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

Optimal. Leaf size=60 \[ a^2 d x+\frac{1}{3} a^2 e x^3+\frac{2}{5} a c d x^5+\frac{2}{7} a c e x^7+\frac{1}{9} c^2 d x^9+\frac{1}{11} c^2 e x^{11} \]

[Out]

a^2*d*x + (a^2*e*x^3)/3 + (2*a*c*d*x^5)/5 + (2*a*c*e*x^7)/7 + (c^2*d*x^9)/9 + (c^2*e*x^11)/11

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Rubi [A]  time = 0.0304607, antiderivative size = 60, 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} \[ a^2 d x+\frac{1}{3} a^2 e x^3+\frac{2}{5} a c d x^5+\frac{2}{7} a c e x^7+\frac{1}{9} c^2 d x^9+\frac{1}{11} c^2 e x^{11} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)*(a + c*x^4)^2,x]

[Out]

a^2*d*x + (a^2*e*x^3)/3 + (2*a*c*d*x^5)/5 + (2*a*c*e*x^7)/7 + (c^2*d*x^9)/9 + (c^2*e*x^11)/11

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 )^2 \, dx &=\int \left (a^2 d+a^2 e x^2+2 a c d x^4+2 a c e x^6+c^2 d x^8+c^2 e x^{10}\right ) \, dx\\ &=a^2 d x+\frac{1}{3} a^2 e x^3+\frac{2}{5} a c d x^5+\frac{2}{7} a c e x^7+\frac{1}{9} c^2 d x^9+\frac{1}{11} c^2 e x^{11}\\ \end{align*}

Mathematica [A]  time = 0.0023163, size = 60, normalized size = 1. \[ a^2 d x+\frac{1}{3} a^2 e x^3+\frac{2}{5} a c d x^5+\frac{2}{7} a c e x^7+\frac{1}{9} c^2 d x^9+\frac{1}{11} c^2 e x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*d*x + (a^2*e*x^3)/3 + (2*a*c*d*x^5)/5 + (2*a*c*e*x^7)/7 + (c^2*d*x^9)/9 + (c^2*e*x^11)/11

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Maple [A]  time = 0.041, size = 51, normalized size = 0.9 \begin{align*}{a}^{2}dx+{\frac{{a}^{2}e{x}^{3}}{3}}+{\frac{2\,acd{x}^{5}}{5}}+{\frac{2\,ace{x}^{7}}{7}}+{\frac{{c}^{2}d{x}^{9}}{9}}+{\frac{{c}^{2}e{x}^{11}}{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*d*x+1/3*a^2*e*x^3+2/5*a*c*d*x^5+2/7*a*c*e*x^7+1/9*c^2*d*x^9+1/11*c^2*e*x^11

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Maxima [A]  time = 0.93175, size = 68, normalized size = 1.13 \begin{align*} \frac{1}{11} \, c^{2} e x^{11} + \frac{1}{9} \, c^{2} d x^{9} + \frac{2}{7} \, a c e x^{7} + \frac{2}{5} \, a c d x^{5} + \frac{1}{3} \, a^{2} e x^{3} + a^{2} d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*c^2*e*x^11 + 1/9*c^2*d*x^9 + 2/7*a*c*e*x^7 + 2/5*a*c*d*x^5 + 1/3*a^2*e*x^3 + a^2*d*x

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Fricas [A]  time = 1.64708, size = 123, normalized size = 2.05 \begin{align*} \frac{1}{11} x^{11} e c^{2} + \frac{1}{9} x^{9} d c^{2} + \frac{2}{7} x^{7} e c a + \frac{2}{5} x^{5} d c a + \frac{1}{3} x^{3} e a^{2} + x d a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*x^11*e*c^2 + 1/9*x^9*d*c^2 + 2/7*x^7*e*c*a + 2/5*x^5*d*c*a + 1/3*x^3*e*a^2 + x*d*a^2

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Sympy [A]  time = 0.067496, size = 60, normalized size = 1. \begin{align*} a^{2} d x + \frac{a^{2} e x^{3}}{3} + \frac{2 a c d x^{5}}{5} + \frac{2 a c e x^{7}}{7} + \frac{c^{2} d x^{9}}{9} + \frac{c^{2} e x^{11}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*d*x + a**2*e*x**3/3 + 2*a*c*d*x**5/5 + 2*a*c*e*x**7/7 + c**2*d*x**9/9 + c**2*e*x**11/11

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Giac [A]  time = 1.1765, size = 72, normalized size = 1.2 \begin{align*} \frac{1}{11} \, c^{2} x^{11} e + \frac{1}{9} \, c^{2} d x^{9} + \frac{2}{7} \, a c x^{7} e + \frac{2}{5} \, a c d x^{5} + \frac{1}{3} \, a^{2} x^{3} e + a^{2} d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*c^2*x^11*e + 1/9*c^2*d*x^9 + 2/7*a*c*x^7*e + 2/5*a*c*d*x^5 + 1/3*a^2*x^3*e + a^2*d*x