### 3.2 $$\int (d+e x^3)^4 (a+b x^3+c x^6) \, dx$$

Optimal. Leaf size=135 $\frac{1}{13} e^2 x^{13} \left (e (a e+4 b d)+6 c d^2\right )+\frac{1}{7} d^2 x^7 \left (6 a e^2+4 b d e+c d^2\right )+\frac{1}{5} d e x^{10} \left (e (2 a e+3 b d)+2 c d^2\right )+\frac{1}{4} d^3 x^4 (4 a e+b d)+a d^4 x+\frac{1}{16} e^3 x^{16} (b e+4 c d)+\frac{1}{19} c e^4 x^{19}$

[Out]

a*d^4*x + (d^3*(b*d + 4*a*e)*x^4)/4 + (d^2*(c*d^2 + 4*b*d*e + 6*a*e^2)*x^7)/7 + (d*e*(2*c*d^2 + e*(3*b*d + 2*a
*e))*x^10)/5 + (e^2*(6*c*d^2 + e*(4*b*d + a*e))*x^13)/13 + (e^3*(4*c*d + b*e)*x^16)/16 + (c*e^4*x^19)/19

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Rubi [A]  time = 0.125073, antiderivative size = 135, 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} $\frac{1}{13} e^2 x^{13} \left (e (a e+4 b d)+6 c d^2\right )+\frac{1}{7} d^2 x^7 \left (6 a e^2+4 b d e+c d^2\right )+\frac{1}{5} d e x^{10} \left (e (2 a e+3 b d)+2 c d^2\right )+\frac{1}{4} d^3 x^4 (4 a e+b d)+a d^4 x+\frac{1}{16} e^3 x^{16} (b e+4 c d)+\frac{1}{19} c e^4 x^{19}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x^3)^4*(a + b*x^3 + c*x^6),x]

[Out]

a*d^4*x + (d^3*(b*d + 4*a*e)*x^4)/4 + (d^2*(c*d^2 + 4*b*d*e + 6*a*e^2)*x^7)/7 + (d*e*(2*c*d^2 + e*(3*b*d + 2*a
*e))*x^10)/5 + (e^2*(6*c*d^2 + e*(4*b*d + a*e))*x^13)/13 + (e^3*(4*c*d + b*e)*x^16)/16 + (c*e^4*x^19)/19

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 )^4 \left (a+b x^3+c x^6\right ) \, dx &=\int \left (a d^4+d^3 (b d+4 a e) x^3+d^2 \left (c d^2+4 b d e+6 a e^2\right ) x^6+2 d e \left (2 c d^2+e (3 b d+2 a e)\right ) x^9+e^2 \left (6 c d^2+e (4 b d+a e)\right ) x^{12}+e^3 (4 c d+b e) x^{15}+c e^4 x^{18}\right ) \, dx\\ &=a d^4 x+\frac{1}{4} d^3 (b d+4 a e) x^4+\frac{1}{7} d^2 \left (c d^2+4 b d e+6 a e^2\right ) x^7+\frac{1}{5} d e \left (2 c d^2+e (3 b d+2 a e)\right ) x^{10}+\frac{1}{13} e^2 \left (6 c d^2+e (4 b d+a e)\right ) x^{13}+\frac{1}{16} e^3 (4 c d+b e) x^{16}+\frac{1}{19} c e^4 x^{19}\\ \end{align*}

Mathematica [A]  time = 0.0361796, size = 135, normalized size = 1. $\frac{1}{13} e^2 x^{13} \left (a e^2+4 b d e+6 c d^2\right )+\frac{1}{5} d e x^{10} \left (2 a e^2+3 b d e+2 c d^2\right )+\frac{1}{7} d^2 x^7 \left (6 a e^2+4 b d e+c d^2\right )+\frac{1}{4} d^3 x^4 (4 a e+b d)+a d^4 x+\frac{1}{16} e^3 x^{16} (b e+4 c d)+\frac{1}{19} c e^4 x^{19}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x^3)^4*(a + b*x^3 + c*x^6),x]

