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

Optimal. Leaf size=42 \[ \frac{1}{3} x^3 (a e+b d)+a d x+\frac{1}{5} x^5 (b e+c d)+\frac{1}{7} c e x^7 \]

[Out]

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

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Rubi [A]  time = 0.0273412, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {1153} \[ \frac{1}{3} x^3 (a e+b d)+a d x+\frac{1}{5} x^5 (b e+c d)+\frac{1}{7} c e x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0081182, size = 42, normalized size = 1. \[ \frac{1}{3} x^3 (a e+b d)+a d x+\frac{1}{5} x^5 (b e+c d)+\frac{1}{7} c e x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 37, normalized size = 0.9 \begin{align*} adx+{\frac{ \left ( ae+bd \right ){x}^{3}}{3}}+{\frac{ \left ( be+cd \right ){x}^{5}}{5}}+{\frac{ce{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.950216, size = 49, normalized size = 1.17 \begin{align*} \frac{1}{7} \, c e x^{7} + \frac{1}{5} \,{\left (c d + b e\right )} x^{5} + \frac{1}{3} \,{\left (b d + a e\right )} x^{3} + a d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.38201, size = 104, normalized size = 2.48 \begin{align*} \frac{1}{7} x^{7} e c + \frac{1}{5} x^{5} d c + \frac{1}{5} x^{5} e b + \frac{1}{3} x^{3} d b + \frac{1}{3} x^{3} e a + x d a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.063286, size = 39, normalized size = 0.93 \begin{align*} a d x + \frac{c e x^{7}}{7} + x^{5} \left (\frac{b e}{5} + \frac{c d}{5}\right ) + x^{3} \left (\frac{a e}{3} + \frac{b d}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13787, size = 58, normalized size = 1.38 \begin{align*} \frac{1}{7} \, c x^{7} e + \frac{1}{5} \, c d x^{5} + \frac{1}{5} \, b x^{5} e + \frac{1}{3} \, b d x^{3} + \frac{1}{3} \, a x^{3} e + a d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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