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

Optimal. Leaf size=135 \[ \frac{1}{9} e^2 x^9 \left (e (a e+4 b d)+6 c d^2\right )+\frac{1}{5} d^2 x^5 \left (6 a e^2+4 b d e+c d^2\right )+\frac{2}{7} d e x^7 \left (e (2 a e+3 b d)+2 c d^2\right )+\frac{1}{3} d^3 x^3 (4 a e+b d)+a d^4 x+\frac{1}{11} e^3 x^{11} (b e+4 c d)+\frac{1}{13} c e^4 x^{13} \]

[Out]

a*d^4*x + (d^3*(b*d + 4*a*e)*x^3)/3 + (d^2*(c*d^2 + 4*b*d*e + 6*a*e^2)*x^5)/5 + (2*d*e*(2*c*d^2 + e*(3*b*d + 2
*a*e))*x^7)/7 + (e^2*(6*c*d^2 + e*(4*b*d + a*e))*x^9)/9 + (e^3*(4*c*d + b*e)*x^11)/11 + (c*e^4*x^13)/13

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Rubi [A]  time = 0.125944, antiderivative size = 135, 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} \[ \frac{1}{9} e^2 x^9 \left (e (a e+4 b d)+6 c d^2\right )+\frac{1}{5} d^2 x^5 \left (6 a e^2+4 b d e+c d^2\right )+\frac{2}{7} d e x^7 \left (e (2 a e+3 b d)+2 c d^2\right )+\frac{1}{3} d^3 x^3 (4 a e+b d)+a d^4 x+\frac{1}{11} e^3 x^{11} (b e+4 c d)+\frac{1}{13} c e^4 x^{13} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*d^4*x + (d^3*(b*d + 4*a*e)*x^3)/3 + (d^2*(c*d^2 + 4*b*d*e + 6*a*e^2)*x^5)/5 + (2*d*e*(2*c*d^2 + e*(3*b*d + 2
*a*e))*x^7)/7 + (e^2*(6*c*d^2 + e*(4*b*d + a*e))*x^9)/9 + (e^3*(4*c*d + b*e)*x^11)/11 + (c*e^4*x^13)/13

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

Mathematica [A]  time = 0.0384811, size = 135, normalized size = 1. \[ \frac{1}{9} e^2 x^9 \left (a e^2+4 b d e+6 c d^2\right )+\frac{2}{7} d e x^7 \left (2 a e^2+3 b d e+2 c d^2\right )+\frac{1}{5} d^2 x^5 \left (6 a e^2+4 b d e+c d^2\right )+\frac{1}{3} d^3 x^3 (4 a e+b d)+a d^4 x+\frac{1}{11} e^3 x^{11} (b e+4 c d)+\frac{1}{13} c e^4 x^{13} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*d^4*x + (d^3*(b*d + 4*a*e)*x^3)/3 + (d^2*(c*d^2 + 4*b*d*e + 6*a*e^2)*x^5)/5 + (2*d*e*(2*c*d^2 + 3*b*d*e + 2*
a*e^2)*x^7)/7 + (e^2*(6*c*d^2 + 4*b*d*e + a*e^2)*x^9)/9 + (e^3*(4*c*d + b*e)*x^11)/11 + (c*e^4*x^13)/13

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Maple [A]  time = 0., size = 136, normalized size = 1. \begin{align*}{\frac{c{e}^{4}{x}^{13}}{13}}+{\frac{ \left ({e}^{4}b+4\,d{e}^{3}c \right ){x}^{11}}{11}}+{\frac{ \left ({e}^{4}a+4\,d{e}^{3}b+6\,{d}^{2}{e}^{2}c \right ){x}^{9}}{9}}+{\frac{ \left ( 4\,d{e}^{3}a+6\,{d}^{2}{e}^{2}b+4\,{d}^{3}ec \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,{d}^{2}{e}^{2}a+4\,{d}^{3}eb+{d}^{4}c \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,{d}^{3}ea+{d}^{4}b \right ){x}^{3}}{3}}+a{d}^{4}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*c*e^4*x^13+1/11*(b*e^4+4*c*d*e^3)*x^11+1/9*(a*e^4+4*b*d*e^3+6*c*d^2*e^2)*x^9+1/7*(4*a*d*e^3+6*b*d^2*e^2+4
*c*d^3*e)*x^7+1/5*(6*a*d^2*e^2+4*b*d^3*e+c*d^4)*x^5+1/3*(4*a*d^3*e+b*d^4)*x^3+a*d^4*x

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Maxima [A]  time = 0.973295, size = 182, normalized size = 1.35 \begin{align*} \frac{1}{13} \, c e^{4} x^{13} + \frac{1}{11} \,{\left (4 \, c d e^{3} + b e^{4}\right )} x^{11} + \frac{1}{9} \,{\left (6 \, c d^{2} e^{2} + 4 \, b d e^{3} + a e^{4}\right )} x^{9} + \frac{2}{7} \,{\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{7} + a d^{4} x + \frac{1}{5} \,{\left (c d^{4} + 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} x^{5} + \frac{1}{3} \,{\left (b d^{4} + 4 \, a d^{3} e\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*c*e^4*x^13 + 1/11*(4*c*d*e^3 + b*e^4)*x^11 + 1/9*(6*c*d^2*e^2 + 4*b*d*e^3 + a*e^4)*x^9 + 2/7*(2*c*d^3*e +
3*b*d^2*e^2 + 2*a*d*e^3)*x^7 + a*d^4*x + 1/5*(c*d^4 + 4*b*d^3*e + 6*a*d^2*e^2)*x^5 + 1/3*(b*d^4 + 4*a*d^3*e)*
x^3

