### 3.493 $$\int (d+e x)^2 (a+c x^2)^4 \, dx$$

Optimal. Leaf size=132 $\frac{2}{5} a^2 c x^5 \left (2 a e^2+3 c d^2\right )+\frac{1}{3} a^3 x^3 \left (a e^2+4 c d^2\right )+a^4 d^2 x+\frac{1}{9} c^3 x^9 \left (4 a e^2+c d^2\right )+\frac{2}{7} a c^2 x^7 \left (3 a e^2+2 c d^2\right )+\frac{d e \left (a+c x^2\right )^5}{5 c}+\frac{1}{11} c^4 e^2 x^{11}$

[Out]

a^4*d^2*x + (a^3*(4*c*d^2 + a*e^2)*x^3)/3 + (2*a^2*c*(3*c*d^2 + 2*a*e^2)*x^5)/5 + (2*a*c^2*(2*c*d^2 + 3*a*e^2)
*x^7)/7 + (c^3*(c*d^2 + 4*a*e^2)*x^9)/9 + (c^4*e^2*x^11)/11 + (d*e*(a + c*x^2)^5)/(5*c)

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Rubi [A]  time = 0.0951265, antiderivative size = 132, 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} $\frac{2}{5} a^2 c x^5 \left (2 a e^2+3 c d^2\right )+\frac{1}{3} a^3 x^3 \left (a e^2+4 c d^2\right )+a^4 d^2 x+\frac{1}{9} c^3 x^9 \left (4 a e^2+c d^2\right )+\frac{2}{7} a c^2 x^7 \left (3 a e^2+2 c d^2\right )+\frac{d e \left (a+c x^2\right )^5}{5 c}+\frac{1}{11} c^4 e^2 x^{11}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^2*(a + c*x^2)^4,x]

[Out]

a^4*d^2*x + (a^3*(4*c*d^2 + a*e^2)*x^3)/3 + (2*a^2*c*(3*c*d^2 + 2*a*e^2)*x^5)/5 + (2*a*c^2*(2*c*d^2 + 3*a*e^2)
*x^7)/7 + (c^3*(c*d^2 + 4*a*e^2)*x^9)/9 + (c^4*e^2*x^11)/11 + (d*e*(a + c*x^2)^5)/(5*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)^2 \left (a+c x^2\right )^4 \, dx &=\frac{d e \left (a+c x^2\right )^5}{5 c}+\int \left (a+c x^2\right )^4 \left (-2 d e x+(d+e x)^2\right ) \, dx\\ &=\frac{d e \left (a+c x^2\right )^5}{5 c}+\int \left (a^4 d^2+a^3 \left (4 c d^2+a e^2\right ) x^2+2 a^2 c \left (3 c d^2+2 a e^2\right ) x^4+2 a c^2 \left (2 c d^2+3 a e^2\right ) x^6+c^3 \left (c d^2+4 a e^2\right ) x^8+c^4 e^2 x^{10}\right ) \, dx\\ &=a^4 d^2 x+\frac{1}{3} a^3 \left (4 c d^2+a e^2\right ) x^3+\frac{2}{5} a^2 c \left (3 c d^2+2 a e^2\right ) x^5+\frac{2}{7} a c^2 \left (2 c d^2+3 a e^2\right ) x^7+\frac{1}{9} c^3 \left (c d^2+4 a e^2\right ) x^9+\frac{1}{11} c^4 e^2 x^{11}+\frac{d e \left (a+c x^2\right )^5}{5 c}\\ \end{align*}

Mathematica [A]  time = 0.0300655, size = 148, normalized size = 1.12 $\frac{2}{35} a^2 c^2 x^5 \left (21 d^2+35 d e x+15 e^2 x^2\right )+\frac{2}{15} a^3 c x^3 \left (10 d^2+15 d e x+6 e^2 x^2\right )+a^4 \left (d^2 x+d e x^2+\frac{e^2 x^3}{3}\right )+\frac{1}{63} a c^3 x^7 \left (36 d^2+63 d e x+28 e^2 x^2\right )+\frac{1}{495} c^4 x^9 \left (55 d^2+99 d e x+45 e^2 x^2\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^2*(a + c*x^2)^4,x]

[Out]

