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

Optimal. Leaf size=75 \[ \frac{\left (d+e x^2\right )^3 \left (a e^2-b d e+c d^2\right )}{6 e^3}-\frac{\left (d+e x^2\right )^4 (2 c d-b e)}{8 e^3}+\frac{c \left (d+e x^2\right )^5}{10 e^3} \]

[Out]

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

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Rubi [A]  time = 0.131203, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {1247, 698} \[ \frac{\left (d+e x^2\right )^3 \left (a e^2-b d e+c d^2\right )}{6 e^3}-\frac{\left (d+e x^2\right )^4 (2 c d-b e)}{8 e^3}+\frac{c \left (d+e x^2\right )^5}{10 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(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]

Rule 698

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

Rubi steps

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

Mathematica [A]  time = 0.0215497, size = 72, normalized size = 0.96 \[ \frac{1}{120} x^2 \left (20 x^4 \left (e (a e+2 b d)+c d^2\right )+30 d x^2 (2 a e+b d)+60 a d^2+15 e x^6 (b e+2 c d)+12 c e^2 x^8\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(60*a*d^2 + 30*d*(b*d + 2*a*e)*x^2 + 20*(c*d^2 + e*(2*b*d + a*e))*x^4 + 15*e*(2*c*d + b*e)*x^6 + 12*c*e^2
*x^8))/120

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Maple [A]  time = 0., size = 73, normalized size = 1. \begin{align*}{\frac{c{e}^{2}{x}^{10}}{10}}+{\frac{ \left ({e}^{2}b+2\,dec \right ){x}^{8}}{8}}+{\frac{ \left ( a{e}^{2}+2\,deb+c{d}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 2\,dea+{d}^{2}b \right ){x}^{4}}{4}}+{\frac{a{d}^{2}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.931862, size = 97, normalized size = 1.29 \begin{align*} \frac{1}{10} \, c e^{2} x^{10} + \frac{1}{8} \,{\left (2 \, c d e + b e^{2}\right )} x^{8} + \frac{1}{6} \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{6} + \frac{1}{2} \, a d^{2} x^{2} + \frac{1}{4} \,{\left (b d^{2} + 2 \, a d e\right )} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.48674, size = 196, normalized size = 2.61 \begin{align*} \frac{1}{10} x^{10} e^{2} c + \frac{1}{4} x^{8} e d c + \frac{1}{8} x^{8} e^{2} b + \frac{1}{6} x^{6} d^{2} c + \frac{1}{3} x^{6} e d b + \frac{1}{6} x^{6} e^{2} a + \frac{1}{4} x^{4} d^{2} b + \frac{1}{2} x^{4} e d a + \frac{1}{2} x^{2} d^{2} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.074472, size = 76, normalized size = 1.01 \begin{align*} \frac{a d^{2} x^{2}}{2} + \frac{c e^{2} x^{10}}{10} + x^{8} \left (\frac{b e^{2}}{8} + \frac{c d e}{4}\right ) + x^{6} \left (\frac{a e^{2}}{6} + \frac{b d e}{3} + \frac{c d^{2}}{6}\right ) + x^{4} \left (\frac{a d e}{2} + \frac{b d^{2}}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.07203, size = 107, normalized size = 1.43 \begin{align*} \frac{1}{10} \, c x^{10} e^{2} + \frac{1}{4} \, c d x^{8} e + \frac{1}{8} \, b x^{8} e^{2} + \frac{1}{6} \, c d^{2} x^{6} + \frac{1}{3} \, b d x^{6} e + \frac{1}{6} \, a x^{6} e^{2} + \frac{1}{4} \, b d^{2} x^{4} + \frac{1}{2} \, a d x^{4} e + \frac{1}{2} \, a d^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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