### 3.3 $$\int (d+e x^3)^3 (a+b x^3+c x^6) \, dx$$

Optimal. Leaf size=103 $\frac{1}{10} e x^{10} \left (e (a e+3 b d)+3 c d^2\right )+\frac{1}{7} d x^7 \left (3 e (a e+b d)+c d^2\right )+\frac{1}{4} d^2 x^4 (3 a e+b d)+a d^3 x+\frac{1}{13} e^2 x^{13} (b e+3 c d)+\frac{1}{16} c e^3 x^{16}$

[Out]

a*d^3*x + (d^2*(b*d + 3*a*e)*x^4)/4 + (d*(c*d^2 + 3*e*(b*d + a*e))*x^7)/7 + (e*(3*c*d^2 + e*(3*b*d + a*e))*x^1
0)/10 + (e^2*(3*c*d + b*e)*x^13)/13 + (c*e^3*x^16)/16

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Rubi [A]  time = 0.0969754, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.045, Rules used = {1407} $\frac{1}{10} e x^{10} \left (e (a e+3 b d)+3 c d^2\right )+\frac{1}{7} d x^7 \left (3 e (a e+b d)+c d^2\right )+\frac{1}{4} d^2 x^4 (3 a e+b d)+a d^3 x+\frac{1}{13} e^2 x^{13} (b e+3 c d)+\frac{1}{16} c e^3 x^{16}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x^3)^3*(a + b*x^3 + c*x^6),x]

[Out]

a*d^3*x + (d^2*(b*d + 3*a*e)*x^4)/4 + (d*(c*d^2 + 3*e*(b*d + a*e))*x^7)/7 + (e*(3*c*d^2 + e*(3*b*d + a*e))*x^1
0)/10 + (e^2*(3*c*d + b*e)*x^13)/13 + (c*e^3*x^16)/16

Rule 1407

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x^n)^q*(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 0]

Rubi steps

\begin{align*} \int \left (d+e x^3\right )^3 \left (a+b x^3+c x^6\right ) \, dx &=\int \left (a d^3+d^2 (b d+3 a e) x^3+d \left (c d^2+3 e (b d+a e)\right ) x^6+e \left (3 c d^2+e (3 b d+a e)\right ) x^9+e^2 (3 c d+b e) x^{12}+c e^3 x^{15}\right ) \, dx\\ &=a d^3 x+\frac{1}{4} d^2 (b d+3 a e) x^4+\frac{1}{7} d \left (c d^2+3 e (b d+a e)\right ) x^7+\frac{1}{10} e \left (3 c d^2+e (3 b d+a e)\right ) x^{10}+\frac{1}{13} e^2 (3 c d+b e) x^{13}+\frac{1}{16} c e^3 x^{16}\\ \end{align*}

Mathematica [A]  time = 0.0285729, size = 104, normalized size = 1.01 $\frac{1}{10} e x^{10} \left (a e^2+3 b d e+3 c d^2\right )+\frac{1}{7} d x^7 \left (3 a e^2+3 b d e+c d^2\right )+\frac{1}{4} d^2 x^4 (3 a e+b d)+a d^3 x+\frac{1}{13} e^2 x^{13} (b e+3 c d)+\frac{1}{16} c e^3 x^{16}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x^3)^3*(a + b*x^3 + c*x^6),x]

[Out]

a*d^3*x + (d^2*(b*d + 3*a*e)*x^4)/4 + (d*(c*d^2 + 3*b*d*e + 3*a*e^2)*x^7)/7 + (e*(3*c*d^2 + 3*b*d*e + a*e^2)*x
^10)/10 + (e^2*(3*c*d + b*e)*x^13)/13 + (c*e^3*x^16)/16

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Maple [A]  time = 0.002, size = 103, normalized size = 1. \begin{align*}{\frac{c{e}^{3}{x}^{16}}{16}}+{\frac{ \left ({e}^{3}b+3\,cd{e}^{2} \right ){x}^{13}}{13}}+{\frac{ \left ({e}^{3}a+3\,d{e}^{2}b+3\,{d}^{2}ec \right ){x}^{10}}{10}}+{\frac{ \left ( 3\,ad{e}^{2}+3\,b{d}^{2}e+c{d}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{d}^{2}ea+{d}^{3}b \right ){x}^{4}}{4}}+a{d}^{3}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^3+d)^3*(c*x^6+b*x^3+a),x)

[Out]

1/16*c*e^3*x^16+1/13*(b*e^3+3*c*d*e^2)*x^13+1/10*(a*e^3+3*b*d*e^2+3*c*d^2*e)*x^10+1/7*(3*a*d*e^2+3*b*d^2*e+c*d
^3)*x^7+1/4*(3*a*d^2*e+b*d^3)*x^4+a*d^3*x

