### 3.5 $$\int (d+e x^3) (a+b x^3+c x^6) \, dx$$

Optimal. Leaf size=42 $\frac{1}{4} x^4 (a e+b d)+a d x+\frac{1}{7} x^7 (b e+c d)+\frac{1}{10} c e x^{10}$

[Out]

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

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Rubi [A]  time = 0.0278242, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.05, Rules used = {1407} $\frac{1}{4} x^4 (a e+b d)+a d x+\frac{1}{7} x^7 (b e+c d)+\frac{1}{10} c e x^{10}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0085175, size = 42, normalized size = 1. $\frac{1}{4} x^4 (a e+b d)+a d x+\frac{1}{7} x^7 (b e+c d)+\frac{1}{10} c e x^{10}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 37, normalized size = 0.9 \begin{align*} adx+{\frac{ \left ( ae+bd \right ){x}^{4}}{4}}+{\frac{ \left ( be+cd \right ){x}^{7}}{7}}+{\frac{ce{x}^{10}}{10}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05804, size = 49, normalized size = 1.17 \begin{align*} \frac{1}{10} \, c e x^{10} + \frac{1}{7} \,{\left (c d + b e\right )} x^{7} + \frac{1}{4} \,{\left (b d + a e\right )} x^{4} + a d x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.12813, size = 107, normalized size = 2.55 \begin{align*} \frac{1}{10} x^{10} e c + \frac{1}{7} x^{7} d c + \frac{1}{7} x^{7} e b + \frac{1}{4} x^{4} d b + \frac{1}{4} x^{4} e a + x d a \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.062923, size = 39, normalized size = 0.93 \begin{align*} a d x + \frac{c e x^{10}}{10} + x^{7} \left (\frac{b e}{7} + \frac{c d}{7}\right ) + x^{4} \left (\frac{a e}{4} + \frac{b d}{4}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.09884, size = 58, normalized size = 1.38 \begin{align*} \frac{1}{10} \, c x^{10} e + \frac{1}{7} \, c d x^{7} + \frac{1}{7} \, b x^{7} e + \frac{1}{4} \, b d x^{4} + \frac{1}{4} \, a x^{4} e + a d x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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