3.3.74 \(\int x^2 (d+e x^2)^2 (a+b x^2+c x^4) \, dx\) [274]

Optimal. Leaf size=78 \[ \frac {1}{3} a d^2 x^3+\frac {1}{5} d (b d+2 a e) x^5+\frac {1}{7} \left (c d^2+e (2 b d+a e)\right ) x^7+\frac {1}{9} e (2 c d+b e) x^9+\frac {1}{11} c e^2 x^{11} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {1275} \begin {gather*} \frac {1}{7} x^7 \left (e (a e+2 b d)+c d^2\right )+\frac {1}{5} d x^5 (2 a e+b d)+\frac {1}{3} a d^2 x^3+\frac {1}{9} e x^9 (b e+2 c d)+\frac {1}{11} c e^2 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1275

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

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 78, normalized size = 1.00 \begin {gather*} \frac {1}{3} a d^2 x^3+\frac {1}{5} d (b d+2 a e) x^5+\frac {1}{7} \left (c d^2+2 b d e+a e^2\right ) x^7+\frac {1}{9} e (2 c d+b e) x^9+\frac {1}{11} c e^2 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.14, size = 73, normalized size = 0.94

method	result	size
default	\(\frac {c \,e^{2} x^{11}}{11}+\frac {\left (e^{2} b +2 c d e \right ) x^{9}}{9}+\frac {\left (a \,e^{2}+2 d e b +c \,d^{2}\right ) x^{7}}{7}+\frac {\left (2 a d e +d^{2} b \right ) x^{5}}{5}+\frac {a \,d^{2} x^{3}}{3}\)	\(73\)
norman	\(\frac {c \,e^{2} x^{11}}{11}+\left (\frac {1}{9} e^{2} b +\frac {2}{9} c d e \right ) x^{9}+\left (\frac {1}{7} a \,e^{2}+\frac {2}{7} d e b +\frac {1}{7} c \,d^{2}\right ) x^{7}+\left (\frac {2}{5} a d e +\frac {1}{5} d^{2} b \right ) x^{5}+\frac {a \,d^{2} x^{3}}{3}\)	\(74\)
gosper	\(\frac {1}{11} c \,e^{2} x^{11}+\frac {1}{9} x^{9} e^{2} b +\frac {2}{9} x^{9} c d e +\frac {1}{7} x^{7} a \,e^{2}+\frac {2}{7} x^{7} d e b +\frac {1}{7} x^{7} c \,d^{2}+\frac {2}{5} x^{5} a d e +\frac {1}{5} x^{5} d^{2} b +\frac {1}{3} a \,d^{2} x^{3}\)	\(80\)
risch	\(\frac {1}{11} c \,e^{2} x^{11}+\frac {1}{9} x^{9} e^{2} b +\frac {2}{9} x^{9} c d e +\frac {1}{7} x^{7} a \,e^{2}+\frac {2}{7} x^{7} d e b +\frac {1}{7} x^{7} c \,d^{2}+\frac {2}{5} x^{5} a d e +\frac {1}{5} x^{5} d^{2} b +\frac {1}{3} a \,d^{2} x^{3}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 72, normalized size = 0.92 \begin {gather*} \frac {1}{11} \, c x^{11} e^{2} + \frac {1}{9} \, {\left (2 \, c d e + b e^{2}\right )} x^{9} + \frac {1}{7} \, {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{7} + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{5} \, {\left (b d^{2} + 2 \, a d e\right )} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.35, size = 77, normalized size = 0.99 \begin {gather*} \frac {1}{7} \, c d^{2} x^{7} + \frac {1}{5} \, b d^{2} x^{5} + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{693} \, {\left (63 \, c x^{11} + 77 \, b x^{9} + 99 \, a x^{7}\right )} e^{2} + \frac {2}{315} \, {\left (35 \, c d x^{9} + 45 \, b d x^{7} + 63 \, a d x^{5}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*c*d^2*x^7 + 1/5*b*d^2*x^5 + 1/3*a*d^2*x^3 + 1/693*(63*c*x^11 + 77*b*x^9 + 99*a*x^7)*e^2 + 2/315*(35*c*d*x^
9 + 45*b*d*x^7 + 63*a*d*x^5)*e

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Sympy [A]
time = 0.01, size = 82, normalized size = 1.05 \begin {gather*} \frac {a d^{2} x^{3}}{3} + \frac {c e^{2} x^{11}}{11} + x^{9} \left (\frac {b e^{2}}{9} + \frac {2 c d e}{9}\right ) + x^{7} \left (\frac {a e^{2}}{7} + \frac {2 b d e}{7} + \frac {c d^{2}}{7}\right ) + x^{5} \cdot \left (\frac {2 a d e}{5} + \frac {b d^{2}}{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 3.52, size = 79, normalized size = 1.01 \begin {gather*} \frac {1}{11} \, c x^{11} e^{2} + \frac {2}{9} \, c d x^{9} e + \frac {1}{9} \, b x^{9} e^{2} + \frac {1}{7} \, c d^{2} x^{7} + \frac {2}{7} \, b d x^{7} e + \frac {1}{7} \, a x^{7} e^{2} + \frac {1}{5} \, b d^{2} x^{5} + \frac {2}{5} \, a d x^{5} e + \frac {1}{3} \, a d^{2} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.03, size = 73, normalized size = 0.94 \begin {gather*} x^7\,\left (\frac {c\,d^2}{7}+\frac {2\,b\,d\,e}{7}+\frac {a\,e^2}{7}\right )+x^5\,\left (\frac {b\,d^2}{5}+\frac {2\,a\,e\,d}{5}\right )+x^9\,\left (\frac {b\,e^2}{9}+\frac {2\,c\,d\,e}{9}\right )+\frac {a\,d^2\,x^3}{3}+\frac {c\,e^2\,x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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