### 3.461 $$\int (d+e x)^3 (a+c x^2)^2 \, dx$$

Optimal. Leaf size=117 $\frac{c (d+e x)^6 \left (a e^2+3 c d^2\right )}{3 e^5}-\frac{4 c d (d+e x)^5 \left (a e^2+c d^2\right )}{5 e^5}+\frac{(d+e x)^4 \left (a e^2+c d^2\right )^2}{4 e^5}+\frac{c^2 (d+e x)^8}{8 e^5}-\frac{4 c^2 d (d+e x)^7}{7 e^5}$

[Out]

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

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Rubi [A]  time = 0.102017, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.059, Rules used = {697} $\frac{c (d+e x)^6 \left (a e^2+3 c d^2\right )}{3 e^5}-\frac{4 c d (d+e x)^5 \left (a e^2+c d^2\right )}{5 e^5}+\frac{(d+e x)^4 \left (a e^2+c d^2\right )^2}{4 e^5}+\frac{c^2 (d+e x)^8}{8 e^5}-\frac{4 c^2 d (d+e x)^7}{7 e^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^3*(a + c*x^2)^2,x]

[Out]

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

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

Mathematica [A]  time = 0.0170777, size = 117, normalized size = 1. $\frac{1}{4} a^2 x \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+\frac{1}{30} a c x^3 \left (45 d^2 e x+20 d^3+36 d e^2 x^2+10 e^3 x^3\right )+\frac{1}{280} c^2 x^5 \left (140 d^2 e x+56 d^3+120 d e^2 x^2+35 e^3 x^3\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(a^2*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3))/4 + (a*c*x^3*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*
x^3))/30 + (c^2*x^5*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3))/280

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Maple [A]  time = 0.043, size = 131, normalized size = 1.1 \begin{align*}{\frac{{c}^{2}{e}^{3}{x}^{8}}{8}}+{\frac{3\,d{e}^{2}{c}^{2}{x}^{7}}{7}}+{\frac{ \left ( 2\,{e}^{3}ac+3\,{d}^{2}e{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 6\,d{e}^{2}ac+{c}^{2}{d}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ({a}^{2}{e}^{3}+6\,{d}^{2}eac \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,d{e}^{2}{a}^{2}+2\,{d}^{3}ac \right ){x}^{3}}{3}}+{\frac{3\,{d}^{2}e{a}^{2}{x}^{2}}{2}}+{d}^{3}{a}^{2}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.18425, size = 176, normalized size = 1.5 \begin{align*} \frac{1}{8} \, c^{2} e^{3} x^{8} + \frac{3}{7} \, c^{2} d e^{2} x^{7} + \frac{3}{2} \, a^{2} d^{2} e x^{2} + \frac{1}{6} \,{\left (3 \, c^{2} d^{2} e + 2 \, a c e^{3}\right )} x^{6} + a^{2} d^{3} x + \frac{1}{5} \,{\left (c^{2} d^{3} + 6 \, a c d e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (6 \, a c d^{2} e + a^{2} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (2 \, a c d^{3} + 3 \, a^{2} 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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.73849, size = 293, normalized size = 2.5 \begin{align*} \frac{1}{8} x^{8} e^{3} c^{2} + \frac{3}{7} x^{7} e^{2} d c^{2} + \frac{1}{2} x^{6} e d^{2} c^{2} + \frac{1}{3} x^{6} e^{3} c a + \frac{1}{5} x^{5} d^{3} c^{2} + \frac{6}{5} x^{5} e^{2} d c a + \frac{3}{2} x^{4} e d^{2} c a + \frac{1}{4} x^{4} e^{3} a^{2} + \frac{2}{3} x^{3} d^{3} c a + x^{3} e^{2} d a^{2} + \frac{3}{2} x^{2} e d^{2} a^{2} + x d^{3} a^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.119764, size = 141, normalized size = 1.21 \begin{align*} a^{2} d^{3} x + \frac{3 a^{2} d^{2} e x^{2}}{2} + \frac{3 c^{2} d e^{2} x^{7}}{7} + \frac{c^{2} e^{3} x^{8}}{8} + x^{6} \left (\frac{a c e^{3}}{3} + \frac{c^{2} d^{2} e}{2}\right ) + x^{5} \left (\frac{6 a c d e^{2}}{5} + \frac{c^{2} d^{3}}{5}\right ) + x^{4} \left (\frac{a^{2} e^{3}}{4} + \frac{3 a c d^{2} e}{2}\right ) + x^{3} \left (a^{2} d e^{2} + \frac{2 a 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)**2,x)

[Out]

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

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Giac [A]  time = 1.37294, size = 173, normalized size = 1.48 \begin{align*} \frac{1}{8} \, c^{2} x^{8} e^{3} + \frac{3}{7} \, c^{2} d x^{7} e^{2} + \frac{1}{2} \, c^{2} d^{2} x^{6} e + \frac{1}{5} \, c^{2} d^{3} x^{5} + \frac{1}{3} \, a c x^{6} e^{3} + \frac{6}{5} \, a c d x^{5} e^{2} + \frac{3}{2} \, a c d^{2} x^{4} e + \frac{2}{3} \, a c d^{3} x^{3} + \frac{1}{4} \, a^{2} x^{4} e^{3} + a^{2} d x^{3} e^{2} + \frac{3}{2} \, a^{2} d^{2} x^{2} e + a^{2} 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)^2,x, algorithm="giac")

[Out]

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