### 3.477 $$\int (d+e x) (a+c x^2)^3 \, dx$$

Optimal. Leaf size=56 $a^2 c d x^3+a^3 d x+\frac{3}{5} a c^2 d x^5+\frac{e \left (a+c x^2\right )^4}{8 c}+\frac{1}{7} c^3 d x^7$

[Out]

a^3*d*x + a^2*c*d*x^3 + (3*a*c^2*d*x^5)/5 + (c^3*d*x^7)/7 + (e*(a + c*x^2)^4)/(8*c)

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Rubi [A]  time = 0.021189, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.133, Rules used = {641, 194} $a^2 c d x^3+a^3 d x+\frac{3}{5} a c^2 d x^5+\frac{e \left (a+c x^2\right )^4}{8 c}+\frac{1}{7} c^3 d x^7$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)*(a + c*x^2)^3,x]

[Out]

a^3*d*x + a^2*c*d*x^3 + (3*a*c^2*d*x^5)/5 + (c^3*d*x^7)/7 + (e*(a + c*x^2)^4)/(8*c)

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int (d+e x) \left (a+c x^2\right )^3 \, dx &=\frac{e \left (a+c x^2\right )^4}{8 c}+d \int \left (a+c x^2\right )^3 \, dx\\ &=\frac{e \left (a+c x^2\right )^4}{8 c}+d \int \left (a^3+3 a^2 c x^2+3 a c^2 x^4+c^3 x^6\right ) \, dx\\ &=a^3 d x+a^2 c d x^3+\frac{3}{5} a c^2 d x^5+\frac{1}{7} c^3 d x^7+\frac{e \left (a+c x^2\right )^4}{8 c}\\ \end{align*}

Mathematica [A]  time = 0.0024985, size = 85, normalized size = 1.52 $a^2 c d x^3+\frac{3}{4} a^2 c e x^4+a^3 d x+\frac{1}{2} a^3 e x^2+\frac{3}{5} a c^2 d x^5+\frac{1}{2} a c^2 e x^6+\frac{1}{7} c^3 d x^7+\frac{1}{8} c^3 e x^8$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)*(a + c*x^2)^3,x]

[Out]

a^3*d*x + (a^3*e*x^2)/2 + a^2*c*d*x^3 + (3*a^2*c*e*x^4)/4 + (3*a*c^2*d*x^5)/5 + (a*c^2*e*x^6)/2 + (c^3*d*x^7)/
7 + (c^3*e*x^8)/8

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Maple [A]  time = 0.042, size = 74, normalized size = 1.3 \begin{align*}{\frac{{c}^{3}e{x}^{8}}{8}}+{\frac{{c}^{3}d{x}^{7}}{7}}+{\frac{ea{c}^{2}{x}^{6}}{2}}+{\frac{3\,a{c}^{2}d{x}^{5}}{5}}+{\frac{3\,e{a}^{2}c{x}^{4}}{4}}+{a}^{2}cd{x}^{3}+{\frac{{a}^{3}e{x}^{2}}{2}}+{a}^{3}dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*c^3*e*x^8+1/7*c^3*d*x^7+1/2*e*a*c^2*x^6+3/5*a*c^2*d*x^5+3/4*e*a^2*c*x^4+a^2*c*d*x^3+1/2*a^3*e*x^2+a^3*d*x

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Maxima [A]  time = 1.17, size = 99, normalized size = 1.77 \begin{align*} \frac{1}{8} \, c^{3} e x^{8} + \frac{1}{7} \, c^{3} d x^{7} + \frac{1}{2} \, a c^{2} e x^{6} + \frac{3}{5} \, a c^{2} d x^{5} + \frac{3}{4} \, a^{2} c e x^{4} + a^{2} c d x^{3} + \frac{1}{2} \, a^{3} e x^{2} + a^{3} d x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*c^3*e*x^8 + 1/7*c^3*d*x^7 + 1/2*a*c^2*e*x^6 + 3/5*a*c^2*d*x^5 + 3/4*a^2*c*e*x^4 + a^2*c*d*x^3 + 1/2*a^3*e*
x^2 + a^3*d*x

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Fricas [A]  time = 1.81113, size = 169, normalized size = 3.02 \begin{align*} \frac{1}{8} x^{8} e c^{3} + \frac{1}{7} x^{7} d c^{3} + \frac{1}{2} x^{6} e c^{2} a + \frac{3}{5} x^{5} d c^{2} a + \frac{3}{4} x^{4} e c a^{2} + x^{3} d c a^{2} + \frac{1}{2} x^{2} e a^{3} + x d a^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^8*e*c^3 + 1/7*x^7*d*c^3 + 1/2*x^6*e*c^2*a + 3/5*x^5*d*c^2*a + 3/4*x^4*e*c*a^2 + x^3*d*c*a^2 + 1/2*x^2*e*
a^3 + x*d*a^3

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Sympy [A]  time = 0.087168, size = 85, normalized size = 1.52 \begin{align*} a^{3} d x + \frac{a^{3} e x^{2}}{2} + a^{2} c d x^{3} + \frac{3 a^{2} c e x^{4}}{4} + \frac{3 a c^{2} d x^{5}}{5} + \frac{a c^{2} e x^{6}}{2} + \frac{c^{3} d x^{7}}{7} + \frac{c^{3} e x^{8}}{8} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*d*x + a**3*e*x**2/2 + a**2*c*d*x**3 + 3*a**2*c*e*x**4/4 + 3*a*c**2*d*x**5/5 + a*c**2*e*x**6/2 + c**3*d*x*
*7/7 + c**3*e*x**8/8

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Giac [A]  time = 1.31346, size = 104, normalized size = 1.86 \begin{align*} \frac{1}{8} \, c^{3} x^{8} e + \frac{1}{7} \, c^{3} d x^{7} + \frac{1}{2} \, a c^{2} x^{6} e + \frac{3}{5} \, a c^{2} d x^{5} + \frac{3}{4} \, a^{2} c x^{4} e + a^{2} c d x^{3} + \frac{1}{2} \, a^{3} x^{2} e + a^{3} d x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*c^3*x^8*e + 1/7*c^3*d*x^7 + 1/2*a*c^2*x^6*e + 3/5*a*c^2*d*x^5 + 3/4*a^2*c*x^4*e + a^2*c*d*x^3 + 1/2*a^3*x^
2*e + a^3*d*x