[Out]

a*d^4*x + (d^3*(b*d + 4*a*e)*x^4)/4 + (d^2*(c*d^2 + 4*b*d*e + 6*a*e^2)*x^7)/7 + (d*e*(2*c*d^2 + 3*b*d*e + 2*a*
e^2)*x^10)/5 + (e^2*(6*c*d^2 + 4*b*d*e + a*e^2)*x^13)/13 + (e^3*(4*c*d + b*e)*x^16)/16 + (c*e^4*x^19)/19

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Maple [A]  time = 0.001, size = 136, normalized size = 1. \begin{align*}{\frac{c{e}^{4}{x}^{19}}{19}}+{\frac{ \left ({e}^{4}b+4\,d{e}^{3}c \right ){x}^{16}}{16}}+{\frac{ \left ({e}^{4}a+4\,d{e}^{3}b+6\,{e}^{2}{d}^{2}c \right ){x}^{13}}{13}}+{\frac{ \left ( 4\,d{e}^{3}a+6\,{e}^{2}{d}^{2}b+4\,{d}^{3}ec \right ){x}^{10}}{10}}+{\frac{ \left ( 6\,{e}^{2}{d}^{2}a+4\,{d}^{3}eb+c{d}^{4} \right ){x}^{7}}{7}}+{\frac{ \left ( 4\,{d}^{3}ea+{d}^{4}b \right ){x}^{4}}{4}}+a{d}^{4}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^3+d)^4*(c*x^6+b*x^3+a),x)

[Out]

1/19*c*e^4*x^19+1/16*(b*e^4+4*c*d*e^3)*x^16+1/13*(a*e^4+4*b*d*e^3+6*c*d^2*e^2)*x^13+1/10*(4*a*d*e^3+6*b*d^2*e^
2+4*c*d^3*e)*x^10+1/7*(6*a*d^2*e^2+4*b*d^3*e+c*d^4)*x^7+1/4*(4*a*d^3*e+b*d^4)*x^4+a*d^4*x

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Maxima [A]  time = 0.98309, size = 182, normalized size = 1.35 \begin{align*} \frac{1}{19} \, c e^{4} x^{19} + \frac{1}{16} \,{\left (4 \, c d e^{3} + b e^{4}\right )} x^{16} + \frac{1}{13} \,{\left (6 \, c d^{2} e^{2} + 4 \, b d e^{3} + a e^{4}\right )} x^{13} + \frac{1}{5} \,{\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{10} + \frac{1}{7} \,{\left (c d^{4} + 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} x^{7} + a d^{4} x + \frac{1}{4} \,{\left (b d^{4} + 4 \, a d^{3} e\right )} x^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^3+d)^4*(c*x^6+b*x^3+a),x, algorithm="maxima")

[Out]

1/19*c*e^4*x^19 + 1/16*(4*c*d*e^3 + b*e^4)*x^16 + 1/13*(6*c*d^2*e^2 + 4*b*d*e^3 + a*e^4)*x^13 + 1/5*(2*c*d^3*e
+ 3*b*d^2*e^2 + 2*a*d*e^3)*x^10 + 1/7*(c*d^4 + 4*b*d^3*e + 6*a*d^2*e^2)*x^7 + a*d^4*x + 1/4*(b*d^4 + 4*a*d^3*
e)*x^4

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Fricas [A]  time = 1.16381, size = 360, normalized size = 2.67 \begin{align*} \frac{1}{19} x^{19} e^{4} c + \frac{1}{4} x^{16} e^{3} d c + \frac{1}{16} x^{16} e^{4} b + \frac{6}{13} x^{13} e^{2} d^{2} c + \frac{4}{13} x^{13} e^{3} d b + \frac{1}{13} x^{13} e^{4} a + \frac{2}{5} x^{10} e d^{3} c + \frac{3}{5} x^{10} e^{2} d^{2} b + \frac{2}{5} x^{10} e^{3} d a + \frac{1}{7} x^{7} d^{4} c + \frac{4}{7} x^{7} e d^{3} b + \frac{6}{7} x^{7} e^{2} d^{2} a + \frac{1}{4} x^{4} d^{4} b + x^{4} e d^{3} a + x d^{4} a \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^3+d)^4*(c*x^6+b*x^3+a),x, algorithm="fricas")

