\(\int (d+e x^2)^4 (a+c x^4) \, dx\) [120]

Optimal result

Integrand size = 17, antiderivative size = 106 \[ \int \left (d+e x^2\right )^4 \left (a+c x^4\right ) \, dx=a d^4 x+\frac {4}{3} a d^3 e x^3+\frac {1}{5} d^2 \left (c d^2+6 a e^2\right ) x^5+\frac {4}{7} d e \left (c d^2+a e^2\right ) x^7+\frac {1}{9} e^2 \left (6 c d^2+a e^2\right ) x^9+\frac {4}{11} c d e^3 x^{11}+\frac {1}{13} c e^4 x^{13} \]

[Out]

a*d^4*x+4/3*a*d^3*e*x^3+1/5*d^2*(6*a*e^2+c*d^2)*x^5+4/7*d*e*(a*e^2+c*d^2)*x^7+1/9*e^2*(a*e^2+6*c*d^2)*x^9+4/11
*c*d*e^3*x^11+1/13*c*e^4*x^13

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 106, 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} \[ \int \left (d+e x^2\right )^4 \left (a+c x^4\right ) \, dx=\frac {1}{9} e^2 x^9 \left (a e^2+6 c d^2\right )+\frac {4}{7} d e x^7 \left (a e^2+c d^2\right )+\frac {1}{5} d^2 x^5 \left (6 a e^2+c d^2\right )+a d^4 x+\frac {4}{3} a d^3 e x^3+\frac {4}{11} c d e^3 x^{11}+\frac {1}{13} c e^4 x^{13} \]

[In]

Int[(d + e*x^2)^4*(a + c*x^4),x]

[Out]

a*d^4*x + (4*a*d^3*e*x^3)/3 + (d^2*(c*d^2 + 6*a*e^2)*x^5)/5 + (4*d*e*(c*d^2 + a*e^2)*x^7)/7 + (e^2*(6*c*d^2 +
a*e^2)*x^9)/9 + (4*c*d*e^3*x^11)/11 + (c*e^4*x^13)/13

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^4+4 a d^3 e x^2+d^2 \left (c d^2+6 a e^2\right ) x^4+4 d e \left (c d^2+a e^2\right ) x^6+e^2 \left (6 c d^2+a e^2\right ) x^8+4 c d e^3 x^{10}+c e^4 x^{12}\right ) \, dx \\ & = a d^4 x+\frac {4}{3} a d^3 e x^3+\frac {1}{5} d^2 \left (c d^2+6 a e^2\right ) x^5+\frac {4}{7} d e \left (c d^2+a e^2\right ) x^7+\frac {1}{9} e^2 \left (6 c d^2+a e^2\right ) x^9+\frac {4}{11} c d e^3 x^{11}+\frac {1}{13} c e^4 x^{13} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^2\right )^4 \left (a+c x^4\right ) \, dx=a d^4 x+\frac {4}{3} a d^3 e x^3+\frac {1}{5} d^2 \left (c d^2+6 a e^2\right ) x^5+\frac {4}{7} d e \left (c d^2+a e^2\right ) x^7+\frac {1}{9} e^2 \left (6 c d^2+a e^2\right ) x^9+\frac {4}{11} c d e^3 x^{11}+\frac {1}{13} c e^4 x^{13} \]

[In]

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

[Out]

a*d^4*x + (4*a*d^3*e*x^3)/3 + (d^2*(c*d^2 + 6*a*e^2)*x^5)/5 + (4*d*e*(c*d^2 + a*e^2)*x^7)/7 + (e^2*(6*c*d^2 +
a*e^2)*x^9)/9 + (4*c*d*e^3*x^11)/11 + (c*e^4*x^13)/13

