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

Optimal. Leaf size=96 \[ a^2 d x+\frac{1}{7} x^7 \left (2 a c e+b^2 e+2 b c d\right )+\frac{1}{5} x^5 \left (2 a b e+2 a c d+b^2 d\right )+\frac{1}{3} a x^3 (a e+2 b d)+\frac{1}{9} c x^9 (2 b e+c d)+\frac{1}{11} c^2 e x^{11} \]

[Out]

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

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Rubi [A]  time = 0.06776, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {1153} \[ a^2 d x+\frac{1}{7} x^7 \left (2 a c e+b^2 e+2 b c d\right )+\frac{1}{5} x^5 \left (2 a b e+2 a c d+b^2 d\right )+\frac{1}{3} a x^3 (a e+2 b d)+\frac{1}{9} c x^9 (2 b e+c d)+\frac{1}{11} c^2 e x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

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

Mathematica [A]  time = 0.0239447, size = 96, normalized size = 1. \[ a^2 d x+\frac{1}{7} x^7 \left (2 a c e+b^2 e+2 b c d\right )+\frac{1}{5} x^5 \left (2 a b e+2 a c d+b^2 d\right )+\frac{1}{3} a x^3 (a e+2 b d)+\frac{1}{9} c x^9 (2 b e+c d)+\frac{1}{11} c^2 e x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 91, normalized size = 1. \begin{align*}{\frac{{c}^{2}e{x}^{11}}{11}}+{\frac{ \left ( 2\,ebc+d{c}^{2} \right ){x}^{9}}{9}}+{\frac{ \left ( 2\,bcd+e \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( d \left ( 2\,ac+{b}^{2} \right ) +2\,abe \right ){x}^{5}}{5}}+{\frac{ \left ( e{a}^{2}+2\,dab \right ){x}^{3}}{3}}+{a}^{2}dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.966369, size = 122, normalized size = 1.27 \begin{align*} \frac{1}{11} \, c^{2} e x^{11} + \frac{1}{9} \,{\left (c^{2} d + 2 \, b c e\right )} x^{9} + \frac{1}{7} \,{\left (2 \, b c d +{\left (b^{2} + 2 \, a c\right )} e\right )} x^{7} + \frac{1}{5} \,{\left (2 \, a b e +{\left (b^{2} + 2 \, a c\right )} d\right )} x^{5} + a^{2} d x + \frac{1}{3} \,{\left (2 \, a b d + a^{2} e\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.33555, size = 252, normalized size = 2.62 \begin{align*} \frac{1}{11} x^{11} e c^{2} + \frac{1}{9} x^{9} d c^{2} + \frac{2}{9} x^{9} e c b + \frac{2}{7} x^{7} d c b + \frac{1}{7} x^{7} e b^{2} + \frac{2}{7} x^{7} e c a + \frac{1}{5} x^{5} d b^{2} + \frac{2}{5} x^{5} d c a + \frac{2}{5} x^{5} e b a + \frac{2}{3} x^{3} d b 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+b*x^2+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.080717, size = 107, normalized size = 1.11 \begin{align*} a^{2} d x + \frac{c^{2} e x^{11}}{11} + x^{9} \left (\frac{2 b c e}{9} + \frac{c^{2} d}{9}\right ) + x^{7} \left (\frac{2 a c e}{7} + \frac{b^{2} e}{7} + \frac{2 b c d}{7}\right ) + x^{5} \left (\frac{2 a b e}{5} + \frac{2 a c d}{5} + \frac{b^{2} d}{5}\right ) + x^{3} \left (\frac{a^{2} e}{3} + \frac{2 a b d}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18721, size = 143, normalized size = 1.49 \begin{align*} \frac{1}{11} \, c^{2} x^{11} e + \frac{1}{9} \, c^{2} d x^{9} + \frac{2}{9} \, b c x^{9} e + \frac{2}{7} \, b c d x^{7} + \frac{1}{7} \, b^{2} x^{7} e + \frac{2}{7} \, a c x^{7} e + \frac{1}{5} \, b^{2} d x^{5} + \frac{2}{5} \, a c d x^{5} + \frac{2}{5} \, a b x^{5} e + \frac{2}{3} \, a b d x^{3} + \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+b*x^2+a)^2,x, algorithm="giac")

[Out]

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