3.452 $$\int (d+e x)^3 (a+c x^2) \, dx$$

Optimal. Leaf size=57 $\frac{(d+e x)^4 \left (a e^2+c d^2\right )}{4 e^3}+\frac{c (d+e x)^6}{6 e^3}-\frac{2 c d (d+e x)^5}{5 e^3}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0455637, 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} $\frac{(d+e x)^4 \left (a e^2+c d^2\right )}{4 e^3}+\frac{c (d+e x)^6}{6 e^3}-\frac{2 c d (d+e x)^5}{5 e^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^3*(a + c*x^2),x]

[Out]

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

Mathematica [A]  time = 0.013662, size = 74, normalized size = 1.3 $\frac{1}{4} a x \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+\frac{1}{60} c x^3 \left (45 d^2 e x+20 d^3+36 d e^2 x^2+10 e^3 x^3\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^3*(a + c*x^2),x]

[Out]

(a*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3))/4 + (c*x^3*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3)
)/60

________________________________________________________________________________________

Maple [A]  time = 0.04, size = 73, normalized size = 1.3 \begin{align*}{\frac{{e}^{3}c{x}^{6}}{6}}+{\frac{3\,d{e}^{2}c{x}^{5}}{5}}+{\frac{ \left ({e}^{3}a+3\,{d}^{2}ec \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,d{e}^{2}a+{d}^{3}c \right ){x}^{3}}{3}}+{\frac{3\,{d}^{2}ea{x}^{2}}{2}}+{d}^{3}ax \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*e^3*c*x^6+3/5*d*e^2*c*x^5+1/4*(a*e^3+3*c*d^2*e)*x^4+1/3*(3*a*d*e^2+c*d^3)*x^3+3/2*d^2*e*a*x^2+d^3*a*x

________________________________________________________________________________________

Maxima [A]  time = 1.15385, size = 97, normalized size = 1.7 \begin{align*} \frac{1}{6} \, c e^{3} x^{6} + \frac{3}{5} \, c d e^{2} x^{5} + \frac{3}{2} \, a d^{2} e x^{2} + a d^{3} x + \frac{1}{4} \,{\left (3 \, c d^{2} e + a e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (c d^{3} + 3 \, a d e^{2}\right )} x^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c*e^3*x^6 + 3/5*c*d*e^2*x^5 + 3/2*a*d^2*e*x^2 + a*d^3*x + 1/4*(3*c*d^2*e + a*e^3)*x^4 + 1/3*(c*d^3 + 3*a*d
*e^2)*x^3

________________________________________________________________________________________

Fricas [A]  time = 1.60084, size = 169, normalized size = 2.96 \begin{align*} \frac{1}{6} x^{6} e^{3} c + \frac{3}{5} x^{5} e^{2} d c + \frac{3}{4} x^{4} e d^{2} c + \frac{1}{4} x^{4} e^{3} a + \frac{1}{3} x^{3} d^{3} c + x^{3} e^{2} d a + \frac{3}{2} x^{2} e d^{2} a + x d^{3} a \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.125414, size = 80, normalized size = 1.4 \begin{align*} a d^{3} x + \frac{3 a d^{2} e x^{2}}{2} + \frac{3 c d e^{2} x^{5}}{5} + \frac{c e^{3} x^{6}}{6} + x^{4} \left (\frac{a e^{3}}{4} + \frac{3 c d^{2} e}{4}\right ) + x^{3} \left (a d e^{2} + \frac{c d^{3}}{3}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.31064, size = 96, normalized size = 1.68 \begin{align*} \frac{1}{6} \, c x^{6} e^{3} + \frac{3}{5} \, c d x^{5} e^{2} + \frac{3}{4} \, c d^{2} x^{4} e + \frac{1}{3} \, c d^{3} x^{3} + \frac{1}{4} \, a x^{4} e^{3} + a d x^{3} e^{2} + \frac{3}{2} \, a d^{2} x^{2} e + a d^{3} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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