = 1.36 \begin{align*} \frac{1}{10} \, b^{2} d x^{10} + \frac{1}{7} \, b^{2} c x^{7} + \frac{2}{7} \, a b d x^{7} + \frac{1}{2} \, a b c x^{4} + \frac{1}{4} \, a^{2} d x^{4} + a^{2} c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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