(2*a^3*c*x^3*(10*d^2 + 15*d*e*x + 6*e^2*x^2))/15 + (2*a^2*c^2*x^5*(21*d^2 + 35*d*e*x + 15*e^2*x^2))/35 + (a*c^
3*x^7*(36*d^2 + 63*d*e*x + 28*e^2*x^2))/63 + (c^4*x^9*(55*d^2 + 99*d*e*x + 45*e^2*x^2))/495 + a^4*(d^2*x + d*e
*x^2 + (e^2*x^3)/3)

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Maple [A]  time = 0.045, size = 170, normalized size = 1.3 \begin{align*}{\frac{{c}^{4}{e}^{2}{x}^{11}}{11}}+{\frac{de{c}^{4}{x}^{10}}{5}}+{\frac{ \left ( 4\,{e}^{2}a{c}^{3}+{d}^{2}{c}^{4} \right ){x}^{9}}{9}}+dea{c}^{3}{x}^{8}+{\frac{ \left ( 6\,{e}^{2}{a}^{2}{c}^{2}+4\,{d}^{2}a{c}^{3} \right ){x}^{7}}{7}}+2\,de{a}^{2}{c}^{2}{x}^{6}+{\frac{ \left ( 4\,{e}^{2}{a}^{3}c+6\,{d}^{2}{a}^{2}{c}^{2} \right ){x}^{5}}{5}}+2\,de{a}^{3}c{x}^{4}+{\frac{ \left ({e}^{2}{a}^{4}+4\,{d}^{2}{a}^{3}c \right ){x}^{3}}{3}}+de{a}^{4}{x}^{2}+{a}^{4}{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)^4,x)

[Out]

1/11*c^4*e^2*x^11+1/5*d*e*c^4*x^10+1/9*(4*a*c^3*e^2+c^4*d^2)*x^9+d*e*a*c^3*x^8+1/7*(6*a^2*c^2*e^2+4*a*c^3*d^2)
*x^7+2*d*e*a^2*c^2*x^6+1/5*(4*a^3*c*e^2+6*a^2*c^2*d^2)*x^5+2*d*e*a^3*c*x^4+1/3*(a^4*e^2+4*a^3*c*d^2)*x^3+d*e*a
^4*x^2+a^4*d^2*x

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Maxima [A]  time = 1.16173, size = 228, normalized size = 1.73 \begin{align*} \frac{1}{11} \, c^{4} e^{2} x^{11} + \frac{1}{5} \, c^{4} d e x^{10} + a c^{3} d e x^{8} + 2 \, a^{2} c^{2} d e x^{6} + 2 \, a^{3} c d e x^{4} + \frac{1}{9} \,{\left (c^{4} d^{2} + 4 \, a c^{3} e^{2}\right )} x^{9} + a^{4} d e x^{2} + \frac{2}{7} \,{\left (2 \, a c^{3} d^{2} + 3 \, a^{2} c^{2} e^{2}\right )} x^{7} + a^{4} d^{2} x + \frac{2}{5} \,{\left (3 \, a^{2} c^{2} d^{2} + 2 \, a^{3} c e^{2}\right )} x^{5} + \frac{1}{3} \,{\left (4 \, a^{3} c d^{2} + a^{4} 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)^4,x, algorithm="maxima")

[Out]

1/11*c^4*e^2*x^11 + 1/5*c^4*d*e*x^10 + a*c^3*d*e*x^8 + 2*a^2*c^2*d*e*x^6 + 2*a^3*c*d*e*x^4 + 1/9*(c^4*d^2 + 4*
a*c^3*e^2)*x^9 + a^4*d*e*x^2 + 2/7*(2*a*c^3*d^2 + 3*a^2*c^2*e^2)*x^7 + a^4*d^2*x + 2/5*(3*a^2*c^2*d^2 + 2*a^3*
c*e^2)*x^5 + 1/3*(4*a^3*c*d^2 + a^4*e^2)*x^3

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Fricas [A]  time = 1.59751, size = 375, normalized size = 2.84 \begin{align*} \frac{1}{11} x^{11} e^{2} c^{4} + \frac{1}{5} x^{10} e d c^{4} + \frac{1}{9} x^{9} d^{2} c^{4} + \frac{4}{9} x^{9} e^{2} c^{3} a + x^{8} e d c^{3} a + \frac{4}{7} x^{7} d^{2} c^{3} a + \frac{6}{7} x^{7} e^{2} c^{2} a^{2} + 2 x^{6} e d c^{2} a^{2} + \frac{6}{5} x^{5} d^{2} c^{2} a^{2} + \frac{4}{5} x^{5} e^{2} c a^{3} + 2 x^{4} e d c a^{3} + \frac{4}{3} x^{3} d^{2} c a^{3} + \frac{1}{3} x^{3} e^{2} a^{4} + x^{2} e d a^{4} + x d^{2} a^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*x^2+a)^4,x, algorithm="fricas")

