\(\int (d+e x^2) (a+b x^2+c x^4)^2 \, dx\) [254]

Optimal result

Integrand size = 22, antiderivative size = 96 \[ \int \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=a^2 d x+\frac {1}{3} a (2 b d+a e) x^3+\frac {1}{5} \left (b^2 d+2 a c d+2 a b e\right ) x^5+\frac {1}{7} \left (2 b c d+b^2 e+2 a c e\right ) x^7+\frac {1}{9} c (c d+2 b e) x^9+\frac {1}{11} c^2 e x^{11} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 96, 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} \[ \int \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=a^2 d x+\frac {1}{7} x^7 \left (2 a c e+b^2 e+2 b c d\right )+\frac {1}{5} x^5 \left (2 a b e+2 a c d+b^2 d\right )+\frac {1}{3} a x^3 (a e+2 b d)+\frac {1}{9} c x^9 (2 b e+c d)+\frac {1}{11} c^2 e x^{11} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=a^2 d x+\frac {1}{3} a (2 b d+a e) x^3+\frac {1}{5} \left (b^2 d+2 a c d+2 a b e\right ) x^5+\frac {1}{7} \left (2 b c d+b^2 e+2 a c e\right ) x^7+\frac {1}{9} c (c d+2 b e) x^9+\frac {1}{11} c^2 e x^{11} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.95

method	result	size
default	\(\frac {c^{2} e \,x^{11}}{11}+\frac {\left (2 e b c +c^{2} d \right ) x^{9}}{9}+\frac {\left (2 b c d +e \left (2 a c +b^{2}\right )\right ) x^{7}}{7}+\frac {\left (d \left (2 a c +b^{2}\right )+2 a b e \right ) x^{5}}{5}+\frac {\left (e \,a^{2}+2 d a b \right ) x^{3}}{3}+a^{2} d x\)	\(91\)
norman	\(\frac {c^{2} e \,x^{11}}{11}+\left (\frac {2}{9} e b c +\frac {1}{9} c^{2} d \right ) x^{9}+\left (\frac {2}{7} a c e +\frac {1}{7} b^{2} e +\frac {2}{7} b c d \right ) x^{7}+\left (\frac {2}{5} a b e +\frac {2}{5} a c d +\frac {1}{5} b^{2} d \right ) x^{5}+\left (\frac {1}{3} e \,a^{2}+\frac {2}{3} d a b \right ) x^{3}+a^{2} d x\)	\(91\)
gosper	\(\frac {1}{11} c^{2} e \,x^{11}+\frac {2}{9} x^{9} e b c +\frac {1}{9} c^{2} d \,x^{9}+\frac {2}{7} a c e \,x^{7}+\frac {1}{7} x^{7} b^{2} e +\frac {2}{7} x^{7} b c d +\frac {2}{5} x^{5} a b e +\frac {2}{5} a c d \,x^{5}+\frac {1}{5} x^{5} b^{2} d +\frac {1}{3} a^{2} e \,x^{3}+\frac {2}{3} x^{3} d a b +a^{2} d x\)	\(101\)
risch	\(\frac {1}{11} c^{2} e \,x^{11}+\frac {2}{9} x^{9} e b c +\frac {1}{9} c^{2} d \,x^{9}+\frac {2}{7} a c e \,x^{7}+\frac {1}{7} x^{7} b^{2} e +\frac {2}{7} x^{7} b c d +\frac {2}{5} x^{5} a b e +\frac {2}{5} a c d \,x^{5}+\frac {1}{5} x^{5} b^{2} d +\frac {1}{3} a^{2} e \,x^{3}+\frac {2}{3} x^{3} d a b +a^{2} d x\)	\(101\)
parallelrisch	\(\frac {1}{11} c^{2} e \,x^{11}+\frac {2}{9} x^{9} e b c +\frac {1}{9} c^{2} d \,x^{9}+\frac {2}{7} a c e \,x^{7}+\frac {1}{7} x^{7} b^{2} e +\frac {2}{7} x^{7} b c d +\frac {2}{5} x^{5} a b e +\frac {2}{5} a c d \,x^{5}+\frac {1}{5} x^{5} b^{2} d +\frac {1}{3} a^{2} e \,x^{3}+\frac {2}{3} x^{3} d a b +a^{2} d x\)	\(101\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94 \[ \int \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{11} \, c^{2} e x^{11} + \frac {1}{9} \, {\left (c^{2} d + 2 \, b c e\right )} x^{9} + \frac {1}{7} \, {\left (2 \, b c d + {\left (b^{2} + 2 \, a c\right )} e\right )} x^{7} + \frac {1}{5} \, {\left (2 \, a b e + {\left (b^{2} + 2 \, a c\right )} d\right )} x^{5} + a^{2} d x + \frac {1}{3} \, {\left (2 \, a b d + a^{2} e\right )} x^{3} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.11 \[ \int \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=a^{2} d x + \frac {c^{2} e x^{11}}{11} + x^{9} \cdot \left (\frac {2 b c e}{9} + \frac {c^{2} d}{9}\right ) + x^{7} \cdot \left (\frac {2 a c e}{7} + \frac {b^{2} e}{7} + \frac {2 b c d}{7}\right ) + x^{5} \cdot \left (\frac {2 a b e}{5} + \frac {2 a c d}{5} + \frac {b^{2} d}{5}\right ) + x^{3} \left (\frac {a^{2} e}{3} + \frac {2 a b d}{3}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94 \[ \int \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{11} \, c^{2} e x^{11} + \frac {1}{9} \, {\left (c^{2} d + 2 \, b c e\right )} x^{9} + \frac {1}{7} \, {\left (2 \, b c d + {\left (b^{2} + 2 \, a c\right )} e\right )} x^{7} + \frac {1}{5} \, {\left (2 \, a b e + {\left (b^{2} + 2 \, a c\right )} d\right )} x^{5} + a^{2} d x + \frac {1}{3} \, {\left (2 \, a b d + a^{2} e\right )} x^{3} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.04 \[ \int \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=\frac {1}{11} \, c^{2} e x^{11} + \frac {1}{9} \, c^{2} d x^{9} + \frac {2}{9} \, b c e x^{9} + \frac {2}{7} \, b c d x^{7} + \frac {1}{7} \, b^{2} e x^{7} + \frac {2}{7} \, a c e x^{7} + \frac {1}{5} \, b^{2} d x^{5} + \frac {2}{5} \, a c d x^{5} + \frac {2}{5} \, a b e x^{5} + \frac {2}{3} \, a b d x^{3} + \frac {1}{3} \, a^{2} e x^{3} + a^{2} d x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94 \[ \int \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx=x^5\,\left (\frac {d\,b^2}{5}+\frac {2\,a\,e\,b}{5}+\frac {2\,a\,c\,d}{5}\right )+x^7\,\left (\frac {e\,b^2}{7}+\frac {2\,c\,d\,b}{7}+\frac {2\,a\,c\,e}{7}\right )+x^3\,\left (\frac {e\,a^2}{3}+\frac {2\,b\,d\,a}{3}\right )+x^9\,\left (\frac {d\,c^2}{9}+\frac {2\,b\,e\,c}{9}\right )+\frac {c^2\,e\,x^{11}}{11}+a^2\,d\,x \]

[In]

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

[Out]

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