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

Optimal. Leaf size=32 \[ a d x+\frac{1}{3} a e x^3+\frac{1}{5} c d x^5+\frac{1}{7} c e x^7 \]

[Out]

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

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Rubi [A]  time = 0.0137964, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {1154} \[ a d x+\frac{1}{3} a e x^3+\frac{1}{5} c d x^5+\frac{1}{7} c e x^7 \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)*(a + c*x^4),x]

[Out]

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

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

Mathematica [A]  time = 0.0016087, size = 32, normalized size = 1. \[ a d x+\frac{1}{3} a e x^3+\frac{1}{5} c d x^5+\frac{1}{7} c e x^7 \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2)*(a + c*x^4),x]

[Out]

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

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Maple [A]  time = 0.041, size = 27, normalized size = 0.8 \begin{align*} adx+{\frac{ae{x}^{3}}{3}}+{\frac{cd{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+a),x)

[Out]

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

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Maxima [A]  time = 0.957647, size = 35, normalized size = 1.09 \begin{align*} \frac{1}{7} \, c e x^{7} + \frac{1}{5} \, c d x^{5} + \frac{1}{3} \, a e 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+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.55392, size = 66, normalized size = 2.06 \begin{align*} \frac{1}{7} x^{7} e c + \frac{1}{5} x^{5} d c + \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+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.057187, size = 29, normalized size = 0.91 \begin{align*} a d x + \frac{a e x^{3}}{3} + \frac{c d x^{5}}{5} + \frac{c e x^{7}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d*x + a*e*x**3/3 + c*d*x**5/5 + c*e*x**7/7

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Giac [A]  time = 1.12466, size = 38, normalized size = 1.19 \begin{align*} \frac{1}{7} \, c x^{7} e + \frac{1}{5} \, c d x^{5} + \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+a),x, algorithm="giac")

[Out]

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