3.273 \(\int x^3 (d+e x^2)^2 (a+b x^2+c x^4) \, dx\)

Optimal. Leaf size=78 \[ \frac{1}{8} x^8 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{6} d x^6 (2 a e+b d)+\frac{1}{4} a d^2 x^4+\frac{1}{10} e x^{10} (b e+2 c d)+\frac{1}{12} c e^2 x^{12} \]

[Out]

(a*d^2*x^4)/4 + (d*(b*d + 2*a*e)*x^6)/6 + ((c*d^2 + e*(2*b*d + a*e))*x^8)/8 + (e*(2*c*d + b*e)*x^10)/10 + (c*e
^2*x^12)/12

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Rubi [A]  time = 0.13454, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {1251, 771} \[ \frac{1}{8} x^8 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{6} d x^6 (2 a e+b d)+\frac{1}{4} a d^2 x^4+\frac{1}{10} e x^{10} (b e+2 c d)+\frac{1}{12} c e^2 x^{12} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(d + e*x^2)^2*(a + b*x^2 + c*x^4),x]

[Out]

(a*d^2*x^4)/4 + (d*(b*d + 2*a*e)*x^6)/6 + ((c*d^2 + e*(2*b*d + a*e))*x^8)/8 + (e*(2*c*d + b*e)*x^10)/10 + (c*e
^2*x^12)/12

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int x^3 \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x (d+e x)^2 \left (a+b x+c x^2\right ) \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a d^2 x+d (b d+2 a e) x^2+\left (c d^2+e (2 b d+a e)\right ) x^3+e (2 c d+b e) x^4+c e^2 x^5\right ) \, dx,x,x^2\right )\\ &=\frac{1}{4} a d^2 x^4+\frac{1}{6} d (b d+2 a e) x^6+\frac{1}{8} \left (c d^2+e (2 b d+a e)\right ) x^8+\frac{1}{10} e (2 c d+b e) x^{10}+\frac{1}{12} c e^2 x^{12}\\ \end{align*}

Mathematica [A]  time = 0.0256374, size = 72, normalized size = 0.92 \[ \frac{1}{120} x^4 \left (15 x^4 \left (e (a e+2 b d)+c d^2\right )+20 d x^2 (2 a e+b d)+30 a d^2+12 e x^6 (b e+2 c d)+10 c e^2 x^8\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(d + e*x^2)^2*(a + b*x^2 + c*x^4),x]

[Out]

(x^4*(30*a*d^2 + 20*d*(b*d + 2*a*e)*x^2 + 15*(c*d^2 + e*(2*b*d + a*e))*x^4 + 12*e*(2*c*d + b*e)*x^6 + 10*c*e^2
*x^8))/120

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Maple [A]  time = 0.001, size = 73, normalized size = 0.9 \begin{align*}{\frac{c{e}^{2}{x}^{12}}{12}}+{\frac{ \left ({e}^{2}b+2\,dec \right ){x}^{10}}{10}}+{\frac{ \left ( a{e}^{2}+2\,deb+c{d}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( 2\,dea+{d}^{2}b \right ){x}^{6}}{6}}+{\frac{a{d}^{2}{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(e*x^2+d)^2*(c*x^4+b*x^2+a),x)

[Out]

1/12*c*e^2*x^12+1/10*(b*e^2+2*c*d*e)*x^10+1/8*(a*e^2+2*b*d*e+c*d^2)*x^8+1/6*(2*a*d*e+b*d^2)*x^6+1/4*a*d^2*x^4

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Maxima [A]  time = 0.955376, size = 97, normalized size = 1.24 \begin{align*} \frac{1}{12} \, c e^{2} x^{12} + \frac{1}{10} \,{\left (2 \, c d e + b e^{2}\right )} x^{10} + \frac{1}{8} \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{8} + \frac{1}{4} \, a d^{2} x^{4} + \frac{1}{6} \,{\left (b d^{2} + 2 \, a d e\right )} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)^2*(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

1/12*c*e^2*x^12 + 1/10*(2*c*d*e + b*e^2)*x^10 + 1/8*(c*d^2 + 2*b*d*e + a*e^2)*x^8 + 1/4*a*d^2*x^4 + 1/6*(b*d^2
+ 2*a*d*e)*x^6

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Fricas [A]  time = 1.50246, size = 200, normalized size = 2.56 \begin{align*} \frac{1}{12} x^{12} e^{2} c + \frac{1}{5} x^{10} e d c + \frac{1}{10} x^{10} e^{2} b + \frac{1}{8} x^{8} d^{2} c + \frac{1}{4} x^{8} e d b + \frac{1}{8} x^{8} e^{2} a + \frac{1}{6} x^{6} d^{2} b + \frac{1}{3} x^{6} e d a + \frac{1}{4} x^{4} d^{2} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)^2*(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

1/12*x^12*e^2*c + 1/5*x^10*e*d*c + 1/10*x^10*e^2*b + 1/8*x^8*d^2*c + 1/4*x^8*e*d*b + 1/8*x^8*e^2*a + 1/6*x^6*d
^2*b + 1/3*x^6*e*d*a + 1/4*x^4*d^2*a

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Sympy [A]  time = 0.074094, size = 76, normalized size = 0.97 \begin{align*} \frac{a d^{2} x^{4}}{4} + \frac{c e^{2} x^{12}}{12} + x^{10} \left (\frac{b e^{2}}{10} + \frac{c d e}{5}\right ) + x^{8} \left (\frac{a e^{2}}{8} + \frac{b d e}{4} + \frac{c d^{2}}{8}\right ) + x^{6} \left (\frac{a d e}{3} + \frac{b d^{2}}{6}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(e*x**2+d)**2*(c*x**4+b*x**2+a),x)

[Out]

a*d**2*x**4/4 + c*e**2*x**12/12 + x**10*(b*e**2/10 + c*d*e/5) + x**8*(a*e**2/8 + b*d*e/4 + c*d**2/8) + x**6*(a
*d*e/3 + b*d**2/6)

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Giac [A]  time = 1.09467, size = 107, normalized size = 1.37 \begin{align*} \frac{1}{12} \, c x^{12} e^{2} + \frac{1}{5} \, c d x^{10} e + \frac{1}{10} \, b x^{10} e^{2} + \frac{1}{8} \, c d^{2} x^{8} + \frac{1}{4} \, b d x^{8} e + \frac{1}{8} \, a x^{8} e^{2} + \frac{1}{6} \, b d^{2} x^{6} + \frac{1}{3} \, a d x^{6} e + \frac{1}{4} \, a d^{2} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)^2*(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

1/12*c*x^12*e^2 + 1/5*c*d*x^10*e + 1/10*b*x^10*e^2 + 1/8*c*d^2*x^8 + 1/4*b*d*x^8*e + 1/8*a*x^8*e^2 + 1/6*b*d^2
*x^6 + 1/3*a*d*x^6*e + 1/4*a*d^2*x^4