### 3.475 $$\int (d+e x)^3 (a+c x^2)^3 \, dx$$

Optimal. Leaf size=161 $a^2 d x^3 \left (a e^2+c d^2\right )+\frac{1}{2} a^2 c e^3 x^6+a^3 d^3 x+\frac{1}{4} a^3 e^3 x^4+\frac{1}{7} c^2 d x^7 \left (9 a e^2+c d^2\right )+\frac{3}{8} a c^2 e^3 x^8+\frac{3}{5} a c d x^5 \left (3 a e^2+c d^2\right )+\frac{3 d^2 e \left (a+c x^2\right )^4}{8 c}+\frac{1}{3} c^3 d e^2 x^9+\frac{1}{10} c^3 e^3 x^{10}$

[Out]

a^3*d^3*x + a^2*d*(c*d^2 + a*e^2)*x^3 + (a^3*e^3*x^4)/4 + (3*a*c*d*(c*d^2 + 3*a*e^2)*x^5)/5 + (a^2*c*e^3*x^6)/
2 + (c^2*d*(c*d^2 + 9*a*e^2)*x^7)/7 + (3*a*c^2*e^3*x^8)/8 + (c^3*d*e^2*x^9)/3 + (c^3*e^3*x^10)/10 + (3*d^2*e*(
a + c*x^2)^4)/(8*c)

________________________________________________________________________________________

Rubi [A]  time = 0.140378, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.118, Rules used = {696, 1810} $a^2 d x^3 \left (a e^2+c d^2\right )+\frac{1}{2} a^2 c e^3 x^6+a^3 d^3 x+\frac{1}{4} a^3 e^3 x^4+\frac{1}{7} c^2 d x^7 \left (9 a e^2+c d^2\right )+\frac{3}{8} a c^2 e^3 x^8+\frac{3}{5} a c d x^5 \left (3 a e^2+c d^2\right )+\frac{3 d^2 e \left (a+c x^2\right )^4}{8 c}+\frac{1}{3} c^3 d e^2 x^9+\frac{1}{10} c^3 e^3 x^{10}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^3*(a + c*x^2)^3,x]

[Out]

a^3*d^3*x + a^2*d*(c*d^2 + a*e^2)*x^3 + (a^3*e^3*x^4)/4 + (3*a*c*d*(c*d^2 + 3*a*e^2)*x^5)/5 + (a^2*c*e^3*x^6)/
2 + (c^2*d*(c*d^2 + 9*a*e^2)*x^7)/7 + (3*a*c^2*e^3*x^8)/8 + (c^3*d*e^2*x^9)/3 + (c^3*e^3*x^10)/10 + (3*d^2*e*(
a + c*x^2)^4)/(8*c)

Rule 696

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*m*d^(m - 1)*(a + c*x^2)^(p + 1))
/(2*c*(p + 1)), x] + Int[((d + e*x)^m - e*m*d^(m - 1)*x)*(a + c*x^2)^p, x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*
d^2 + a*e^2, 0] && IGtQ[p, 1] && IGtQ[m, 0] && LeQ[m, p]

Rule 1810

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin{align*} \int (d+e x)^3 \left (a+c x^2\right )^3 \, dx &=\frac{3 d^2 e \left (a+c x^2\right )^4}{8 c}+\int \left (a+c x^2\right )^3 \left (-3 d^2 e x+(d+e x)^3\right ) \, dx\\ &=\frac{3 d^2 e \left (a+c x^2\right )^4}{8 c}+\int \left (a^3 d^3+3 a^2 d \left (c d^2+a e^2\right ) x^2+a^3 e^3 x^3+3 a c d \left (c d^2+3 a e^2\right ) x^4+3 a^2 c e^3 x^5+c^2 d \left (c d^2+9 a e^2\right ) x^6+3 a c^2 e^3 x^7+3 c^3 d e^2 x^8+c^3 e^3 x^9\right ) \, dx\\ &=a^3 d^3 x+a^2 d \left (c d^2+a e^2\right ) x^3+\frac{1}{4} a^3 e^3 x^4+\frac{3}{5} a c d \left (c d^2+3 a e^2\right ) x^5+\frac{1}{2} a^2 c e^3 x^6+\frac{1}{7} c^2 d \left (c d^2+9 a e^2\right ) x^7+\frac{3}{8} a c^2 e^3 x^8+\frac{1}{3} c^3 d e^2 x^9+\frac{1}{10} c^3 e^3 x^{10}+\frac{3 d^2 e \left (a+c x^2\right )^4}{8 c}\\ \end{align*}

