### 3.4 $$\int (d+e x^3)^2 (a+b x^3+c x^6) \, dx$$

Optimal. Leaf size=73 $\frac{1}{7} x^7 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{4} d x^4 (2 a e+b d)+a d^2 x+\frac{1}{10} e x^{10} (b e+2 c d)+\frac{1}{13} c e^2 x^{13}$

[Out]

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

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Rubi [A]  time = 0.0622015, antiderivative size = 73, 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} $\frac{1}{7} x^7 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{4} d x^4 (2 a e+b d)+a d^2 x+\frac{1}{10} e x^{10} (b e+2 c d)+\frac{1}{13} c e^2 x^{13}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x^3)^2*(a + b*x^3 + c*x^6),x]

[Out]

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

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

Mathematica [A]  time = 0.0218051, size = 73, normalized size = 1. $\frac{1}{7} x^7 \left (a e^2+2 b d e+c d^2\right )+\frac{1}{4} d x^4 (2 a e+b d)+a d^2 x+\frac{1}{10} e x^{10} (b e+2 c d)+\frac{1}{13} c e^2 x^{13}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x^3)^2*(a + b*x^3 + c*x^6),x]

[Out]

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

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Maple [A]  time = 0., size = 70, normalized size = 1. \begin{align*}{\frac{c{e}^{2}{x}^{13}}{13}}+{\frac{ \left ( b{e}^{2}+2\,dec \right ){x}^{10}}{10}}+{\frac{ \left ( a{e}^{2}+2\,bde+c{d}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 2\,dea+b{d}^{2} \right ){x}^{4}}{4}}+a{d}^{2}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^3+d)^2*(c*x^6+b*x^3+a),x)

[Out]

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

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Maxima [A]  time = 0.962235, size = 93, normalized size = 1.27 \begin{align*} \frac{1}{13} \, c e^{2} x^{13} + \frac{1}{10} \,{\left (2 \, c d e + b e^{2}\right )} x^{10} + \frac{1}{7} \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{7} + \frac{1}{4} \,{\left (b d^{2} + 2 \, a d e\right )} x^{4} + a d^{2} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^3+d)^2*(c*x^6+b*x^3+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.09655, size = 192, normalized size = 2.63 \begin{align*} \frac{1}{13} x^{13} e^{2} c + \frac{1}{5} x^{10} e d c + \frac{1}{10} x^{10} e^{2} b + \frac{1}{7} x^{7} d^{2} c + \frac{2}{7} x^{7} e d b + \frac{1}{7} x^{7} e^{2} a + \frac{1}{4} x^{4} d^{2} b + \frac{1}{2} x^{4} e d a + x d^{2} a \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^3+d)^2*(c*x^6+b*x^3+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.074794, size = 75, normalized size = 1.03 \begin{align*} a d^{2} x + \frac{c e^{2} x^{13}}{13} + x^{10} \left (\frac{b e^{2}}{10} + \frac{c d e}{5}\right ) + x^{7} \left (\frac{a e^{2}}{7} + \frac{2 b d e}{7} + \frac{c d^{2}}{7}\right ) + x^{4} \left (\frac{a d e}{2} + \frac{b d^{2}}{4}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**3+d)**2*(c*x**6+b*x**3+a),x)

[Out]

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

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Giac [A]  time = 1.16215, size = 103, normalized size = 1.41 \begin{align*} \frac{1}{13} \, c x^{13} e^{2} + \frac{1}{5} \, c d x^{10} e + \frac{1}{10} \, b x^{10} e^{2} + \frac{1}{7} \, c d^{2} x^{7} + \frac{2}{7} \, b d x^{7} e + \frac{1}{7} \, a x^{7} e^{2} + \frac{1}{4} \, b d^{2} x^{4} + \frac{1}{2} \, a d x^{4} e + a d^{2} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^3+d)^2*(c*x^6+b*x^3+a),x, algorithm="giac")

[Out]

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