\(\int (d+e x^2)^2 (a+c x^4) \, dx\) [122]

Optimal result

Integrand size = 17, antiderivative size = 56 \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right ) \, dx=a d^2 x+\frac {2}{3} a d e x^3+\frac {1}{5} \left (c d^2+a e^2\right ) x^5+\frac {2}{7} c d e x^7+\frac {1}{9} c e^2 x^9 \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1168} \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right ) \, dx=\frac {1}{5} x^5 \left (a e^2+c d^2\right )+a d^2 x+\frac {2}{3} a d e x^3+\frac {2}{7} c d e x^7+\frac {1}{9} c e^2 x^9 \]

[In]

Int[(d + e*x^2)^2*(a + c*x^4),x]

[Out]

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

Rule 1168

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right ) \, dx=a d^2 x+\frac {2}{3} a d e x^3+\frac {1}{5} \left (c d^2+a e^2\right ) x^5+\frac {2}{7} c d e x^7+\frac {1}{9} c e^2 x^9 \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88

method	result	size
default	\(a \,d^{2} x +\frac {2 a d e \,x^{3}}{3}+\frac {\left (a \,e^{2}+c \,d^{2}\right ) x^{5}}{5}+\frac {2 c d e \,x^{7}}{7}+\frac {c \,e^{2} x^{9}}{9}\)	\(49\)
norman	\(\frac {c \,e^{2} x^{9}}{9}+\frac {2 c d e \,x^{7}}{7}+\left (\frac {a \,e^{2}}{5}+\frac {c \,d^{2}}{5}\right ) x^{5}+\frac {2 a d e \,x^{3}}{3}+a \,d^{2} x\)	\(50\)
gosper	\(\frac {1}{9} c \,e^{2} x^{9}+\frac {2}{7} c d e \,x^{7}+\frac {1}{5} x^{5} a \,e^{2}+\frac {1}{5} x^{5} c \,d^{2}+\frac {2}{3} a d e \,x^{3}+a \,d^{2} x\)	\(51\)
risch	\(\frac {1}{9} c \,e^{2} x^{9}+\frac {2}{7} c d e \,x^{7}+\frac {1}{5} x^{5} a \,e^{2}+\frac {1}{5} x^{5} c \,d^{2}+\frac {2}{3} a d e \,x^{3}+a \,d^{2} x\)	\(51\)
parallelrisch	\(\frac {1}{9} c \,e^{2} x^{9}+\frac {2}{7} c d e \,x^{7}+\frac {1}{5} x^{5} a \,e^{2}+\frac {1}{5} x^{5} c \,d^{2}+\frac {2}{3} a d e \,x^{3}+a \,d^{2} x\)	\(51\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86 \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right ) \, dx=\frac {1}{9} \, c e^{2} x^{9} + \frac {2}{7} \, c d e x^{7} + \frac {2}{3} \, a d e x^{3} + \frac {1}{5} \, {\left (c d^{2} + a e^{2}\right )} x^{5} + a d^{2} x \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right ) \, dx=a d^{2} x + \frac {2 a d e x^{3}}{3} + \frac {2 c d e x^{7}}{7} + \frac {c e^{2} x^{9}}{9} + x^{5} \left (\frac {a e^{2}}{5} + \frac {c d^{2}}{5}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86 \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right ) \, dx=\frac {1}{9} \, c e^{2} x^{9} + \frac {2}{7} \, c d e x^{7} + \frac {2}{3} \, a d e x^{3} + \frac {1}{5} \, {\left (c d^{2} + a e^{2}\right )} x^{5} + a d^{2} x \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.89 \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right ) \, dx=\frac {1}{9} \, c e^{2} x^{9} + \frac {2}{7} \, c d e x^{7} + \frac {1}{5} \, c d^{2} x^{5} + \frac {1}{5} \, a e^{2} x^{5} + \frac {2}{3} \, a d e x^{3} + a d^{2} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88 \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right ) \, dx=x^5\,\left (\frac {c\,d^2}{5}+\frac {a\,e^2}{5}\right )+\frac {c\,e^2\,x^9}{9}+a\,d^2\,x+\frac {2\,a\,d\,e\,x^3}{3}+\frac {2\,c\,d\,e\,x^7}{7} \]

[In]

int((a + c*x^4)*(d + e*x^2)^2,x)

[Out]

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