\(\int (a+c x^4)^2 \, dx\) [131]

Optimal result

Integrand size = 9, antiderivative size = 25 \[ \int \left (a+c x^4\right )^2 \, dx=a^2 x+\frac {2}{5} a c x^5+\frac {c^2 x^9}{9} \]

[Out]

a^2*x+2/5*a*c*x^5+1/9*c^2*x^9

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {200} \[ \int \left (a+c x^4\right )^2 \, dx=a^2 x+\frac {2}{5} a c x^5+\frac {c^2 x^9}{9} \]

[In]

Int[(a + c*x^4)^2,x]

[Out]

a^2*x + (2*a*c*x^5)/5 + (c^2*x^9)/9

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^2+2 a c x^4+c^2 x^8\right ) \, dx \\ & = a^2 x+\frac {2}{5} a c x^5+\frac {c^2 x^9}{9} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \left (a+c x^4\right )^2 \, dx=a^2 x+\frac {2}{5} a c x^5+\frac {c^2 x^9}{9} \]

[In]

Integrate[(a + c*x^4)^2,x]

[Out]

a^2*x + (2*a*c*x^5)/5 + (c^2*x^9)/9

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88

method	result	size
gosper	\(a^{2} x +\frac {2}{5} x^{5} a c +\frac {1}{9} c^{2} x^{9}\)	\(22\)
default	\(a^{2} x +\frac {2}{5} x^{5} a c +\frac {1}{9} c^{2} x^{9}\)	\(22\)
norman	\(a^{2} x +\frac {2}{5} x^{5} a c +\frac {1}{9} c^{2} x^{9}\)	\(22\)
risch	\(a^{2} x +\frac {2}{5} x^{5} a c +\frac {1}{9} c^{2} x^{9}\)	\(22\)
parallelrisch	\(a^{2} x +\frac {2}{5} x^{5} a c +\frac {1}{9} c^{2} x^{9}\)	\(22\)

[In]

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

[Out]

a^2*x+2/5*x^5*a*c+1/9*c^2*x^9

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \left (a+c x^4\right )^2 \, dx=\frac {1}{9} \, c^{2} x^{9} + \frac {2}{5} \, a c x^{5} + a^{2} x \]

[In]

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

[Out]

1/9*c^2*x^9 + 2/5*a*c*x^5 + a^2*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \left (a+c x^4\right )^2 \, dx=a^{2} x + \frac {2 a c x^{5}}{5} + \frac {c^{2} x^{9}}{9} \]

[In]

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

[Out]

a**2*x + 2*a*c*x**5/5 + c**2*x**9/9

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \left (a+c x^4\right )^2 \, dx=\frac {1}{9} \, c^{2} x^{9} + \frac {2}{5} \, a c x^{5} + a^{2} x \]

[In]

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

[Out]

1/9*c^2*x^9 + 2/5*a*c*x^5 + a^2*x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \left (a+c x^4\right )^2 \, dx=\frac {1}{9} \, c^{2} x^{9} + \frac {2}{5} \, a c x^{5} + a^{2} x \]

[In]

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

[Out]

1/9*c^2*x^9 + 2/5*a*c*x^5 + a^2*x

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \left (a+c x^4\right )^2 \, dx=a^2\,x+\frac {2\,a\,c\,x^5}{5}+\frac {c^2\,x^9}{9} \]

[In]

int((a + c*x^4)^2,x)

[Out]

a^2*x + (c^2*x^9)/9 + (2*a*c*x^5)/5