[Out]

1/19*x^19*e^4*c + 1/4*x^16*e^3*d*c + 1/16*x^16*e^4*b + 6/13*x^13*e^2*d^2*c + 4/13*x^13*e^3*d*b + 1/13*x^13*e^4
*a + 2/5*x^10*e*d^3*c + 3/5*x^10*e^2*d^2*b + 2/5*x^10*e^3*d*a + 1/7*x^7*d^4*c + 4/7*x^7*e*d^3*b + 6/7*x^7*e^2*
d^2*a + 1/4*x^4*d^4*b + x^4*e*d^3*a + x*d^4*a

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Sympy [A]  time = 0.089108, size = 151, normalized size = 1.12 \begin{align*} a d^{4} x + \frac{c e^{4} x^{19}}{19} + x^{16} \left (\frac{b e^{4}}{16} + \frac{c d e^{3}}{4}\right ) + x^{13} \left (\frac{a e^{4}}{13} + \frac{4 b d e^{3}}{13} + \frac{6 c d^{2} e^{2}}{13}\right ) + x^{10} \left (\frac{2 a d e^{3}}{5} + \frac{3 b d^{2} e^{2}}{5} + \frac{2 c d^{3} e}{5}\right ) + x^{7} \left (\frac{6 a d^{2} e^{2}}{7} + \frac{4 b d^{3} e}{7} + \frac{c d^{4}}{7}\right ) + x^{4} \left (a d^{3} e + \frac{b d^{4}}{4}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**3+d)**4*(c*x**6+b*x**3+a),x)

[Out]

a*d**4*x + c*e**4*x**19/19 + x**16*(b*e**4/16 + c*d*e**3/4) + x**13*(a*e**4/13 + 4*b*d*e**3/13 + 6*c*d**2*e**2
/13) + x**10*(2*a*d*e**3/5 + 3*b*d**2*e**2/5 + 2*c*d**3*e/5) + x**7*(6*a*d**2*e**2/7 + 4*b*d**3*e/7 + c*d**4/7
) + x**4*(a*d**3*e + b*d**4/4)

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Giac [A]  time = 1.11388, size = 190, normalized size = 1.41 \begin{align*} \frac{1}{19} \, c x^{19} e^{4} + \frac{1}{4} \, c d x^{16} e^{3} + \frac{1}{16} \, b x^{16} e^{4} + \frac{6}{13} \, c d^{2} x^{13} e^{2} + \frac{4}{13} \, b d x^{13} e^{3} + \frac{1}{13} \, a x^{13} e^{4} + \frac{2}{5} \, c d^{3} x^{10} e + \frac{3}{5} \, b d^{2} x^{10} e^{2} + \frac{2}{5} \, a d x^{10} e^{3} + \frac{1}{7} \, c d^{4} x^{7} + \frac{4}{7} \, b d^{3} x^{7} e + \frac{6}{7} \, a d^{2} x^{7} e^{2} + \frac{1}{4} \, b d^{4} x^{4} + a d^{3} x^{4} e + a d^{4} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^3+d)^4*(c*x^6+b*x^3+a),x, algorithm="giac")

[Out]

1/19*c*x^19*e^4 + 1/4*c*d*x^16*e^3 + 1/16*b*x^16*e^4 + 6/13*c*d^2*x^13*e^2 + 4/13*b*d*x^13*e^3 + 1/13*a*x^13*e
^4 + 2/5*c*d^3*x^10*e + 3/5*b*d^2*x^10*e^2 + 2/5*a*d*x^10*e^3 + 1/7*c*d^4*x^7 + 4/7*b*d^3*x^7*e + 6/7*a*d^2*x^
7*e^2 + 1/4*b*d^4*x^4 + a*d^3*x^4*e + a*d^4*x