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

Optimal. Leaf size=97 \[ a^2 d^2 x+\frac{2}{3} a^2 d e x^3+\frac{1}{9} c x^9 \left (2 a e^2+c d^2\right )+\frac{1}{5} a x^5 \left (a e^2+2 c d^2\right )+\frac{4}{7} a c d e x^7+\frac{2}{11} c^2 d e x^{11}+\frac{1}{13} c^2 e^2 x^{13} \]

[Out]

a^2*d^2*x + (2*a^2*d*e*x^3)/3 + (a*(2*c*d^2 + a*e^2)*x^5)/5 + (4*a*c*d*e*x^7)/7 + (c*(c*d^2 + 2*a*e^2)*x^9)/9
+ (2*c^2*d*e*x^11)/11 + (c^2*e^2*x^13)/13

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Rubi [A]  time = 0.0683933, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {1154} \[ a^2 d^2 x+\frac{2}{3} a^2 d e x^3+\frac{1}{9} c x^9 \left (2 a e^2+c d^2\right )+\frac{1}{5} a x^5 \left (a e^2+2 c d^2\right )+\frac{4}{7} a c d e x^7+\frac{2}{11} c^2 d e x^{11}+\frac{1}{13} c^2 e^2 x^{13} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^2*(a + c*x^4)^2,x]

[Out]

a^2*d^2*x + (2*a^2*d*e*x^3)/3 + (a*(2*c*d^2 + a*e^2)*x^5)/5 + (4*a*c*d*e*x^7)/7 + (c*(c*d^2 + 2*a*e^2)*x^9)/9
+ (2*c^2*d*e*x^11)/11 + (c^2*e^2*x^13)/13

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 )^2 \left (a+c x^4\right )^2 \, dx &=\int \left (a^2 d^2+2 a^2 d e x^2+a \left (2 c d^2+a e^2\right ) x^4+4 a c d e x^6+c \left (c d^2+2 a e^2\right ) x^8+2 c^2 d e x^{10}+c^2 e^2 x^{12}\right ) \, dx\\ &=a^2 d^2 x+\frac{2}{3} a^2 d e x^3+\frac{1}{5} a \left (2 c d^2+a e^2\right ) x^5+\frac{4}{7} a c d e x^7+\frac{1}{9} c \left (c d^2+2 a e^2\right ) x^9+\frac{2}{11} c^2 d e x^{11}+\frac{1}{13} c^2 e^2 x^{13}\\ \end{align*}

Mathematica [A]  time = 0.0175475, size = 97, normalized size = 1. \[ a^2 d^2 x+\frac{2}{3} a^2 d e x^3+\frac{1}{9} c x^9 \left (2 a e^2+c d^2\right )+\frac{1}{5} a x^5 \left (a e^2+2 c d^2\right )+\frac{4}{7} a c d e x^7+\frac{2}{11} c^2 d e x^{11}+\frac{1}{13} c^2 e^2 x^{13} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*d^2*x + (2*a^2*d*e*x^3)/3 + (a*(2*c*d^2 + a*e^2)*x^5)/5 + (4*a*c*d*e*x^7)/7 + (c*(c*d^2 + 2*a*e^2)*x^9)/9
+ (2*c^2*d*e*x^11)/11 + (c^2*e^2*x^13)/13

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Maple [A]  time = 0.041, size = 90, normalized size = 0.9 \begin{align*}{\frac{{c}^{2}{e}^{2}{x}^{13}}{13}}+{\frac{2\,{c}^{2}de{x}^{11}}{11}}+{\frac{ \left ( 2\,ac{e}^{2}+{c}^{2}{d}^{2} \right ){x}^{9}}{9}}+{\frac{4\,acde{x}^{7}}{7}}+{\frac{ \left ({e}^{2}{a}^{2}+2\,ac{d}^{2} \right ){x}^{5}}{5}}+{\frac{2\,{a}^{2}de{x}^{3}}{3}}+{a}^{2}{d}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*c^2*e^2*x^13+2/11*c^2*d*e*x^11+1/9*(2*a*c*e^2+c^2*d^2)*x^9+4/7*a*c*d*e*x^7+1/5*(a^2*e^2+2*a*c*d^2)*x^5+2/
3*a^2*d*e*x^3+a^2*d^2*x

