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

Optimal. Leaf size=70 \[ \frac{1}{4} c^2 x^4 (3 a d+b c)+\frac{1}{10} d^2 x^{10} (a d+3 b c)+\frac{3}{7} c d x^7 (a d+b c)+a c^3 x+\frac{1}{13} b d^3 x^{13} \]

[Out]

a*c^3*x + (c^2*(b*c + 3*a*d)*x^4)/4 + (3*c*d*(b*c + a*d)*x^7)/7 + (d^2*(3*b*c + a*d)*x^10)/10 + (b*d^3*x^13)/1
3

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Rubi [A]  time = 0.0426525, antiderivative size = 70, 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} \[ \frac{1}{4} c^2 x^4 (3 a d+b c)+\frac{1}{10} d^2 x^{10} (a d+3 b c)+\frac{3}{7} c d x^7 (a d+b c)+a c^3 x+\frac{1}{13} b d^3 x^{13} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)*(c + d*x^3)^3,x]

[Out]

a*c^3*x + (c^2*(b*c + 3*a*d)*x^4)/4 + (3*c*d*(b*c + a*d)*x^7)/7 + (d^2*(3*b*c + a*d)*x^10)/10 + (b*d^3*x^13)/1
3

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 )^3 \, dx &=\int \left (a c^3+c^2 (b c+3 a d) x^3+3 c d (b c+a d) x^6+d^2 (3 b c+a d) x^9+b d^3 x^{12}\right ) \, dx\\ &=a c^3 x+\frac{1}{4} c^2 (b c+3 a d) x^4+\frac{3}{7} c d (b c+a d) x^7+\frac{1}{10} d^2 (3 b c+a d) x^{10}+\frac{1}{13} b d^3 x^{13}\\ \end{align*}

Mathematica [A]  time = 0.0125842, size = 70, normalized size = 1. \[ \frac{1}{4} c^2 x^4 (3 a d+b c)+\frac{1}{10} d^2 x^{10} (a d+3 b c)+\frac{3}{7} c d x^7 (a d+b c)+a c^3 x+\frac{1}{13} b d^3 x^{13} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*c^3*x + (c^2*(b*c + 3*a*d)*x^4)/4 + (3*c*d*(b*c + a*d)*x^7)/7 + (d^2*(3*b*c + a*d)*x^10)/10 + (b*d^3*x^13)/1
3

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Maple [A]  time = 0.001, size = 73, normalized size = 1. \begin{align*}{\frac{b{d}^{3}{x}^{13}}{13}}+{\frac{ \left ( a{d}^{3}+3\,bc{d}^{2} \right ){x}^{10}}{10}}+{\frac{ \left ( 3\,ac{d}^{2}+3\,b{c}^{2}d \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,a{c}^{2}d+b{c}^{3} \right ){x}^{4}}{4}}+a{c}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*b*d^3*x^13+1/10*(a*d^3+3*b*c*d^2)*x^10+1/7*(3*a*c*d^2+3*b*c^2*d)*x^7+1/4*(3*a*c^2*d+b*c^3)*x^4+a*c^3*x

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Maxima [A]  time = 0.955029, size = 95, normalized size = 1.36 \begin{align*} \frac{1}{13} \, b d^{3} x^{13} + \frac{1}{10} \,{\left (3 \, b c d^{2} + a d^{3}\right )} x^{10} + \frac{3}{7} \,{\left (b c^{2} d + a c d^{2}\right )} x^{7} + a c^{3} x + \frac{1}{4} \,{\left (b c^{3} + 3 \, a c^{2} d\right )} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*b*d^3*x^13 + 1/10*(3*b*c*d^2 + a*d^3)*x^10 + 3/7*(b*c^2*d + a*c*d^2)*x^7 + a*c^3*x + 1/4*(b*c^3 + 3*a*c^2
*d)*x^4

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Fricas [A]  time = 1.34321, size = 182, normalized size = 2.6 \begin{align*} \frac{1}{13} x^{13} d^{3} b + \frac{3}{10} x^{10} d^{2} c b + \frac{1}{10} x^{10} d^{3} a + \frac{3}{7} x^{7} d c^{2} b + \frac{3}{7} x^{7} d^{2} c a + \frac{1}{4} x^{4} c^{3} b + \frac{3}{4} x^{4} d c^{2} a + x c^{3} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*x^13*d^3*b + 3/10*x^10*d^2*c*b + 1/10*x^10*d^3*a + 3/7*x^7*d*c^2*b + 3/7*x^7*d^2*c*a + 1/4*x^4*c^3*b + 3/
4*x^4*d*c^2*a + x*c^3*a

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Sympy [A]  time = 0.077301, size = 80, normalized size = 1.14 \begin{align*} a c^{3} x + \frac{b d^{3} x^{13}}{13} + x^{10} \left (\frac{a d^{3}}{10} + \frac{3 b c d^{2}}{10}\right ) + x^{7} \left (\frac{3 a c d^{2}}{7} + \frac{3 b c^{2} d}{7}\right ) + x^{4} \left (\frac{3 a c^{2} d}{4} + \frac{b c^{3}}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c**3*x + b*d**3*x**13/13 + x**10*(a*d**3/10 + 3*b*c*d**2/10) + x**7*(3*a*c*d**2/7 + 3*b*c**2*d/7) + x**4*(3*
a*c**2*d/4 + b*c**3/4)

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Giac [A]  time = 1.10375, size = 100, normalized size = 1.43 \begin{align*} \frac{1}{13} \, b d^{3} x^{13} + \frac{3}{10} \, b c d^{2} x^{10} + \frac{1}{10} \, a d^{3} x^{10} + \frac{3}{7} \, b c^{2} d x^{7} + \frac{3}{7} \, a c d^{2} x^{7} + \frac{1}{4} \, b c^{3} x^{4} + \frac{3}{4} \, a c^{2} d x^{4} + a c^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*b*d^3*x^13 + 3/10*b*c*d^2*x^10 + 1/10*a*d^3*x^10 + 3/7*b*c^2*d*x^7 + 3/7*a*c*d^2*x^7 + 1/4*b*c^3*x^4 + 3/
4*a*c^2*d*x^4 + a*c^3*x