\(\int x (a+b x^4)^2 \, dx\) [625]

Optimal result

Integrand size = 11, antiderivative size = 30 \[ \int x \left (a+b x^4\right )^2 \, dx=\frac {a^2 x^2}{2}+\frac {1}{3} a b x^6+\frac {b^2 x^{10}}{10} \]

[Out]

1/2*a^2*x^2+1/3*a*b*x^6+1/10*b^2*x^10

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {276} \[ \int x \left (a+b x^4\right )^2 \, dx=\frac {a^2 x^2}{2}+\frac {1}{3} a b x^6+\frac {b^2 x^{10}}{10} \]

[In]

Int[x*(a + b*x^4)^2,x]

[Out]

(a^2*x^2)/2 + (a*b*x^6)/3 + (b^2*x^10)/10

Rule 276

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

Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 x+2 a b x^5+b^2 x^9\right ) \, dx \\ & = \frac {a^2 x^2}{2}+\frac {1}{3} a b x^6+\frac {b^2 x^{10}}{10} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int x \left (a+b x^4\right )^2 \, dx=\frac {a^2 x^2}{2}+\frac {1}{3} a b x^6+\frac {b^2 x^{10}}{10} \]

[In]

Integrate[x*(a + b*x^4)^2,x]

[Out]

(a^2*x^2)/2 + (a*b*x^6)/3 + (b^2*x^10)/10

Maple [A] (verified)

Time = 4.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83

method	result	size
gosper	\(\frac {1}{2} a^{2} x^{2}+\frac {1}{3} a b \,x^{6}+\frac {1}{10} b^{2} x^{10}\)	\(25\)
default	\(\frac {1}{2} a^{2} x^{2}+\frac {1}{3} a b \,x^{6}+\frac {1}{10} b^{2} x^{10}\)	\(25\)
norman	\(\frac {1}{2} a^{2} x^{2}+\frac {1}{3} a b \,x^{6}+\frac {1}{10} b^{2} x^{10}\)	\(25\)
risch	\(\frac {1}{2} a^{2} x^{2}+\frac {1}{3} a b \,x^{6}+\frac {1}{10} b^{2} x^{10}\)	\(25\)
parallelrisch	\(\frac {1}{2} a^{2} x^{2}+\frac {1}{3} a b \,x^{6}+\frac {1}{10} b^{2} x^{10}\)	\(25\)

[In]

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

[Out]

1/2*a^2*x^2+1/3*a*b*x^6+1/10*b^2*x^10

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int x \left (a+b x^4\right )^2 \, dx=\frac {1}{10} \, b^{2} x^{10} + \frac {1}{3} \, a b x^{6} + \frac {1}{2} \, a^{2} x^{2} \]

[In]

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

[Out]

1/10*b^2*x^10 + 1/3*a*b*x^6 + 1/2*a^2*x^2

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int x \left (a+b x^4\right )^2 \, dx=\frac {a^{2} x^{2}}{2} + \frac {a b x^{6}}{3} + \frac {b^{2} x^{10}}{10} \]

[In]

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

[Out]

a**2*x**2/2 + a*b*x**6/3 + b**2*x**10/10

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int x \left (a+b x^4\right )^2 \, dx=\frac {1}{10} \, b^{2} x^{10} + \frac {1}{3} \, a b x^{6} + \frac {1}{2} \, a^{2} x^{2} \]

[In]

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

[Out]

1/10*b^2*x^10 + 1/3*a*b*x^6 + 1/2*a^2*x^2

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int x \left (a+b x^4\right )^2 \, dx=\frac {1}{10} \, b^{2} x^{10} + \frac {1}{3} \, a b x^{6} + \frac {1}{2} \, a^{2} x^{2} \]

[In]

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

[Out]

1/10*b^2*x^10 + 1/3*a*b*x^6 + 1/2*a^2*x^2

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int x \left (a+b x^4\right )^2 \, dx=\frac {a^2\,x^2}{2}+\frac {a\,b\,x^6}{3}+\frac {b^2\,x^{10}}{10} \]

[In]

int(x*(a + b*x^4)^2,x)

[Out]

(a^2*x^2)/2 + (b^2*x^10)/10 + (a*b*x^6)/3