\(\int (d+e x^2)^3 (a+b x^2+c x^4) \, dx\) [245]

Optimal result

Integrand size = 22, antiderivative size = 103 \[ \int \left (d+e x^2\right )^3 \left (a+b x^2+c x^4\right ) \, dx=a d^3 x+\frac {1}{3} d^2 (b d+3 a e) x^3+\frac {1}{5} d \left (c d^2+3 e (b d+a e)\right ) x^5+\frac {1}{7} e \left (3 c d^2+e (3 b d+a e)\right ) x^7+\frac {1}{9} e^2 (3 c d+b e) x^9+\frac {1}{11} c e^3 x^{11} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 103, 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} \[ \int \left (d+e x^2\right )^3 \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{7} e x^7 \left (e (a e+3 b d)+3 c d^2\right )+\frac {1}{5} d x^5 \left (3 e (a e+b d)+c d^2\right )+\frac {1}{3} d^2 x^3 (3 a e+b d)+a d^3 x+\frac {1}{9} e^2 x^9 (b e+3 c d)+\frac {1}{11} c e^3 x^{11} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.01 \[ \int \left (d+e x^2\right )^3 \left (a+b x^2+c x^4\right ) \, dx=a d^3 x+\frac {1}{3} d^2 (b d+3 a e) x^3+\frac {1}{5} d \left (c d^2+3 b d e+3 a e^2\right ) x^5+\frac {1}{7} e \left (3 c d^2+3 b d e+a e^2\right ) x^7+\frac {1}{9} e^2 (3 c d+b e) x^9+\frac {1}{11} c e^3 x^{11} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.99

