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

Optimal. Leaf size=73 \[ \frac{1}{5} x^5 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{3} d x^3 (2 a e+b d)+a d^2 x+\frac{1}{7} e x^7 (b e+2 c d)+\frac{1}{9} c e^2 x^9 \]

[Out]

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

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Rubi [A]  time = 0.044485, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {1153} \[ \frac{1}{5} x^5 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{3} d x^3 (2 a e+b d)+a d^2 x+\frac{1}{7} e x^7 (b e+2 c d)+\frac{1}{9} c e^2 x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1153

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

Rubi steps

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

Mathematica [A]  time = 0.0161226, size = 73, normalized size = 1. \[ \frac{1}{5} x^5 \left (a e^2+2 b d e+c d^2\right )+\frac{1}{3} d x^3 (2 a e+b d)+a d^2 x+\frac{1}{7} e x^7 (b e+2 c d)+\frac{1}{9} c e^2 x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 70, normalized size = 1. \begin{align*}{\frac{c{e}^{2}{x}^{9}}{9}}+{\frac{ \left ({e}^{2}b+2\,dec \right ){x}^{7}}{7}}+{\frac{ \left ( a{e}^{2}+2\,deb+c{d}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,dea+{d}^{2}b \right ){x}^{3}}{3}}+a{d}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.946527, size = 93, normalized size = 1.27 \begin{align*} \frac{1}{9} \, c e^{2} x^{9} + \frac{1}{7} \,{\left (2 \, c d e + b e^{2}\right )} x^{7} + \frac{1}{5} \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{5} + a d^{2} x + \frac{1}{3} \,{\left (b d^{2} + 2 \, a d e\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.52848, size = 185, normalized size = 2.53 \begin{align*} \frac{1}{9} x^{9} e^{2} c + \frac{2}{7} x^{7} e d c + \frac{1}{7} x^{7} e^{2} b + \frac{1}{5} x^{5} d^{2} c + \frac{2}{5} x^{5} e d b + \frac{1}{5} x^{5} e^{2} a + \frac{1}{3} x^{3} d^{2} b + \frac{2}{3} x^{3} e d a + x d^{2} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9*e^2*c + 2/7*x^7*e*d*c + 1/7*x^7*e^2*b + 1/5*x^5*d^2*c + 2/5*x^5*e*d*b + 1/5*x^5*e^2*a + 1/3*x^3*d^2*b
+ 2/3*x^3*e*d*a + x*d^2*a

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Sympy [A]  time = 0.07416, size = 78, normalized size = 1.07 \begin{align*} a d^{2} x + \frac{c e^{2} x^{9}}{9} + x^{7} \left (\frac{b e^{2}}{7} + \frac{2 c d e}{7}\right ) + x^{5} \left (\frac{a e^{2}}{5} + \frac{2 b d e}{5} + \frac{c d^{2}}{5}\right ) + x^{3} \left (\frac{2 a d e}{3} + \frac{b d^{2}}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.11682, size = 103, normalized size = 1.41 \begin{align*} \frac{1}{9} \, c x^{9} e^{2} + \frac{2}{7} \, c d x^{7} e + \frac{1}{7} \, b x^{7} e^{2} + \frac{1}{5} \, c d^{2} x^{5} + \frac{2}{5} \, b d x^{5} e + \frac{1}{5} \, a x^{5} e^{2} + \frac{1}{3} \, b d^{2} x^{3} + \frac{2}{3} \, a d x^{3} e + a d^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*c*x^9*e^2 + 2/7*c*d*x^7*e + 1/7*b*x^7*e^2 + 1/5*c*d^2*x^5 + 2/5*b*d*x^5*e + 1/5*a*x^5*e^2 + 1/3*b*d^2*x^3
+ 2/3*a*d*x^3*e + a*d^2*x