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

Optimal. Leaf size=56 \[ \frac{1}{5} x^5 \left (a e^2+c d^2\right )+a d^2 x+\frac{2}{3} a d e x^3+\frac{2}{7} c d e x^7+\frac{1}{9} c e^2 x^9 \]

[Out]

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

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Rubi [A]  time = 0.0321749, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {1154} \[ \frac{1}{5} x^5 \left (a e^2+c d^2\right )+a d^2 x+\frac{2}{3} a d e x^3+\frac{2}{7} c d e x^7+\frac{1}{9} c e^2 x^9 \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^2*(a + c*x^4),x]

[Out]

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

Rule 1154

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

Rubi steps

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

Mathematica [A]  time = 0.0114073, size = 56, normalized size = 1. \[ \frac{1}{5} x^5 \left (a e^2+c d^2\right )+a d^2 x+\frac{2}{3} a d e x^3+\frac{2}{7} c d e x^7+\frac{1}{9} c e^2 x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.042, size = 49, normalized size = 0.9 \begin{align*} a{d}^{2}x+{\frac{2\,ade{x}^{3}}{3}}+{\frac{ \left ( a{e}^{2}+c{d}^{2} \right ){x}^{5}}{5}}+{\frac{2\,cde{x}^{7}}{7}}+{\frac{c{e}^{2}{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.975884, size = 65, normalized size = 1.16 \begin{align*} \frac{1}{9} \, c e^{2} x^{9} + \frac{2}{7} \, c d e x^{7} + \frac{2}{3} \, a d e x^{3} + \frac{1}{5} \,{\left (c d^{2} + a e^{2}\right )} x^{5} + 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+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.36688, size = 120, normalized size = 2.14 \begin{align*} \frac{1}{9} x^{9} e^{2} c + \frac{2}{7} x^{7} e d c + \frac{1}{5} x^{5} d^{2} c + \frac{1}{5} x^{5} e^{2} a + \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+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.065501, size = 56, normalized size = 1. \begin{align*} a d^{2} x + \frac{2 a d e x^{3}}{3} + \frac{2 c d e x^{7}}{7} + \frac{c e^{2} x^{9}}{9} + x^{5} \left (\frac{a e^{2}}{5} + \frac{c d^{2}}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12523, size = 68, normalized size = 1.21 \begin{align*} \frac{1}{9} \, c x^{9} e^{2} + \frac{2}{7} \, c d x^{7} e + \frac{1}{5} \, c d^{2} x^{5} + \frac{1}{5} \, a x^{5} e^{2} + \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+a),x, algorithm="giac")

[Out]

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