method	result	size
norman	\(\frac {c \,e^{3} x^{11}}{11}+\left (\frac {1}{9} e^{3} b +\frac {1}{3} d \,e^{2} c \right ) x^{9}+\left (\frac {1}{7} a \,e^{3}+\frac {3}{7} d \,e^{2} b +\frac {3}{7} c \,d^{2} e \right ) x^{7}+\left (\frac {3}{5} d \,e^{2} a +\frac {3}{5} d^{2} e b +\frac {1}{5} d^{3} c \right ) x^{5}+\left (d^{2} e a +\frac {1}{3} d^{3} b \right ) x^{3}+a \,d^{3} x\)	\(102\)
default	\(\frac {c \,e^{3} x^{11}}{11}+\frac {\left (e^{3} b +3 d \,e^{2} c \right ) x^{9}}{9}+\frac {\left (a \,e^{3}+3 d \,e^{2} b +3 c \,d^{2} e \right ) x^{7}}{7}+\frac {\left (3 d \,e^{2} a +3 d^{2} e b +d^{3} c \right ) x^{5}}{5}+\frac {\left (3 d^{2} e a +d^{3} b \right ) x^{3}}{3}+a \,d^{3} x\)	\(103\)
gosper	\(\frac {1}{11} c \,e^{3} x^{11}+\frac {1}{9} x^{9} e^{3} b +\frac {1}{3} c d \,e^{2} x^{9}+\frac {1}{7} x^{7} a \,e^{3}+\frac {3}{7} x^{7} d \,e^{2} b +\frac {3}{7} x^{7} c \,d^{2} e +\frac {3}{5} x^{5} d \,e^{2} a +\frac {3}{5} x^{5} d^{2} e b +\frac {1}{5} x^{5} d^{3} c +a \,d^{2} e \,x^{3}+\frac {1}{3} x^{3} d^{3} b +a \,d^{3} x\)	\(112\)
risch	\(\frac {1}{11} c \,e^{3} x^{11}+\frac {1}{9} x^{9} e^{3} b +\frac {1}{3} c d \,e^{2} x^{9}+\frac {1}{7} x^{7} a \,e^{3}+\frac {3}{7} x^{7} d \,e^{2} b +\frac {3}{7} x^{7} c \,d^{2} e +\frac {3}{5} x^{5} d \,e^{2} a +\frac {3}{5} x^{5} d^{2} e b +\frac {1}{5} x^{5} d^{3} c +a \,d^{2} e \,x^{3}+\frac {1}{3} x^{3} d^{3} b +a \,d^{3} x\)	\(112\)
parallelrisch	\(\frac {1}{11} c \,e^{3} x^{11}+\frac {1}{9} x^{9} e^{3} b +\frac {1}{3} c d \,e^{2} x^{9}+\frac {1}{7} x^{7} a \,e^{3}+\frac {3}{7} x^{7} d \,e^{2} b +\frac {3}{7} x^{7} c \,d^{2} e +\frac {3}{5} x^{5} d \,e^{2} a +\frac {3}{5} x^{5} d^{2} e b +\frac {1}{5} x^{5} d^{3} c +a \,d^{2} e \,x^{3}+\frac {1}{3} x^{3} d^{3} b +a \,d^{3} x\)	\(112\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.99 \[ \int \left (d+e x^2\right )^3 \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{11} \, c e^{3} x^{11} + \frac {1}{9} \, {\left (3 \, c d e^{2} + b e^{3}\right )} x^{9} + \frac {1}{7} \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} x^{7} + \frac {1}{5} \, {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} x^{5} + a d^{3} x + \frac {1}{3} \, {\left (b d^{3} + 3 \, a d^{2} e\right )} x^{3} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.09 \[ \int \left (d+e x^2\right )^3 \left (a+b x^2+c x^4\right ) \, dx=a d^{3} x + \frac {c e^{3} x^{11}}{11} + x^{9} \left (\frac {b e^{3}}{9} + \frac {c d e^{2}}{3}\right ) + x^{7} \left (\frac {a e^{3}}{7} + \frac {3 b d e^{2}}{7} + \frac {3 c d^{2} e}{7}\right ) + x^{5} \cdot \left (\frac {3 a d e^{2}}{5} + \frac {3 b d^{2} e}{5} + \frac {c d^{3}}{5}\right ) + x^{3} \left (a d^{2} e + \frac {b d^{3}}{3}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.99 \[ \int \left (d+e x^2\right )^3 \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{11} \, c e^{3} x^{11} + \frac {1}{9} \, {\left (3 \, c d e^{2} + b e^{3}\right )} x^{9} + \frac {1}{7} \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} x^{7} + \frac {1}{5} \, {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} x^{5} + a d^{3} x + \frac {1}{3} \, {\left (b d^{3} + 3 \, a d^{2} e\right )} x^{3} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.08 \[ \int \left (d+e x^2\right )^3 \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{11} \, c e^{3} x^{11} + \frac {1}{3} \, c d e^{2} x^{9} + \frac {1}{9} \, b e^{3} x^{9} + \frac {3}{7} \, c d^{2} e x^{7} + \frac {3}{7} \, b d e^{2} x^{7} + \frac {1}{7} \, a e^{3} x^{7} + \frac {1}{5} \, c d^{3} x^{5} + \frac {3}{5} \, b d^{2} e x^{5} + \frac {3}{5} \, a d e^{2} x^{5} + \frac {1}{3} \, b d^{3} x^{3} + a d^{2} e x^{3} + a d^{3} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 7.49 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.98 \[ \int \left (d+e x^2\right )^3 \left (a+b x^2+c x^4\right ) \, dx=x^3\,\left (\frac {b\,d^3}{3}+a\,e\,d^2\right )+x^9\,\left (\frac {b\,e^3}{9}+\frac {c\,d\,e^2}{3}\right )+x^5\,\left (\frac {c\,d^3}{5}+\frac {3\,b\,d^2\,e}{5}+\frac {3\,a\,d\,e^2}{5}\right )+x^7\,\left (\frac {3\,c\,d^2\,e}{7}+\frac {3\,b\,d\,e^2}{7}+\frac {a\,e^3}{7}\right )+\frac {c\,e^3\,x^{11}}{11}+a\,d^3\,x \]

[In]

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

[Out]

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