3.3.73 \(\int x^3 (d+e x^2)^2 (a+b x^2+c x^4) \, dx\) [273]

Optimal. Leaf size=78 \[ \frac {1}{4} a d^2 x^4+\frac {1}{6} d (b d+2 a e) x^6+\frac {1}{8} \left (c d^2+e (2 b d+a e)\right ) x^8+\frac {1}{10} e (2 c d+b e) x^{10}+\frac {1}{12} c e^2 x^{12} \]

[Out]

1/4*a*d^2*x^4+1/6*d*(2*a*e+b*d)*x^6+1/8*(c*d^2+e*(a*e+2*b*d))*x^8+1/10*e*(b*e+2*c*d)*x^10+1/12*c*e^2*x^12

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Rubi [A]
time = 0.09, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1265, 785} \begin {gather*} \frac {1}{8} x^8 \left (e (a e+2 b d)+c d^2\right )+\frac {1}{6} d x^6 (2 a e+b d)+\frac {1}{4} a d^2 x^4+\frac {1}{10} e x^{10} (b e+2 c d)+\frac {1}{12} c e^2 x^{12} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(d + e*x^2)^2*(a + b*x^2 + c*x^4),x]

[Out]

(a*d^2*x^4)/4 + (d*(b*d + 2*a*e)*x^6)/6 + ((c*d^2 + e*(2*b*d + a*e))*x^8)/8 + (e*(2*c*d + b*e)*x^10)/10 + (c*e
^2*x^12)/12

Rule 785

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 72, normalized size = 0.92 \begin {gather*} \frac {1}{120} x^4 \left (30 a d^2+20 d (b d+2 a e) x^2+15 \left (c d^2+e (2 b d+a e)\right ) x^4+12 e (2 c d+b e) x^6+10 c e^2 x^8\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(d + e*x^2)^2*(a + b*x^2 + c*x^4),x]

[Out]

(x^4*(30*a*d^2 + 20*d*(b*d + 2*a*e)*x^2 + 15*(c*d^2 + e*(2*b*d + a*e))*x^4 + 12*e*(2*c*d + b*e)*x^6 + 10*c*e^2
*x^8))/120

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Maple [A]
time = 0.17, size = 73, normalized size = 0.94

method	result	size
default	\(\frac {c \,e^{2} x^{12}}{12}+\frac {\left (e^{2} b +2 c d e \right ) x^{10}}{10}+\frac {\left (a \,e^{2}+2 d e b +c \,d^{2}\right ) x^{8}}{8}+\frac {\left (2 a d e +d^{2} b \right ) x^{6}}{6}+\frac {a \,d^{2} x^{4}}{4}\)	\(73\)
norman	\(\frac {c \,e^{2} x^{12}}{12}+\left (\frac {1}{10} e^{2} b +\frac {1}{5} c d e \right ) x^{10}+\left (\frac {1}{8} a \,e^{2}+\frac {1}{4} d e b +\frac {1}{8} c \,d^{2}\right ) x^{8}+\left (\frac {1}{3} a d e +\frac {1}{6} d^{2} b \right ) x^{6}+\frac {a \,d^{2} x^{4}}{4}\)	\(74\)
gosper	\(\frac {1}{12} c \,e^{2} x^{12}+\frac {1}{10} x^{10} e^{2} b +\frac {1}{5} x^{10} c d e +\frac {1}{8} x^{8} a \,e^{2}+\frac {1}{4} x^{8} d e b +\frac {1}{8} x^{8} c \,d^{2}+\frac {1}{3} x^{6} a d e +\frac {1}{6} x^{6} d^{2} b +\frac {1}{4} a \,d^{2} x^{4}\)	\(80\)
risch	\(\frac {1}{12} c \,e^{2} x^{12}+\frac {1}{10} x^{10} e^{2} b +\frac {1}{5} x^{10} c d e +\frac {1}{8} x^{8} a \,e^{2}+\frac {1}{4} x^{8} d e b +\frac {1}{8} x^{8} c \,d^{2}+\frac {1}{3} x^{6} a d e +\frac {1}{6} x^{6} d^{2} b +\frac {1}{4} a \,d^{2} x^{4}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(e*x^2+d)^2*(c*x^4+b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

1/12*c*e^2*x^12+1/10*(b*e^2+2*c*d*e)*x^10+1/8*(a*e^2+2*b*d*e+c*d^2)*x^8+1/6*(2*a*d*e+b*d^2)*x^6+1/4*a*d^2*x^4