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.91

method	result	size
norman	\(\frac {c \,e^{4} x^{13}}{13}+\frac {4 c d \,e^{3} x^{11}}{11}+\left (\frac {1}{9} e^{4} a +\frac {2}{3} e^{2} d^{2} c \right ) x^{9}+\left (\frac {4}{7} d \,e^{3} a +\frac {4}{7} d^{3} e c \right ) x^{7}+\left (\frac {6}{5} e^{2} d^{2} a +\frac {1}{5} d^{4} c \right ) x^{5}+\frac {4 a \,d^{3} e \,x^{3}}{3}+a \,d^{4} x\)	\(96\)
default	\(\frac {c \,e^{4} x^{13}}{13}+\frac {4 c d \,e^{3} x^{11}}{11}+\frac {\left (e^{4} a +6 e^{2} d^{2} c \right ) x^{9}}{9}+\frac {\left (4 d \,e^{3} a +4 d^{3} e c \right ) x^{7}}{7}+\frac {\left (6 e^{2} d^{2} a +d^{4} c \right ) x^{5}}{5}+\frac {4 a \,d^{3} e \,x^{3}}{3}+a \,d^{4} x\)	\(97\)
gosper	\(\frac {1}{13} c \,e^{4} x^{13}+\frac {4}{11} c d \,e^{3} x^{11}+\frac {1}{9} x^{9} e^{4} a +\frac {2}{3} x^{9} e^{2} d^{2} c +\frac {4}{7} x^{7} d \,e^{3} a +\frac {4}{7} x^{7} d^{3} e c +\frac {6}{5} x^{5} e^{2} d^{2} a +\frac {1}{5} x^{5} d^{4} c +\frac {4}{3} a \,d^{3} e \,x^{3}+a \,d^{4} x\)	\(99\)
risch	\(\frac {1}{13} c \,e^{4} x^{13}+\frac {4}{11} c d \,e^{3} x^{11}+\frac {1}{9} x^{9} e^{4} a +\frac {2}{3} x^{9} e^{2} d^{2} c +\frac {4}{7} x^{7} d \,e^{3} a +\frac {4}{7} x^{7} d^{3} e c +\frac {6}{5} x^{5} e^{2} d^{2} a +\frac {1}{5} x^{5} d^{4} c +\frac {4}{3} a \,d^{3} e \,x^{3}+a \,d^{4} x\)	\(99\)
parallelrisch	\(\frac {1}{13} c \,e^{4} x^{13}+\frac {4}{11} c d \,e^{3} x^{11}+\frac {1}{9} x^{9} e^{4} a +\frac {2}{3} x^{9} e^{2} d^{2} c +\frac {4}{7} x^{7} d \,e^{3} a +\frac {4}{7} x^{7} d^{3} e c +\frac {6}{5} x^{5} e^{2} d^{2} a +\frac {1}{5} x^{5} d^{4} c +\frac {4}{3} a \,d^{3} e \,x^{3}+a \,d^{4} x\)	\(99\)

[In]

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

[Out]

1/13*c*e^4*x^13+4/11*c*d*e^3*x^11+(1/9*e^4*a+2/3*e^2*d^2*c)*x^9+(4/7*d*e^3*a+4/7*d^3*e*c)*x^7+(6/5*e^2*d^2*a+1
/5*d^4*c)*x^5+4/3*a*d^3*e*x^3+a*d^4*x

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.89 \[ \int \left (d+e x^2\right )^4 \left (a+c x^4\right ) \, dx=\frac {1}{13} \, c e^{4} x^{13} + \frac {4}{11} \, c d e^{3} x^{11} + \frac {1}{9} \, {\left (6 \, c d^{2} e^{2} + a e^{4}\right )} x^{9} + \frac {4}{3} \, a d^{3} e x^{3} + \frac {4}{7} \, {\left (c d^{3} e + a d e^{3}\right )} x^{7} + a d^{4} x + \frac {1}{5} \, {\left (c d^{4} + 6 \, a d^{2} e^{2}\right )} x^{5} \]

[In]

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

[Out]

1/13*c*e^4*x^13 + 4/11*c*d*e^3*x^11 + 1/9*(6*c*d^2*e^2 + a*e^4)*x^9 + 4/3*a*d^3*e*x^3 + 4/7*(c*d^3*e + a*d*e^3
)*x^7 + a*d^4*x + 1/5*(c*d^4 + 6*a*d^2*e^2)*x^5

