3.137 \(\int (a+c x^2) (e+f x^4)^2 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0266964, 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} \[ a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)*(e + f*x^4)^2,x]

[Out]

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

Mathematica [A]  time = 0.0022818, size = 60, normalized size = 1. \[ a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.974872, size = 68, normalized size = 1.13 \begin{align*} \frac{1}{11} \, c f^{2} x^{11} + \frac{1}{9} \, a f^{2} x^{9} + \frac{2}{7} \, c e f x^{7} + \frac{2}{5} \, a e f x^{5} + \frac{1}{3} \, c e^{2} x^{3} + a e^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.0562, size = 123, normalized size = 2.05 \begin{align*} \frac{1}{11} x^{11} f^{2} c + \frac{1}{9} x^{9} f^{2} a + \frac{2}{7} x^{7} f e c + \frac{2}{5} x^{5} f e a + \frac{1}{3} x^{3} e^{2} c + x e^{2} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.069098, size = 60, normalized size = 1. \begin{align*} a e^{2} x + \frac{2 a e f x^{5}}{5} + \frac{a f^{2} x^{9}}{9} + \frac{c e^{2} x^{3}}{3} + \frac{2 c e f x^{7}}{7} + \frac{c f^{2} x^{11}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.04692, size = 68, normalized size = 1.13 \begin{align*} \frac{1}{11} \, c f^{2} x^{11} + \frac{1}{9} \, a f^{2} x^{9} + \frac{2}{7} \, c f x^{7} e + \frac{2}{5} \, a f x^{5} e + \frac{1}{3} \, c x^{3} e^{2} + a x e^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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