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

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Rubi [A]  time = 0.04, antiderivative size = 70, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 70, normalized size = 1.00 \[ \frac {1}{4} c^2 x^4 (3 a d+b c)+\frac {1}{10} d^2 x^{10} (a d+3 b c)+\frac {3}{7} c d x^7 (a d+b c)+a c^3 x+\frac {1}{13} b d^3 x^{13} \]

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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fricas [A]  time = 0.36, size = 74, normalized size = 1.06 \[ \frac {1}{13} x^{13} d^{3} b + \frac {3}{10} x^{10} d^{2} c b + \frac {1}{10} x^{10} d^{3} a + \frac {3}{7} x^{7} d c^{2} b + \frac {3}{7} x^{7} d^{2} c a + \frac {1}{4} x^{4} c^{3} b + \frac {3}{4} x^{4} d c^{2} a + x c^{3} a \]

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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giac [A]  time = 0.15, size = 74, normalized size = 1.06 \[ \frac {1}{13} \, b d^{3} x^{13} + \frac {3}{10} \, b c d^{2} x^{10} + \frac {1}{10} \, a d^{3} x^{10} + \frac {3}{7} \, b c^{2} d x^{7} + \frac {3}{7} \, a c d^{2} x^{7} + \frac {1}{4} \, b c^{3} x^{4} + \frac {3}{4} \, a c^{2} d x^{4} + a c^{3} x \]

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

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maple [A]  time = 0.04, size = 73, normalized size = 1.04 \[ \frac {b \,d^{3} x^{13}}{13}+\frac {\left (a \,d^{3}+3 b c \,d^{2}\right ) x^{10}}{10}+\frac {\left (3 a c \,d^{2}+3 b \,c^{2} d \right ) x^{7}}{7}+a \,c^{3} x +\frac {\left (3 a \,c^{2} d +b \,c^{3}\right ) x^{4}}{4} \]

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 = 0.56, size = 70, normalized size = 1.00 \[ \frac {1}{13} \, b d^{3} x^{13} + \frac {1}{10} \, {\left (3 \, b c d^{2} + a d^{3}\right )} x^{10} + \frac {3}{7} \, {\left (b c^{2} d + a c d^{2}\right )} x^{7} + a c^{3} x + \frac {1}{4} \, {\left (b c^{3} + 3 \, a c^{2} d\right )} x^{4} \]

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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mupad [B]  time = 0.03, size = 66, normalized size = 0.94 \[ x^4\,\left (\frac {b\,c^3}{4}+\frac {3\,a\,d\,c^2}{4}\right )+x^{10}\,\left (\frac {a\,d^3}{10}+\frac {3\,b\,c\,d^2}{10}\right )+\frac {b\,d^3\,x^{13}}{13}+a\,c^3\,x+\frac {3\,c\,d\,x^7\,\left (a\,d+b\,c\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)*(c + d*x^3)^3,x)

[Out]

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

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sympy [A]  time = 0.08, size = 80, normalized size = 1.14 \[ a c^{3} x + \frac {b d^{3} x^{13}}{13} + x^{10} \left (\frac {a d^{3}}{10} + \frac {3 b c d^{2}}{10}\right ) + x^{7} \left (\frac {3 a c d^{2}}{7} + \frac {3 b c^{2} d}{7}\right ) + x^{4} \left (\frac {3 a c^{2} d}{4} + \frac {b c^{3}}{4}\right ) \]

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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