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

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Rubi [A]  time = 0.13, antiderivative size = 135, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.04, size = 135, normalized size = 1.00 \[ \frac {1}{9} e^2 x^9 \left (a e^2+4 b d e+6 c d^2\right )+\frac {2}{7} d e x^7 \left (2 a e^2+3 b d e+2 c d^2\right )+\frac {1}{5} d^2 x^5 \left (6 a e^2+4 b d e+c d^2\right )+\frac {1}{3} d^3 x^3 (4 a e+b d)+a d^4 x+\frac {1}{11} e^3 x^{11} (b e+4 c d)+\frac {1}{13} c e^4 x^{13} \]

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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fricas [A]  time = 0.68, size = 148, normalized size = 1.10 \[ \frac {1}{13} x^{13} e^{4} c + \frac {4}{11} x^{11} e^{3} d c + \frac {1}{11} x^{11} e^{4} b + \frac {2}{3} x^{9} e^{2} d^{2} c + \frac {4}{9} x^{9} e^{3} d b + \frac {1}{9} x^{9} e^{4} a + \frac {4}{7} x^{7} e d^{3} c + \frac {6}{7} x^{7} e^{2} d^{2} b + \frac {4}{7} x^{7} e^{3} d a + \frac {1}{5} x^{5} d^{4} c + \frac {4}{5} x^{5} e d^{3} b + \frac {6}{5} x^{5} e^{2} d^{2} a + \frac {1}{3} x^{3} d^{4} b + \frac {4}{3} x^{3} e d^{3} a + x d^{4} a \]

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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giac [A]  time = 0.15, size = 142, normalized size = 1.05 \[ \frac {1}{13} \, c x^{13} e^{4} + \frac {4}{11} \, c d x^{11} e^{3} + \frac {1}{11} \, b x^{11} e^{4} + \frac {2}{3} \, c d^{2} x^{9} e^{2} + \frac {4}{9} \, b d x^{9} e^{3} + \frac {4}{7} \, c d^{3} x^{7} e + \frac {1}{9} \, a x^{9} e^{4} + \frac {6}{7} \, b d^{2} x^{7} e^{2} + \frac {1}{5} \, c d^{4} x^{5} + \frac {4}{7} \, a d x^{7} e^{3} + \frac {4}{5} \, b d^{3} x^{5} e + \frac {6}{5} \, a d^{2} x^{5} e^{2} + \frac {1}{3} \, b d^{4} x^{3} + \frac {4}{3} \, a d^{3} x^{3} e + a d^{4} x \]

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

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maple [A]  time = 0.00, size = 136, normalized size = 1.01 \[ \frac {c \,e^{4} x^{13}}{13}+\frac {\left (e^{4} b +4 d \,e^{3} c \right ) x^{11}}{11}+\frac {\left (e^{4} a +4 d \,e^{3} b +6 d^{2} e^{2} c \right ) x^{9}}{9}+\frac {\left (4 d \,e^{3} a +6 d^{2} e^{2} b +4 d^{3} e c \right ) x^{7}}{7}+a \,d^{4} x +\frac {\left (6 d^{2} e^{2} a +4 d^{3} e b +d^{4} c \right ) x^{5}}{5}+\frac {\left (4 d^{3} e a +d^{4} b \right ) x^{3}}{3} \]

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 = 0.96, size = 135, normalized size = 1.00 \[ \frac {1}{13} \, c e^{4} x^{13} + \frac {1}{11} \, {\left (4 \, c d e^{3} + b e^{4}\right )} x^{11} + \frac {1}{9} \, {\left (6 \, c d^{2} e^{2} + 4 \, b d e^{3} + a e^{4}\right )} x^{9} + \frac {2}{7} \, {\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{7} + a d^{4} x + \frac {1}{5} \, {\left (c d^{4} + 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} x^{5} + \frac {1}{3} \, {\left (b d^{4} + 4 \, a d^{3} e\right )} x^{3} \]

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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mupad [B]  time = 0.06, size = 131, normalized size = 0.97 \[ x^3\,\left (\frac {b\,d^4}{3}+\frac {4\,a\,e\,d^3}{3}\right )+x^{11}\,\left (\frac {b\,e^4}{11}+\frac {4\,c\,d\,e^3}{11}\right )+x^5\,\left (\frac {c\,d^4}{5}+\frac {4\,b\,d^3\,e}{5}+\frac {6\,a\,d^2\,e^2}{5}\right )+x^9\,\left (\frac {2\,c\,d^2\,e^2}{3}+\frac {4\,b\,d\,e^3}{9}+\frac {a\,e^4}{9}\right )+\frac {c\,e^4\,x^{13}}{13}+a\,d^4\,x+\frac {2\,d\,e\,x^7\,\left (2\,c\,d^2+3\,b\,d\,e+2\,a\,e^2\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.11, size = 156, normalized size = 1.16 \[ a d^{4} x + \frac {c e^{4} x^{13}}{13} + x^{11} \left (\frac {b e^{4}}{11} + \frac {4 c d e^{3}}{11}\right ) + x^{9} \left (\frac {a e^{4}}{9} + \frac {4 b d e^{3}}{9} + \frac {2 c d^{2} e^{2}}{3}\right ) + x^{7} \left (\frac {4 a d e^{3}}{7} + \frac {6 b d^{2} e^{2}}{7} + \frac {4 c d^{3} e}{7}\right ) + x^{5} \left (\frac {6 a d^{2} e^{2}}{5} + \frac {4 b d^{3} e}{5} + \frac {c d^{4}}{5}\right ) + x^{3} \left (\frac {4 a d^{3} e}{3} + \frac {b d^{4}}{3}\right ) \]

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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