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

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Rubi [A]  time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 32, normalized size = 1.00 \[ a d x+\frac {1}{3} a e x^3+\frac {1}{5} c d x^5+\frac {1}{7} c e x^7 \]

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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fricas [A]  time = 0.35, size = 26, normalized size = 0.81 \[ \frac {1}{7} x^{7} e c + \frac {1}{5} x^{5} d c + \frac {1}{3} x^{3} e a + x d a \]

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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giac [A]  time = 0.18, size = 28, normalized size = 0.88 \[ \frac {1}{7} \, c x^{7} e + \frac {1}{5} \, c d x^{5} + \frac {1}{3} \, a x^{3} e + a d x \]

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

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maple [A]  time = 0.00, size = 27, normalized size = 0.84 \[ \frac {1}{7} c e \,x^{7}+\frac {1}{5} c d \,x^{5}+\frac {1}{3} a e \,x^{3}+a d x \]

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 = 1.06, size = 26, normalized size = 0.81 \[ \frac {1}{7} \, c e x^{7} + \frac {1}{5} \, c d x^{5} + \frac {1}{3} \, a e x^{3} + a d x \]

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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mupad [B]  time = 0.04, size = 26, normalized size = 0.81 \[ \frac {c\,e\,x^7}{7}+\frac {c\,d\,x^5}{5}+\frac {a\,e\,x^3}{3}+a\,d\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)*(d + e*x^2),x)

[Out]

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

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sympy [A]  time = 0.08, size = 29, normalized size = 0.91 \[ a d x + \frac {a e x^{3}}{3} + \frac {c d x^{5}}{5} + \frac {c e x^{7}}{7} \]

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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