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Maxima [A]  time = 1.05264, size = 138, normalized size = 1.34 \begin{align*} \frac{1}{16} \, c e^{3} x^{16} + \frac{1}{13} \,{\left (3 \, c d e^{2} + b e^{3}\right )} x^{13} + \frac{1}{10} \,{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} x^{10} + \frac{1}{7} \,{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} x^{7} + a d^{3} x + \frac{1}{4} \,{\left (b d^{3} + 3 \, a d^{2} e\right )} x^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^3+d)^3*(c*x^6+b*x^3+a),x, algorithm="maxima")

[Out]

1/16*c*e^3*x^16 + 1/13*(3*c*d*e^2 + b*e^3)*x^13 + 1/10*(3*c*d^2*e + 3*b*d*e^2 + a*e^3)*x^10 + 1/7*(c*d^3 + 3*b
*d^2*e + 3*a*d*e^2)*x^7 + a*d^3*x + 1/4*(b*d^3 + 3*a*d^2*e)*x^4

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Fricas [A]  time = 1.15749, size = 282, normalized size = 2.74 \begin{align*} \frac{1}{16} x^{16} e^{3} c + \frac{3}{13} x^{13} e^{2} d c + \frac{1}{13} x^{13} e^{3} b + \frac{3}{10} x^{10} e d^{2} c + \frac{3}{10} x^{10} e^{2} d b + \frac{1}{10} x^{10} e^{3} a + \frac{1}{7} x^{7} d^{3} c + \frac{3}{7} x^{7} e d^{2} b + \frac{3}{7} x^{7} e^{2} d a + \frac{1}{4} x^{4} d^{3} b + \frac{3}{4} x^{4} e d^{2} a + x d^{3} a \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^3+d)^3*(c*x^6+b*x^3+a),x, algorithm="fricas")

[Out]

1/16*x^16*e^3*c + 3/13*x^13*e^2*d*c + 1/13*x^13*e^3*b + 3/10*x^10*e*d^2*c + 3/10*x^10*e^2*d*b + 1/10*x^10*e^3*
a + 1/7*x^7*d^3*c + 3/7*x^7*e*d^2*b + 3/7*x^7*e^2*d*a + 1/4*x^4*d^3*b + 3/4*x^4*e*d^2*a + x*d^3*a

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Sympy [A]  time = 0.082, size = 117, normalized size = 1.14 \begin{align*} a d^{3} x + \frac{c e^{3} x^{16}}{16} + x^{13} \left (\frac{b e^{3}}{13} + \frac{3 c d e^{2}}{13}\right ) + x^{10} \left (\frac{a e^{3}}{10} + \frac{3 b d e^{2}}{10} + \frac{3 c d^{2} e}{10}\right ) + x^{7} \left (\frac{3 a d e^{2}}{7} + \frac{3 b d^{2} e}{7} + \frac{c d^{3}}{7}\right ) + x^{4} \left (\frac{3 a d^{2} e}{4} + \frac{b d^{3}}{4}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**3+d)**3*(c*x**6+b*x**3+a),x)

[Out]

a*d**3*x + c*e**3*x**16/16 + x**13*(b*e**3/13 + 3*c*d*e**2/13) + x**10*(a*e**3/10 + 3*b*d*e**2/10 + 3*c*d**2*e
/10) + x**7*(3*a*d*e**2/7 + 3*b*d**2*e/7 + c*d**3/7) + x**4*(3*a*d**2*e/4 + b*d**3/4)

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Giac [A]  time = 1.07675, size = 147, normalized size = 1.43 \begin{align*} \frac{1}{16} \, c x^{16} e^{3} + \frac{3}{13} \, c d x^{13} e^{2} + \frac{1}{13} \, b x^{13} e^{3} + \frac{3}{10} \, c d^{2} x^{10} e + \frac{3}{10} \, b d x^{10} e^{2} + \frac{1}{10} \, a x^{10} e^{3} + \frac{1}{7} \, c d^{3} x^{7} + \frac{3}{7} \, b d^{2} x^{7} e + \frac{3}{7} \, a d x^{7} e^{2} + \frac{1}{4} \, b d^{3} x^{4} + \frac{3}{4} \, a d^{2} x^{4} e + a d^{3} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^3+d)^3*(c*x^6+b*x^3+a),x, algorithm="giac")

[Out]

1/16*c*x^16*e^3 + 3/13*c*d*x^13*e^2 + 1/13*b*x^13*e^3 + 3/10*c*d^2*x^10*e + 3/10*b*d*x^10*e^2 + 1/10*a*x^10*e^
3 + 1/7*c*d^3*x^7 + 3/7*b*d^2*x^7*e + 3/7*a*d*x^7*e^2 + 1/4*b*d^3*x^4 + 3/4*a*d^2*x^4*e + a*d^3*x