\(\int (d+e x^2) (a+c x^4)^2 \, dx\) [130]

Optimal result

Integrand size = 17, antiderivative size = 60 \[ \int \left (d+e x^2\right ) \left (a+c x^4\right )^2 \, dx=a^2 d x+\frac {1}{3} a^2 e x^3+\frac {2}{5} a c d x^5+\frac {2}{7} a c e x^7+\frac {1}{9} c^2 d x^9+\frac {1}{11} c^2 e x^{11} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 60, 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} \[ \int \left (d+e x^2\right ) \left (a+c x^4\right )^2 \, dx=a^2 d x+\frac {1}{3} a^2 e x^3+\frac {2}{5} a c d x^5+\frac {2}{7} a c e x^7+\frac {1}{9} c^2 d x^9+\frac {1}{11} c^2 e x^{11} \]

[In]

Int[(d + e*x^2)*(a + c*x^4)^2,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^2\right ) \left (a+c x^4\right )^2 \, dx=a^2 d x+\frac {1}{3} a^2 e x^3+\frac {2}{5} a c d x^5+\frac {2}{7} a c e x^7+\frac {1}{9} c^2 d x^9+\frac {1}{11} c^2 e x^{11} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.85

method	result	size
gosper	\(a^{2} d x +\frac {1}{3} a^{2} e \,x^{3}+\frac {2}{5} a c d \,x^{5}+\frac {2}{7} a c e \,x^{7}+\frac {1}{9} c^{2} d \,x^{9}+\frac {1}{11} c^{2} e \,x^{11}\)	\(51\)
default	\(a^{2} d x +\frac {1}{3} a^{2} e \,x^{3}+\frac {2}{5} a c d \,x^{5}+\frac {2}{7} a c e \,x^{7}+\frac {1}{9} c^{2} d \,x^{9}+\frac {1}{11} c^{2} e \,x^{11}\)	\(51\)
norman	\(a^{2} d x +\frac {1}{3} a^{2} e \,x^{3}+\frac {2}{5} a c d \,x^{5}+\frac {2}{7} a c e \,x^{7}+\frac {1}{9} c^{2} d \,x^{9}+\frac {1}{11} c^{2} e \,x^{11}\)	\(51\)
risch	\(a^{2} d x +\frac {1}{3} a^{2} e \,x^{3}+\frac {2}{5} a c d \,x^{5}+\frac {2}{7} a c e \,x^{7}+\frac {1}{9} c^{2} d \,x^{9}+\frac {1}{11} c^{2} e \,x^{11}\)	\(51\)
parallelrisch	\(a^{2} d x +\frac {1}{3} a^{2} e \,x^{3}+\frac {2}{5} a c d \,x^{5}+\frac {2}{7} a c e \,x^{7}+\frac {1}{9} c^{2} d \,x^{9}+\frac {1}{11} c^{2} e \,x^{11}\)	\(51\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.83 \[ \int \left (d+e x^2\right ) \left (a+c x^4\right )^2 \, dx=\frac {1}{11} \, c^{2} e x^{11} + \frac {1}{9} \, c^{2} d x^{9} + \frac {2}{7} \, a c e x^{7} + \frac {2}{5} \, a c d x^{5} + \frac {1}{3} \, a^{2} e x^{3} + a^{2} d x \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^2\right ) \left (a+c x^4\right )^2 \, dx=a^{2} d x + \frac {a^{2} e x^{3}}{3} + \frac {2 a c d x^{5}}{5} + \frac {2 a c e x^{7}}{7} + \frac {c^{2} d x^{9}}{9} + \frac {c^{2} e x^{11}}{11} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.83 \[ \int \left (d+e x^2\right ) \left (a+c x^4\right )^2 \, dx=\frac {1}{11} \, c^{2} e x^{11} + \frac {1}{9} \, c^{2} d x^{9} + \frac {2}{7} \, a c e x^{7} + \frac {2}{5} \, a c d x^{5} + \frac {1}{3} \, a^{2} e x^{3} + a^{2} d x \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.83 \[ \int \left (d+e x^2\right ) \left (a+c x^4\right )^2 \, dx=\frac {1}{11} \, c^{2} e x^{11} + \frac {1}{9} \, c^{2} d x^{9} + \frac {2}{7} \, a c e x^{7} + \frac {2}{5} \, a c d x^{5} + \frac {1}{3} \, a^{2} e x^{3} + a^{2} d x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.83 \[ \int \left (d+e x^2\right ) \left (a+c x^4\right )^2 \, dx=\frac {e\,a^2\,x^3}{3}+d\,a^2\,x+\frac {2\,e\,a\,c\,x^7}{7}+\frac {2\,d\,a\,c\,x^5}{5}+\frac {e\,c^2\,x^{11}}{11}+\frac {d\,c^2\,x^9}{9} \]

[In]

int((a + c*x^4)^2*(d + e*x^2),x)

[Out]

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