### 3.453 $$\int (d+e x)^2 (a+c x^2) \, dx$$

Optimal. Leaf size=57 $\frac{(d+e x)^3 \left (a e^2+c d^2\right )}{3 e^3}+\frac{c (d+e x)^5}{5 e^3}-\frac{c d (d+e x)^4}{2 e^3}$

[Out]

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

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Rubi [A]  time = 0.0320099, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.067, Rules used = {697} $\frac{(d+e x)^3 \left (a e^2+c d^2\right )}{3 e^3}+\frac{c (d+e x)^5}{5 e^3}-\frac{c d (d+e x)^4}{2 e^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^2*(a + c*x^2),x]

[Out]

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

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0083728, size = 53, normalized size = 0.93 $\frac{1}{3} x^3 \left (a e^2+c d^2\right )+a d^2 x+a d e x^2+\frac{1}{2} c d e x^4+\frac{1}{5} c e^2 x^5$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^2*(a + c*x^2),x]

[Out]

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

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Maple [A]  time = 0.042, size = 48, normalized size = 0.8 \begin{align*}{\frac{c{e}^{2}{x}^{5}}{5}}+{\frac{cde{x}^{4}}{2}}+{\frac{ \left ( a{e}^{2}+c{d}^{2} \right ){x}^{3}}{3}}+ade{x}^{2}+a{d}^{2}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c*e^2*x^5+1/2*c*d*e*x^4+1/3*(a*e^2+c*d^2)*x^3+a*d*e*x^2+a*d^2*x

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Maxima [A]  time = 1.29003, size = 63, normalized size = 1.11 \begin{align*} \frac{1}{5} \, c e^{2} x^{5} + \frac{1}{2} \, c d e x^{4} + a d e x^{2} + a d^{2} x + \frac{1}{3} \,{\left (c d^{2} + a e^{2}\right )} x^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c*e^2*x^5 + 1/2*c*d*e*x^4 + a*d*e*x^2 + a*d^2*x + 1/3*(c*d^2 + a*e^2)*x^3

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Fricas [A]  time = 1.71241, size = 115, normalized size = 2.02 \begin{align*} \frac{1}{5} x^{5} e^{2} c + \frac{1}{2} x^{4} e d c + \frac{1}{3} x^{3} d^{2} c + \frac{1}{3} x^{3} e^{2} a + x^{2} e d a + x d^{2} a \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.109106, size = 51, normalized size = 0.89 \begin{align*} a d^{2} x + a d e x^{2} + \frac{c d e x^{4}}{2} + \frac{c e^{2} x^{5}}{5} + x^{3} \left (\frac{a e^{2}}{3} + \frac{c d^{2}}{3}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.25979, size = 66, normalized size = 1.16 \begin{align*} \frac{1}{5} \, c x^{5} e^{2} + \frac{1}{2} \, c d x^{4} e + \frac{1}{3} \, c d^{2} x^{3} + \frac{1}{3} \, a x^{3} e^{2} + a d x^{2} e + a d^{2} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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