\(\int (d+e x^2)^3 (a+c x^4) \, dx\) [121]

Optimal result

Integrand size = 17, antiderivative size = 79 \[ \int \left (d+e x^2\right )^3 \left (a+c x^4\right ) \, dx=a d^3 x+a d^2 e x^3+\frac {1}{5} d \left (c d^2+3 a e^2\right ) x^5+\frac {1}{7} e \left (3 c d^2+a e^2\right ) x^7+\frac {1}{3} c d e^2 x^9+\frac {1}{11} c e^3 x^{11} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 79, 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} \[ \int \left (d+e x^2\right )^3 \left (a+c x^4\right ) \, dx=\frac {1}{7} e x^7 \left (a e^2+3 c d^2\right )+\frac {1}{5} d x^5 \left (3 a e^2+c d^2\right )+a d^3 x+a d^2 e x^3+\frac {1}{3} c d e^2 x^9+\frac {1}{11} c e^3 x^{11} \]

[In]

Int[(d + e*x^2)^3*(a + c*x^4),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^2\right )^3 \left (a+c x^4\right ) \, dx=a d^3 x+a d^2 e x^3+\frac {1}{5} d \left (c d^2+3 a e^2\right ) x^5+\frac {1}{7} e \left (3 c d^2+a e^2\right ) x^7+\frac {1}{3} c d e^2 x^9+\frac {1}{11} c e^3 x^{11} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.91

method	result	size
default	\(\frac {c \,e^{3} x^{11}}{11}+\frac {c d \,e^{2} x^{9}}{3}+\frac {\left (a \,e^{3}+3 c \,d^{2} e \right ) x^{7}}{7}+\frac {\left (3 d \,e^{2} a +d^{3} c \right ) x^{5}}{5}+a \,d^{2} e \,x^{3}+a \,d^{3} x\)	\(72\)
norman	\(\frac {c \,e^{3} x^{11}}{11}+\frac {c d \,e^{2} x^{9}}{3}+\left (\frac {1}{7} a \,e^{3}+\frac {3}{7} c \,d^{2} e \right ) x^{7}+\left (\frac {3}{5} d \,e^{2} a +\frac {1}{5} d^{3} c \right ) x^{5}+a \,d^{2} e \,x^{3}+a \,d^{3} x\)	\(72\)
gosper	\(\frac {1}{11} c \,e^{3} x^{11}+\frac {1}{3} c d \,e^{2} x^{9}+\frac {1}{7} x^{7} a \,e^{3}+\frac {3}{7} x^{7} c \,d^{2} e +\frac {3}{5} x^{5} d \,e^{2} a +\frac {1}{5} x^{5} d^{3} c +a \,d^{2} e \,x^{3}+a \,d^{3} x\)	\(74\)
risch	\(\frac {1}{11} c \,e^{3} x^{11}+\frac {1}{3} c d \,e^{2} x^{9}+\frac {1}{7} x^{7} a \,e^{3}+\frac {3}{7} x^{7} c \,d^{2} e +\frac {3}{5} x^{5} d \,e^{2} a +\frac {1}{5} x^{5} d^{3} c +a \,d^{2} e \,x^{3}+a \,d^{3} x\)	\(74\)
parallelrisch	\(\frac {1}{11} c \,e^{3} x^{11}+\frac {1}{3} c d \,e^{2} x^{9}+\frac {1}{7} x^{7} a \,e^{3}+\frac {3}{7} x^{7} c \,d^{2} e +\frac {3}{5} x^{5} d \,e^{2} a +\frac {1}{5} x^{5} d^{3} c +a \,d^{2} e \,x^{3}+a \,d^{3} x\)	\(74\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.90 \[ \int \left (d+e x^2\right )^3 \left (a+c x^4\right ) \, dx=\frac {1}{11} \, c e^{3} x^{11} + \frac {1}{3} \, c d e^{2} x^{9} + \frac {1}{7} \, {\left (3 \, c d^{2} e + a e^{3}\right )} x^{7} + a d^{2} e x^{3} + \frac {1}{5} \, {\left (c d^{3} + 3 \, a d e^{2}\right )} x^{5} + a d^{3} x \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.99 \[ \int \left (d+e x^2\right )^3 \left (a+c x^4\right ) \, dx=a d^{3} x + a d^{2} e x^{3} + \frac {c d e^{2} x^{9}}{3} + \frac {c e^{3} x^{11}}{11} + x^{7} \left (\frac {a e^{3}}{7} + \frac {3 c d^{2} e}{7}\right ) + x^{5} \cdot \left (\frac {3 a d e^{2}}{5} + \frac {c d^{3}}{5}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.90 \[ \int \left (d+e x^2\right )^3 \left (a+c x^4\right ) \, dx=\frac {1}{11} \, c e^{3} x^{11} + \frac {1}{3} \, c d e^{2} x^{9} + \frac {1}{7} \, {\left (3 \, c d^{2} e + a e^{3}\right )} x^{7} + a d^{2} e x^{3} + \frac {1}{5} \, {\left (c d^{3} + 3 \, a d e^{2}\right )} x^{5} + a d^{3} x \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.92 \[ \int \left (d+e x^2\right )^3 \left (a+c x^4\right ) \, dx=\frac {1}{11} \, c e^{3} x^{11} + \frac {1}{3} \, c d e^{2} x^{9} + \frac {3}{7} \, c d^{2} e x^{7} + \frac {1}{7} \, a e^{3} x^{7} + \frac {1}{5} \, c d^{3} x^{5} + \frac {3}{5} \, a d e^{2} x^{5} + a d^{2} e x^{3} + a d^{3} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 13.90 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.90 \[ \int \left (d+e x^2\right )^3 \left (a+c x^4\right ) \, dx=x^5\,\left (\frac {c\,d^3}{5}+\frac {3\,a\,d\,e^2}{5}\right )+x^7\,\left (\frac {3\,c\,d^2\,e}{7}+\frac {a\,e^3}{7}\right )+\frac {c\,e^3\,x^{11}}{11}+a\,d^3\,x+a\,d^2\,e\,x^3+\frac {c\,d\,e^2\,x^9}{3} \]

[In]

int((a + c*x^4)*(d + e*x^2)^3,x)

[Out]

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