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Fricas [A]  time = 1.34899, size = 355, normalized size = 2.63 \begin{align*} \frac{1}{13} x^{13} e^{4} c + \frac{4}{11} x^{11} e^{3} d c + \frac{1}{11} x^{11} e^{4} b + \frac{2}{3} x^{9} e^{2} d^{2} c + \frac{4}{9} x^{9} e^{3} d b + \frac{1}{9} x^{9} e^{4} a + \frac{4}{7} x^{7} e d^{3} c + \frac{6}{7} x^{7} e^{2} d^{2} b + \frac{4}{7} x^{7} e^{3} d a + \frac{1}{5} x^{5} d^{4} c + \frac{4}{5} x^{5} e d^{3} b + \frac{6}{5} x^{5} e^{2} d^{2} a + \frac{1}{3} x^{3} d^{4} b + \frac{4}{3} x^{3} e d^{3} a + x d^{4} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*x^13*e^4*c + 4/11*x^11*e^3*d*c + 1/11*x^11*e^4*b + 2/3*x^9*e^2*d^2*c + 4/9*x^9*e^3*d*b + 1/9*x^9*e^4*a +
4/7*x^7*e*d^3*c + 6/7*x^7*e^2*d^2*b + 4/7*x^7*e^3*d*a + 1/5*x^5*d^4*c + 4/5*x^5*e*d^3*b + 6/5*x^5*e^2*d^2*a +
1/3*x^3*d^4*b + 4/3*x^3*e*d^3*a + x*d^4*a

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Sympy [A]  time = 0.093139, size = 156, normalized size = 1.16 \begin{align*} a d^{4} x + \frac{c e^{4} x^{13}}{13} + x^{11} \left (\frac{b e^{4}}{11} + \frac{4 c d e^{3}}{11}\right ) + x^{9} \left (\frac{a e^{4}}{9} + \frac{4 b d e^{3}}{9} + \frac{2 c d^{2} e^{2}}{3}\right ) + x^{7} \left (\frac{4 a d e^{3}}{7} + \frac{6 b d^{2} e^{2}}{7} + \frac{4 c d^{3} e}{7}\right ) + x^{5} \left (\frac{6 a d^{2} e^{2}}{5} + \frac{4 b d^{3} e}{5} + \frac{c d^{4}}{5}\right ) + x^{3} \left (\frac{4 a d^{3} e}{3} + \frac{b d^{4}}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d**4*x + c*e**4*x**13/13 + x**11*(b*e**4/11 + 4*c*d*e**3/11) + x**9*(a*e**4/9 + 4*b*d*e**3/9 + 2*c*d**2*e**2
/3) + x**7*(4*a*d*e**3/7 + 6*b*d**2*e**2/7 + 4*c*d**3*e/7) + x**5*(6*a*d**2*e**2/5 + 4*b*d**3*e/5 + c*d**4/5)
+ x**3*(4*a*d**3*e/3 + b*d**4/3)

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Giac [A]  time = 1.20255, size = 192, normalized size = 1.42 \begin{align*} \frac{1}{13} \, c x^{13} e^{4} + \frac{4}{11} \, c d x^{11} e^{3} + \frac{1}{11} \, b x^{11} e^{4} + \frac{2}{3} \, c d^{2} x^{9} e^{2} + \frac{4}{9} \, b d x^{9} e^{3} + \frac{4}{7} \, c d^{3} x^{7} e + \frac{1}{9} \, a x^{9} e^{4} + \frac{6}{7} \, b d^{2} x^{7} e^{2} + \frac{1}{5} \, c d^{4} x^{5} + \frac{4}{7} \, a d x^{7} e^{3} + \frac{4}{5} \, b d^{3} x^{5} e + \frac{6}{5} \, a d^{2} x^{5} e^{2} + \frac{1}{3} \, b d^{4} x^{3} + \frac{4}{3} \, a d^{3} x^{3} e + a d^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*c*x^13*e^4 + 4/11*c*d*x^11*e^3 + 1/11*b*x^11*e^4 + 2/3*c*d^2*x^9*e^2 + 4/9*b*d*x^9*e^3 + 4/7*c*d^3*x^7*e
+ 1/9*a*x^9*e^4 + 6/7*b*d^2*x^7*e^2 + 1/5*c*d^4*x^5 + 4/7*a*d*x^7*e^3 + 4/5*b*d^3*x^5*e + 6/5*a*d^2*x^5*e^2 +
1/3*b*d^4*x^3 + 4/3*a*d^3*x^3*e + a*d^4*x