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.04 \[ \int \left (d+e x^2\right )^4 \left (a+c x^4\right ) \, dx=a d^{4} x + \frac {4 a d^{3} e x^{3}}{3} + \frac {4 c d e^{3} x^{11}}{11} + \frac {c e^{4} x^{13}}{13} + x^{9} \left (\frac {a e^{4}}{9} + \frac {2 c d^{2} e^{2}}{3}\right ) + x^{7} \cdot \left (\frac {4 a d e^{3}}{7} + \frac {4 c d^{3} e}{7}\right ) + x^{5} \cdot \left (\frac {6 a d^{2} e^{2}}{5} + \frac {c d^{4}}{5}\right ) \]

[In]

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

[Out]

a*d**4*x + 4*a*d**3*e*x**3/3 + 4*c*d*e**3*x**11/11 + c*e**4*x**13/13 + x**9*(a*e**4/9 + 2*c*d**2*e**2/3) + x**
7*(4*a*d*e**3/7 + 4*c*d**3*e/7) + x**5*(6*a*d**2*e**2/5 + c*d**4/5)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.89 \[ \int \left (d+e x^2\right )^4 \left (a+c x^4\right ) \, dx=\frac {1}{13} \, c e^{4} x^{13} + \frac {4}{11} \, c d e^{3} x^{11} + \frac {1}{9} \, {\left (6 \, c d^{2} e^{2} + a e^{4}\right )} x^{9} + \frac {4}{3} \, a d^{3} e x^{3} + \frac {4}{7} \, {\left (c d^{3} e + a d e^{3}\right )} x^{7} + a d^{4} x + \frac {1}{5} \, {\left (c d^{4} + 6 \, a d^{2} e^{2}\right )} x^{5} \]

[In]

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

[Out]

1/13*c*e^4*x^13 + 4/11*c*d*e^3*x^11 + 1/9*(6*c*d^2*e^2 + a*e^4)*x^9 + 4/3*a*d^3*e*x^3 + 4/7*(c*d^3*e + a*d*e^3
)*x^7 + a*d^4*x + 1/5*(c*d^4 + 6*a*d^2*e^2)*x^5

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.92 \[ \int \left (d+e x^2\right )^4 \left (a+c x^4\right ) \, dx=\frac {1}{13} \, c e^{4} x^{13} + \frac {4}{11} \, c d e^{3} x^{11} + \frac {2}{3} \, c d^{2} e^{2} x^{9} + \frac {1}{9} \, a e^{4} x^{9} + \frac {4}{7} \, c d^{3} e x^{7} + \frac {4}{7} \, a d e^{3} x^{7} + \frac {1}{5} \, c d^{4} x^{5} + \frac {6}{5} \, a d^{2} e^{2} x^{5} + \frac {4}{3} \, a d^{3} e x^{3} + a d^{4} x \]

[In]

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

[Out]

1/13*c*e^4*x^13 + 4/11*c*d*e^3*x^11 + 2/3*c*d^2*e^2*x^9 + 1/9*a*e^4*x^9 + 4/7*c*d^3*e*x^7 + 4/7*a*d*e^3*x^7 +
1/5*c*d^4*x^5 + 6/5*a*d^2*e^2*x^5 + 4/3*a*d^3*e*x^3 + a*d^4*x

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.90 \[ \int \left (d+e x^2\right )^4 \left (a+c x^4\right ) \, dx=x^5\,\left (\frac {c\,d^4}{5}+\frac {6\,a\,d^2\,e^2}{5}\right )+x^9\,\left (\frac {2\,c\,d^2\,e^2}{3}+\frac {a\,e^4}{9}\right )+x^7\,\left (\frac {4\,c\,d^3\,e}{7}+\frac {4\,a\,d\,e^3}{7}\right )+\frac {c\,e^4\,x^{13}}{13}+a\,d^4\,x+\frac {4\,a\,d^3\,e\,x^3}{3}+\frac {4\,c\,d\,e^3\,x^{11}}{11} \]

[In]

int((a + c*x^4)*(d + e*x^2)^4,x)

[Out]

x^5*((c*d^4)/5 + (6*a*d^2*e^2)/5) + x^9*((a*e^4)/9 + (2*c*d^2*e^2)/3) + x^7*((4*a*d*e^3)/7 + (4*c*d^3*e)/7) +
(c*e^4*x^13)/13 + a*d^4*x + (4*a*d^3*e*x^3)/3 + (4*c*d*e^3*x^11)/11