Mathematica [A]  time = 0.0475745, size = 155, normalized size = 0.96 $\frac{1}{840} x \left (42 a^2 c x^2 \left (45 d^2 e x+20 d^3+36 d e^2 x^2+10 e^3 x^3\right )+210 a^3 \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+9 a c^2 x^4 \left (140 d^2 e x+56 d^3+120 d e^2 x^2+35 e^3 x^3\right )+c^3 x^6 \left (315 d^2 e x+120 d^3+280 d e^2 x^2+84 e^3 x^3\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x*(210*a^3*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 42*a^2*c*x^2*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 1
0*e^3*x^3) + 9*a*c^2*x^4*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3) + c^3*x^6*(120*d^3 + 315*d^2*e*x
+ 280*d*e^2*x^2 + 84*e^3*x^3)))/840

________________________________________________________________________________________

Maple [A]  time = 0.039, size = 189, normalized size = 1.2 \begin{align*}{\frac{{c}^{3}{e}^{3}{x}^{10}}{10}}+{\frac{{c}^{3}d{e}^{2}{x}^{9}}{3}}+{\frac{ \left ( 3\,{e}^{3}a{c}^{2}+3\,{d}^{2}e{c}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 9\,d{e}^{2}a{c}^{2}+{c}^{3}{d}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{a}^{2}c{e}^{3}+9\,{d}^{2}ea{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 9\,d{e}^{2}{a}^{2}c+3\,{d}^{3}a{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ({e}^{3}{a}^{3}+9\,{d}^{2}e{a}^{2}c \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,d{e}^{2}{a}^{3}+3\,{d}^{3}{a}^{2}c \right ){x}^{3}}{3}}+{\frac{3\,{d}^{2}e{a}^{3}{x}^{2}}{2}}+{a}^{3}{d}^{3}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*c^3*e^3*x^10+1/3*c^3*d*e^2*x^9+1/8*(3*a*c^2*e^3+3*c^3*d^2*e)*x^8+1/7*(9*a*c^2*d*e^2+c^3*d^3)*x^7+1/6*(3*a
^2*c*e^3+9*a*c^2*d^2*e)*x^6+1/5*(9*a^2*c*d*e^2+3*a*c^2*d^3)*x^5+1/4*(a^3*e^3+9*a^2*c*d^2*e)*x^4+1/3*(3*a^3*d*e
^2+3*a^2*c*d^3)*x^3+3/2*d^2*e*a^3*x^2+a^3*d^3*x

________________________________________________________________________________________

Maxima [A]  time = 1.16501, size = 244, normalized size = 1.52 \begin{align*} \frac{1}{10} \, c^{3} e^{3} x^{10} + \frac{1}{3} \, c^{3} d e^{2} x^{9} + \frac{3}{8} \,{\left (c^{3} d^{2} e + a c^{2} e^{3}\right )} x^{8} + \frac{3}{2} \, a^{3} d^{2} e x^{2} + \frac{1}{7} \,{\left (c^{3} d^{3} + 9 \, a c^{2} d e^{2}\right )} x^{7} + a^{3} d^{3} x + \frac{1}{2} \,{\left (3 \, a c^{2} d^{2} e + a^{2} c e^{3}\right )} x^{6} + \frac{3}{5} \,{\left (a c^{2} d^{3} + 3 \, a^{2} c d e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (9 \, a^{2} c d^{2} e + a^{3} e^{3}\right )} x^{4} +{\left (a^{2} c d^{3} + a^{3} 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)^3,x, algorithm="maxima")

[Out]

1/10*c^3*e^3*x^10 + 1/3*c^3*d*e^2*x^9 + 3/8*(c^3*d^2*e + a*c^2*e^3)*x^8 + 3/2*a^3*d^2*e*x^2 + 1/7*(c^3*d^3 + 9
*a*c^2*d*e^2)*x^7 + a^3*d^3*x + 1/2*(3*a*c^2*d^2*e + a^2*c*e^3)*x^6 + 3/5*(a*c^2*d^3 + 3*a^2*c*d*e^2)*x^5 + 1/
4*(9*a^2*c*d^2*e + a^3*e^3)*x^4 + (a^2*c*d^3 + a^3*d*e^2)*x^3

