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

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

[Out]

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

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Rubi [A]  time = 0.0284518, 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} \[ \frac{1}{7} d x^7 (a d+2 b c)+\frac{1}{4} c x^4 (2 a d+b c)+a c^2 x+\frac{1}{10} b d^2 x^{10} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)*(c + d*x^3)^2,x]

[Out]

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

Mathematica [A]  time = 0.0081618, size = 50, normalized size = 1. \[ \frac{1}{7} d x^7 (a d+2 b c)+\frac{1}{4} c x^4 (2 a d+b c)+a c^2 x+\frac{1}{10} b d^2 x^{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 49, normalized size = 1. \begin{align*}{\frac{b{d}^{2}{x}^{10}}{10}}+{\frac{ \left ( a{d}^{2}+2\,bcd \right ){x}^{7}}{7}}+{\frac{ \left ( 2\,acd+b{c}^{2} \right ){x}^{4}}{4}}+a{c}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.940585, size = 65, normalized size = 1.3 \begin{align*} \frac{1}{10} \, b d^{2} x^{10} + \frac{1}{7} \,{\left (2 \, b c d + a d^{2}\right )} x^{7} + \frac{1}{4} \,{\left (b c^{2} + 2 \, a c d\right )} x^{4} + a c^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.28691, size = 123, normalized size = 2.46 \begin{align*} \frac{1}{10} x^{10} d^{2} b + \frac{2}{7} x^{7} d c b + \frac{1}{7} x^{7} d^{2} a + \frac{1}{4} x^{4} c^{2} b + \frac{1}{2} x^{4} d c a + x c^{2} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.101341, size = 51, normalized size = 1.02 \begin{align*} a c^{2} x + \frac{b d^{2} x^{10}}{10} + x^{7} \left (\frac{a d^{2}}{7} + \frac{2 b c d}{7}\right ) + x^{4} \left (\frac{a c d}{2} + \frac{b c^{2}}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.08964, size = 68, normalized size = 1.36 \begin{align*} \frac{1}{10} \, b d^{2} x^{10} + \frac{2}{7} \, b c d x^{7} + \frac{1}{7} \, a d^{2} x^{7} + \frac{1}{4} \, b c^{2} x^{4} + \frac{1}{2} \, a c d x^{4} + a c^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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