\(\int (d+e x^2)^2 (a+b x^2+c x^4) \, dx\) [246]

Optimal result

Integrand size = 22, antiderivative size = 73 \[ \int \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right ) \, dx=a d^2 x+\frac {1}{3} d (b d+2 a e) x^3+\frac {1}{5} \left (c d^2+e (2 b d+a e)\right ) x^5+\frac {1}{7} e (2 c d+b e) x^7+\frac {1}{9} c e^2 x^9 \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1167} \[ \int \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{5} x^5 \left (e (a e+2 b d)+c d^2\right )+\frac {1}{3} d x^3 (2 a e+b d)+a d^2 x+\frac {1}{7} e x^7 (b e+2 c d)+\frac {1}{9} c e^2 x^9 \]

[In]

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

[Out]

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

Rule 1167

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right ) \, dx=a d^2 x+\frac {1}{3} d (b d+2 a e) x^3+\frac {1}{5} \left (c d^2+2 b d e+a e^2\right ) x^5+\frac {1}{7} e (2 c d+b e) x^7+\frac {1}{9} c e^2 x^9 \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.96

method	result	size
default	\(\frac {c \,e^{2} x^{9}}{9}+\frac {\left (b \,e^{2}+2 d c e \right ) x^{7}}{7}+\frac {\left (a \,e^{2}+2 b d e +c \,d^{2}\right ) x^{5}}{5}+\frac {\left (2 e d a +b \,d^{2}\right ) x^{3}}{3}+a \,d^{2} x\)	\(70\)
norman	\(\frac {c \,e^{2} x^{9}}{9}+\left (\frac {1}{7} b \,e^{2}+\frac {2}{7} d c e \right ) x^{7}+\left (\frac {1}{5} a \,e^{2}+\frac {2}{5} b d e +\frac {1}{5} c \,d^{2}\right ) x^{5}+\left (\frac {2}{3} e d a +\frac {1}{3} b \,d^{2}\right ) x^{3}+a \,d^{2} x\)	\(71\)
gosper	\(\frac {1}{9} c \,e^{2} x^{9}+\frac {1}{7} x^{7} b \,e^{2}+\frac {2}{7} c d e \,x^{7}+\frac {1}{5} x^{5} a \,e^{2}+\frac {2}{5} x^{5} b d e +\frac {1}{5} x^{5} c \,d^{2}+\frac {2}{3} a d e \,x^{3}+\frac {1}{3} x^{3} b \,d^{2}+a \,d^{2} x\)	\(77\)
risch	\(\frac {1}{9} c \,e^{2} x^{9}+\frac {1}{7} x^{7} b \,e^{2}+\frac {2}{7} c d e \,x^{7}+\frac {1}{5} x^{5} a \,e^{2}+\frac {2}{5} x^{5} b d e +\frac {1}{5} x^{5} c \,d^{2}+\frac {2}{3} a d e \,x^{3}+\frac {1}{3} x^{3} b \,d^{2}+a \,d^{2} x\)	\(77\)
parallelrisch	\(\frac {1}{9} c \,e^{2} x^{9}+\frac {1}{7} x^{7} b \,e^{2}+\frac {2}{7} c d e \,x^{7}+\frac {1}{5} x^{5} a \,e^{2}+\frac {2}{5} x^{5} b d e +\frac {1}{5} x^{5} c \,d^{2}+\frac {2}{3} a d e \,x^{3}+\frac {1}{3} x^{3} b \,d^{2}+a \,d^{2} x\)	\(77\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.95 \[ \int \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{9} \, c e^{2} x^{9} + \frac {1}{7} \, {\left (2 \, c d e + b e^{2}\right )} x^{7} + \frac {1}{5} \, {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{5} + a d^{2} x + \frac {1}{3} \, {\left (b d^{2} + 2 \, a d e\right )} x^{3} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.07 \[ \int \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right ) \, dx=a d^{2} x + \frac {c e^{2} x^{9}}{9} + x^{7} \left (\frac {b e^{2}}{7} + \frac {2 c d e}{7}\right ) + x^{5} \left (\frac {a e^{2}}{5} + \frac {2 b d e}{5} + \frac {c d^{2}}{5}\right ) + x^{3} \cdot \left (\frac {2 a d e}{3} + \frac {b d^{2}}{3}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.95 \[ \int \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{9} \, c e^{2} x^{9} + \frac {1}{7} \, {\left (2 \, c d e + b e^{2}\right )} x^{7} + \frac {1}{5} \, {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{5} + a d^{2} x + \frac {1}{3} \, {\left (b d^{2} + 2 \, a d e\right )} x^{3} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.04 \[ \int \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{9} \, c e^{2} x^{9} + \frac {2}{7} \, c d e x^{7} + \frac {1}{7} \, b e^{2} x^{7} + \frac {1}{5} \, c d^{2} x^{5} + \frac {2}{5} \, b d e x^{5} + \frac {1}{5} \, a e^{2} x^{5} + \frac {1}{3} \, b d^{2} x^{3} + \frac {2}{3} \, a d e x^{3} + a d^{2} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 7.51 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.96 \[ \int \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right ) \, dx=x^5\,\left (\frac {c\,d^2}{5}+\frac {2\,b\,d\,e}{5}+\frac {a\,e^2}{5}\right )+x^3\,\left (\frac {b\,d^2}{3}+\frac {2\,a\,e\,d}{3}\right )+x^7\,\left (\frac {b\,e^2}{7}+\frac {2\,c\,d\,e}{7}\right )+\frac {c\,e^2\,x^9}{9}+a\,d^2\,x \]

[In]

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

[Out]

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