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

Optimal. Leaf size=104 \[ \frac{1}{3} a^2 x^3 \left (a e^2+3 c d^2\right )+a^3 d^2 x+\frac{1}{7} c^2 x^7 \left (3 a e^2+c d^2\right )+\frac{3}{5} a c x^5 \left (a e^2+c d^2\right )+\frac{d e \left (a+c x^2\right )^4}{4 c}+\frac{1}{9} c^3 e^2 x^9 \]

[Out]

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

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Rubi [A]  time = 0.0630388, antiderivative size = 104, 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} \[ \frac{1}{3} a^2 x^3 \left (a e^2+3 c d^2\right )+a^3 d^2 x+\frac{1}{7} c^2 x^7 \left (3 a e^2+c d^2\right )+\frac{3}{5} a c x^5 \left (a e^2+c d^2\right )+\frac{d e \left (a+c x^2\right )^4}{4 c}+\frac{1}{9} c^3 e^2 x^9 \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*(a + c*x^2)^3,x]

[Out]

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

Mathematica [A]  time = 0.0231168, size = 116, normalized size = 1.12 \[ \frac{1}{10} a^2 c x^3 \left (10 d^2+15 d e x+6 e^2 x^2\right )+a^3 \left (d^2 x+d e x^2+\frac{e^2 x^3}{3}\right )+\frac{1}{35} a c^2 x^5 \left (21 d^2+35 d e x+15 e^2 x^2\right )+\frac{1}{252} c^3 x^7 \left (36 d^2+63 d e x+28 e^2 x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*c*x^3*(10*d^2 + 15*d*e*x + 6*e^2*x^2))/10 + (a*c^2*x^5*(21*d^2 + 35*d*e*x + 15*e^2*x^2))/35 + (c^3*x^7*(3
6*d^2 + 63*d*e*x + 28*e^2*x^2))/252 + a^3*(d^2*x + d*e*x^2 + (e^2*x^3)/3)

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Maple [A]  time = 0.043, size = 129, normalized size = 1.2 \begin{align*}{\frac{{c}^{3}{e}^{2}{x}^{9}}{9}}+{\frac{de{c}^{3}{x}^{8}}{4}}+{\frac{ \left ( 3\,{e}^{2}a{c}^{2}+{d}^{2}{c}^{3} \right ){x}^{7}}{7}}+a{c}^{2}de{x}^{6}+{\frac{ \left ( 3\,{e}^{2}{a}^{2}c+3\,{d}^{2}a{c}^{2} \right ){x}^{5}}{5}}+{\frac{3\,de{a}^{2}c{x}^{4}}{2}}+{\frac{ \left ({e}^{2}{a}^{3}+3\,{a}^{2}c{d}^{2} \right ){x}^{3}}{3}}+de{a}^{3}{x}^{2}+{a}^{3}{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)^3,x)

[Out]

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

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Maxima [A]  time = 1.14436, size = 170, normalized size = 1.63 \begin{align*} \frac{1}{9} \, c^{3} e^{2} x^{9} + \frac{1}{4} \, c^{3} d e x^{8} + a c^{2} d e x^{6} + \frac{3}{2} \, a^{2} c d e x^{4} + \frac{1}{7} \,{\left (c^{3} d^{2} + 3 \, a c^{2} e^{2}\right )} x^{7} + a^{3} d e x^{2} + a^{3} d^{2} x + \frac{3}{5} \,{\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} x^{5} + \frac{1}{3} \,{\left (3 \, a^{2} c d^{2} + a^{3} 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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.60838, size = 282, normalized size = 2.71 \begin{align*} \frac{1}{9} x^{9} e^{2} c^{3} + \frac{1}{4} x^{8} e d c^{3} + \frac{1}{7} x^{7} d^{2} c^{3} + \frac{3}{7} x^{7} e^{2} c^{2} a + x^{6} e d c^{2} a + \frac{3}{5} x^{5} d^{2} c^{2} a + \frac{3}{5} x^{5} e^{2} c a^{2} + \frac{3}{2} x^{4} e d c a^{2} + x^{3} d^{2} c a^{2} + \frac{1}{3} x^{3} e^{2} a^{3} + x^{2} e d a^{3} + x d^{2} a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.09911, size = 139, normalized size = 1.34 \begin{align*} a^{3} d^{2} x + a^{3} d e x^{2} + \frac{3 a^{2} c d e x^{4}}{2} + a c^{2} d e x^{6} + \frac{c^{3} d e x^{8}}{4} + \frac{c^{3} e^{2} x^{9}}{9} + x^{7} \left (\frac{3 a c^{2} e^{2}}{7} + \frac{c^{3} d^{2}}{7}\right ) + x^{5} \left (\frac{3 a^{2} c e^{2}}{5} + \frac{3 a c^{2} d^{2}}{5}\right ) + x^{3} \left (\frac{a^{3} e^{2}}{3} + a^{2} c d^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.28823, size = 174, normalized size = 1.67 \begin{align*} \frac{1}{9} \, c^{3} x^{9} e^{2} + \frac{1}{4} \, c^{3} d x^{8} e + \frac{1}{7} \, c^{3} d^{2} x^{7} + \frac{3}{7} \, a c^{2} x^{7} e^{2} + a c^{2} d x^{6} e + \frac{3}{5} \, a c^{2} d^{2} x^{5} + \frac{3}{5} \, a^{2} c x^{5} e^{2} + \frac{3}{2} \, a^{2} c d x^{4} e + a^{2} c d^{2} x^{3} + \frac{1}{3} \, a^{3} x^{3} e^{2} + a^{3} d x^{2} e + a^{3} 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)^3,x, algorithm="giac")

[Out]

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