[Out]

1/11*x^11*e^2*c^4 + 1/5*x^10*e*d*c^4 + 1/9*x^9*d^2*c^4 + 4/9*x^9*e^2*c^3*a + x^8*e*d*c^3*a + 4/7*x^7*d^2*c^3*a
+ 6/7*x^7*e^2*c^2*a^2 + 2*x^6*e*d*c^2*a^2 + 6/5*x^5*d^2*c^2*a^2 + 4/5*x^5*e^2*c*a^3 + 2*x^4*e*d*c*a^3 + 4/3*x
^3*d^2*c*a^3 + 1/3*x^3*e^2*a^4 + x^2*e*d*a^4 + x*d^2*a^4

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Sympy [A]  time = 0.107205, size = 187, normalized size = 1.42 \begin{align*} a^{4} d^{2} x + a^{4} d e x^{2} + 2 a^{3} c d e x^{4} + 2 a^{2} c^{2} d e x^{6} + a c^{3} d e x^{8} + \frac{c^{4} d e x^{10}}{5} + \frac{c^{4} e^{2} x^{11}}{11} + x^{9} \left (\frac{4 a c^{3} e^{2}}{9} + \frac{c^{4} d^{2}}{9}\right ) + x^{7} \left (\frac{6 a^{2} c^{2} e^{2}}{7} + \frac{4 a c^{3} d^{2}}{7}\right ) + x^{5} \left (\frac{4 a^{3} c e^{2}}{5} + \frac{6 a^{2} c^{2} d^{2}}{5}\right ) + x^{3} \left (\frac{a^{4} e^{2}}{3} + \frac{4 a^{3} 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)**4,x)

[Out]

a**4*d**2*x + a**4*d*e*x**2 + 2*a**3*c*d*e*x**4 + 2*a**2*c**2*d*e*x**6 + a*c**3*d*e*x**8 + c**4*d*e*x**10/5 +
c**4*e**2*x**11/11 + x**9*(4*a*c**3*e**2/9 + c**4*d**2/9) + x**7*(6*a**2*c**2*e**2/7 + 4*a*c**3*d**2/7) + x**5
*(4*a**3*c*e**2/5 + 6*a**2*c**2*d**2/5) + x**3*(a**4*e**2/3 + 4*a**3*c*d**2/3)

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Giac [A]  time = 1.3557, size = 231, normalized size = 1.75 \begin{align*} \frac{1}{11} \, c^{4} x^{11} e^{2} + \frac{1}{5} \, c^{4} d x^{10} e + \frac{1}{9} \, c^{4} d^{2} x^{9} + \frac{4}{9} \, a c^{3} x^{9} e^{2} + a c^{3} d x^{8} e + \frac{4}{7} \, a c^{3} d^{2} x^{7} + \frac{6}{7} \, a^{2} c^{2} x^{7} e^{2} + 2 \, a^{2} c^{2} d x^{6} e + \frac{6}{5} \, a^{2} c^{2} d^{2} x^{5} + \frac{4}{5} \, a^{3} c x^{5} e^{2} + 2 \, a^{3} c d x^{4} e + \frac{4}{3} \, a^{3} c d^{2} x^{3} + \frac{1}{3} \, a^{4} x^{3} e^{2} + a^{4} d x^{2} e + a^{4} 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)^4,x, algorithm="giac")

[Out]

1/11*c^4*x^11*e^2 + 1/5*c^4*d*x^10*e + 1/9*c^4*d^2*x^9 + 4/9*a*c^3*x^9*e^2 + a*c^3*d*x^8*e + 4/7*a*c^3*d^2*x^7
+ 6/7*a^2*c^2*x^7*e^2 + 2*a^2*c^2*d*x^6*e + 6/5*a^2*c^2*d^2*x^5 + 4/5*a^3*c*x^5*e^2 + 2*a^3*c*d*x^4*e + 4/3*a
^3*c*d^2*x^3 + 1/3*a^4*x^3*e^2 + a^4*d*x^2*e + a^4*d^2*x