________________________________________________________________________________________

Fricas [A]  time = 1.70597, size = 414, normalized size = 2.57 \begin{align*} \frac{1}{10} x^{10} e^{3} c^{3} + \frac{1}{3} x^{9} e^{2} d c^{3} + \frac{3}{8} x^{8} e d^{2} c^{3} + \frac{3}{8} x^{8} e^{3} c^{2} a + \frac{1}{7} x^{7} d^{3} c^{3} + \frac{9}{7} x^{7} e^{2} d c^{2} a + \frac{3}{2} x^{6} e d^{2} c^{2} a + \frac{1}{2} x^{6} e^{3} c a^{2} + \frac{3}{5} x^{5} d^{3} c^{2} a + \frac{9}{5} x^{5} e^{2} d c a^{2} + \frac{9}{4} x^{4} e d^{2} c a^{2} + \frac{1}{4} x^{4} e^{3} a^{3} + x^{3} d^{3} c a^{2} + x^{3} e^{2} d a^{3} + \frac{3}{2} x^{2} e d^{2} a^{3} + x d^{3} a^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*x^10*e^3*c^3 + 1/3*x^9*e^2*d*c^3 + 3/8*x^8*e*d^2*c^3 + 3/8*x^8*e^3*c^2*a + 1/7*x^7*d^3*c^3 + 9/7*x^7*e^2*
d*c^2*a + 3/2*x^6*e*d^2*c^2*a + 1/2*x^6*e^3*c*a^2 + 3/5*x^5*d^3*c^2*a + 9/5*x^5*e^2*d*c*a^2 + 9/4*x^4*e*d^2*c*
a^2 + 1/4*x^4*e^3*a^3 + x^3*d^3*c*a^2 + x^3*e^2*d*a^3 + 3/2*x^2*e*d^2*a^3 + x*d^3*a^3

________________________________________________________________________________________

Sympy [A]  time = 0.122793, size = 202, normalized size = 1.25 \begin{align*} a^{3} d^{3} x + \frac{3 a^{3} d^{2} e x^{2}}{2} + \frac{c^{3} d e^{2} x^{9}}{3} + \frac{c^{3} e^{3} x^{10}}{10} + x^{8} \left (\frac{3 a c^{2} e^{3}}{8} + \frac{3 c^{3} d^{2} e}{8}\right ) + x^{7} \left (\frac{9 a c^{2} d e^{2}}{7} + \frac{c^{3} d^{3}}{7}\right ) + x^{6} \left (\frac{a^{2} c e^{3}}{2} + \frac{3 a c^{2} d^{2} e}{2}\right ) + x^{5} \left (\frac{9 a^{2} c d e^{2}}{5} + \frac{3 a c^{2} d^{3}}{5}\right ) + x^{4} \left (\frac{a^{3} e^{3}}{4} + \frac{9 a^{2} c d^{2} e}{4}\right ) + x^{3} \left (a^{3} d e^{2} + a^{2} c d^{3}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*d**3*x + 3*a**3*d**2*e*x**2/2 + c**3*d*e**2*x**9/3 + c**3*e**3*x**10/10 + x**8*(3*a*c**2*e**3/8 + 3*c**3*
d**2*e/8) + x**7*(9*a*c**2*d*e**2/7 + c**3*d**3/7) + x**6*(a**2*c*e**3/2 + 3*a*c**2*d**2*e/2) + x**5*(9*a**2*c
*d*e**2/5 + 3*a*c**2*d**3/5) + x**4*(a**3*e**3/4 + 9*a**2*c*d**2*e/4) + x**3*(a**3*d*e**2 + a**2*c*d**3)

________________________________________________________________________________________

Giac [A]  time = 1.31344, size = 248, normalized size = 1.54 \begin{align*} \frac{1}{10} \, c^{3} x^{10} e^{3} + \frac{1}{3} \, c^{3} d x^{9} e^{2} + \frac{3}{8} \, c^{3} d^{2} x^{8} e + \frac{1}{7} \, c^{3} d^{3} x^{7} + \frac{3}{8} \, a c^{2} x^{8} e^{3} + \frac{9}{7} \, a c^{2} d x^{7} e^{2} + \frac{3}{2} \, a c^{2} d^{2} x^{6} e + \frac{3}{5} \, a c^{2} d^{3} x^{5} + \frac{1}{2} \, a^{2} c x^{6} e^{3} + \frac{9}{5} \, a^{2} c d x^{5} e^{2} + \frac{9}{4} \, a^{2} c d^{2} x^{4} e + a^{2} c d^{3} x^{3} + \frac{1}{4} \, a^{3} x^{4} e^{3} + a^{3} d x^{3} e^{2} + \frac{3}{2} \, a^{3} d^{2} x^{2} e + a^{3} 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)^3,x, algorithm="giac")

[Out]

1/10*c^3*x^10*e^3 + 1/3*c^3*d*x^9*e^2 + 3/8*c^3*d^2*x^8*e + 1/7*c^3*d^3*x^7 + 3/8*a*c^2*x^8*e^3 + 9/7*a*c^2*d*
x^7*e^2 + 3/2*a*c^2*d^2*x^6*e + 3/5*a*c^2*d^3*x^5 + 1/2*a^2*c*x^6*e^3 + 9/5*a^2*c*d*x^5*e^2 + 9/4*a^2*c*d^2*x^
4*e + a^2*c*d^3*x^3 + 1/4*a^3*x^4*e^3 + a^3*d*x^3*e^2 + 3/2*a^3*d^2*x^2*e + a^3*d^3*x