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Maxima [A]
time = 0.27, size = 72, normalized size = 0.92 \begin {gather*} \frac {1}{12} \, c x^{12} e^{2} + \frac {1}{10} \, {\left (2 \, c d e + b e^{2}\right )} x^{10} + \frac {1}{8} \, {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{8} + \frac {1}{4} \, a d^{2} x^{4} + \frac {1}{6} \, {\left (b d^{2} + 2 \, a d e\right )} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)^2*(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

1/12*c*x^12*e^2 + 1/10*(2*c*d*e + b*e^2)*x^10 + 1/8*(c*d^2 + 2*b*d*e + a*e^2)*x^8 + 1/4*a*d^2*x^4 + 1/6*(b*d^2
+ 2*a*d*e)*x^6

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Fricas [A]
time = 0.34, size = 77, normalized size = 0.99 \begin {gather*} \frac {1}{8} \, c d^{2} x^{8} + \frac {1}{6} \, b d^{2} x^{6} + \frac {1}{4} \, a d^{2} x^{4} + \frac {1}{120} \, {\left (10 \, c x^{12} + 12 \, b x^{10} + 15 \, a x^{8}\right )} e^{2} + \frac {1}{60} \, {\left (12 \, c d x^{10} + 15 \, b d x^{8} + 20 \, a d x^{6}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)^2*(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

1/8*c*d^2*x^8 + 1/6*b*d^2*x^6 + 1/4*a*d^2*x^4 + 1/120*(10*c*x^12 + 12*b*x^10 + 15*a*x^8)*e^2 + 1/60*(12*c*d*x^
10 + 15*b*d*x^8 + 20*a*d*x^6)*e

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Sympy [A]
time = 0.01, size = 76, normalized size = 0.97 \begin {gather*} \frac {a d^{2} x^{4}}{4} + \frac {c e^{2} x^{12}}{12} + x^{10} \left (\frac {b e^{2}}{10} + \frac {c d e}{5}\right ) + x^{8} \left (\frac {a e^{2}}{8} + \frac {b d e}{4} + \frac {c d^{2}}{8}\right ) + x^{6} \left (\frac {a d e}{3} + \frac {b d^{2}}{6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(e*x**2+d)**2*(c*x**4+b*x**2+a),x)

[Out]

a*d**2*x**4/4 + c*e**2*x**12/12 + x**10*(b*e**2/10 + c*d*e/5) + x**8*(a*e**2/8 + b*d*e/4 + c*d**2/8) + x**6*(a
*d*e/3 + b*d**2/6)

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Giac [A]
time = 3.85, size = 79, normalized size = 1.01 \begin {gather*} \frac {1}{12} \, c x^{12} e^{2} + \frac {1}{5} \, c d x^{10} e + \frac {1}{10} \, b x^{10} e^{2} + \frac {1}{8} \, c d^{2} x^{8} + \frac {1}{4} \, b d x^{8} e + \frac {1}{8} \, a x^{8} e^{2} + \frac {1}{6} \, b d^{2} x^{6} + \frac {1}{3} \, a d x^{6} e + \frac {1}{4} \, a d^{2} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)^2*(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

1/12*c*x^12*e^2 + 1/5*c*d*x^10*e + 1/10*b*x^10*e^2 + 1/8*c*d^2*x^8 + 1/4*b*d*x^8*e + 1/8*a*x^8*e^2 + 1/6*b*d^2
*x^6 + 1/3*a*d*x^6*e + 1/4*a*d^2*x^4

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Mupad [B]
time = 0.04, size = 73, normalized size = 0.94 \begin {gather*} x^8\,\left (\frac {c\,d^2}{8}+\frac {b\,d\,e}{4}+\frac {a\,e^2}{8}\right )+x^6\,\left (\frac {b\,d^2}{6}+\frac {a\,e\,d}{3}\right )+x^{10}\,\left (\frac {b\,e^2}{10}+\frac {c\,d\,e}{5}\right )+\frac {a\,d^2\,x^4}{4}+\frac {c\,e^2\,x^{12}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(d + e*x^2)^2*(a + b*x^2 + c*x^4),x)

[Out]

x^8*((a*e^2)/8 + (c*d^2)/8 + (b*d*e)/4) + x^6*((b*d^2)/6 + (a*d*e)/3) + x^10*((b*e^2)/10 + (c*d*e)/5) + (a*d^2
*x^4)/4 + (c*e^2*x^12)/12

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