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Maxima [A]  time = 1.02734, size = 120, normalized size = 1.24 \begin{align*} \frac{1}{13} \, c^{2} e^{2} x^{13} + \frac{2}{11} \, c^{2} d e x^{11} + \frac{4}{7} \, a c d e x^{7} + \frac{1}{9} \,{\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} x^{9} + \frac{2}{3} \, a^{2} d e x^{3} + \frac{1}{5} \,{\left (2 \, a c d^{2} + a^{2} e^{2}\right )} x^{5} + a^{2} d^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*c^2*e^2*x^13 + 2/11*c^2*d*e*x^11 + 4/7*a*c*d*e*x^7 + 1/9*(c^2*d^2 + 2*a*c*e^2)*x^9 + 2/3*a^2*d*e*x^3 + 1/
5*(2*a*c*d^2 + a^2*e^2)*x^5 + a^2*d^2*x

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Fricas [A]  time = 1.48504, size = 215, normalized size = 2.22 \begin{align*} \frac{1}{13} x^{13} e^{2} c^{2} + \frac{2}{11} x^{11} e d c^{2} + \frac{1}{9} x^{9} d^{2} c^{2} + \frac{2}{9} x^{9} e^{2} c a + \frac{4}{7} x^{7} e d c a + \frac{2}{5} x^{5} d^{2} c a + \frac{1}{5} x^{5} e^{2} a^{2} + \frac{2}{3} x^{3} e d a^{2} + x d^{2} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*x^13*e^2*c^2 + 2/11*x^11*e*d*c^2 + 1/9*x^9*d^2*c^2 + 2/9*x^9*e^2*c*a + 4/7*x^7*e*d*c*a + 2/5*x^5*d^2*c*a
+ 1/5*x^5*e^2*a^2 + 2/3*x^3*e*d*a^2 + x*d^2*a^2

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Sympy [A]  time = 0.079009, size = 104, normalized size = 1.07 \begin{align*} a^{2} d^{2} x + \frac{2 a^{2} d e x^{3}}{3} + \frac{4 a c d e x^{7}}{7} + \frac{2 c^{2} d e x^{11}}{11} + \frac{c^{2} e^{2} x^{13}}{13} + x^{9} \left (\frac{2 a c e^{2}}{9} + \frac{c^{2} d^{2}}{9}\right ) + x^{5} \left (\frac{a^{2} e^{2}}{5} + \frac{2 a c d^{2}}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*d**2*x + 2*a**2*d*e*x**3/3 + 4*a*c*d*e*x**7/7 + 2*c**2*d*e*x**11/11 + c**2*e**2*x**13/13 + x**9*(2*a*c*e*
*2/9 + c**2*d**2/9) + x**5*(a**2*e**2/5 + 2*a*c*d**2/5)

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Giac [A]  time = 1.16473, size = 123, normalized size = 1.27 \begin{align*} \frac{1}{13} \, c^{2} x^{13} e^{2} + \frac{2}{11} \, c^{2} d x^{11} e + \frac{1}{9} \, c^{2} d^{2} x^{9} + \frac{2}{9} \, a c x^{9} e^{2} + \frac{4}{7} \, a c d x^{7} e + \frac{2}{5} \, a c d^{2} x^{5} + \frac{1}{5} \, a^{2} x^{5} e^{2} + \frac{2}{3} \, a^{2} d x^{3} e + a^{2} d^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*c^2*x^13*e^2 + 2/11*c^2*d*x^11*e + 1/9*c^2*d^2*x^9 + 2/9*a*c*x^9*e^2 + 4/7*a*c*d*x^7*e + 2/5*a*c*d^2*x^5
+ 1/5*a^2*x^5*e^2 + 2/3*a^2*d*x^3*e + a^2*d^2*x