3.58 \(\int x (a+b x^2)^5 \, dx\)

Optimal. Leaf size=16 \[ \frac {\left (a+b x^2\right )^6}{12 b} \]

[Out]

1/12*(b*x^2+a)^6/b

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {261} \[ \frac {\left (a+b x^2\right )^6}{12 b} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)^5,x]

[Out]

(a + b*x^2)^6/(12*b)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin {align*} \int x \left (a+b x^2\right )^5 \, dx &=\frac {\left (a+b x^2\right )^6}{12 b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \[ \frac {\left (a+b x^2\right )^6}{12 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^2)^5,x]

[Out]

(a + b*x^2)^6/(12*b)

________________________________________________________________________________________

fricas [B]  time = 0.77, size = 57, normalized size = 3.56 \[ \frac {1}{12} x^{12} b^{5} + \frac {1}{2} x^{10} b^{4} a + \frac {5}{4} x^{8} b^{3} a^{2} + \frac {5}{3} x^{6} b^{2} a^{3} + \frac {5}{4} x^{4} b a^{4} + \frac {1}{2} x^{2} a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^5,x, algorithm="fricas")

[Out]

1/12*x^12*b^5 + 1/2*x^10*b^4*a + 5/4*x^8*b^3*a^2 + 5/3*x^6*b^2*a^3 + 5/4*x^4*b*a^4 + 1/2*x^2*a^5

________________________________________________________________________________________

giac [A]  time = 1.06, size = 14, normalized size = 0.88 \[ \frac {{\left (b x^{2} + a\right )}^{6}}{12 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^5,x, algorithm="giac")

[Out]

1/12*(b*x^2 + a)^6/b

________________________________________________________________________________________

maple [B]  time = 0.00, size = 58, normalized size = 3.62 \[ \frac {1}{12} b^{5} x^{12}+\frac {1}{2} a \,b^{4} x^{10}+\frac {5}{4} a^{2} b^{3} x^{8}+\frac {5}{3} a^{3} b^{2} x^{6}+\frac {5}{4} a^{4} b \,x^{4}+\frac {1}{2} a^{5} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x^2+a)^5,x)

[Out]

1/12*b^5*x^12+1/2*a*b^4*x^10+5/4*a^2*b^3*x^8+5/3*a^3*b^2*x^6+5/4*a^4*b*x^4+1/2*a^5*x^2

________________________________________________________________________________________

maxima [A]  time = 1.40, size = 14, normalized size = 0.88 \[ \frac {{\left (b x^{2} + a\right )}^{6}}{12 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^5,x, algorithm="maxima")

[Out]

1/12*(b*x^2 + a)^6/b

________________________________________________________________________________________

mupad [B]  time = 0.02, size = 57, normalized size = 3.56 \[ \frac {a^5\,x^2}{2}+\frac {5\,a^4\,b\,x^4}{4}+\frac {5\,a^3\,b^2\,x^6}{3}+\frac {5\,a^2\,b^3\,x^8}{4}+\frac {a\,b^4\,x^{10}}{2}+\frac {b^5\,x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x^2)^5,x)

[Out]

(a^5*x^2)/2 + (b^5*x^12)/12 + (5*a^4*b*x^4)/4 + (a*b^4*x^10)/2 + (5*a^3*b^2*x^6)/3 + (5*a^2*b^3*x^8)/4

________________________________________________________________________________________

sympy [B]  time = 0.08, size = 65, normalized size = 4.06 \[ \frac {a^{5} x^{2}}{2} + \frac {5 a^{4} b x^{4}}{4} + \frac {5 a^{3} b^{2} x^{6}}{3} + \frac {5 a^{2} b^{3} x^{8}}{4} + \frac {a b^{4} x^{10}}{2} + \frac {b^{5} x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x**2+a)**5,x)

[Out]

a**5*x**2/2 + 5*a**4*b*x**4/4 + 5*a**3*b**2*x**6/3 + 5*a**2*b**3*x**8/4 + a*b**4*x**10/2 + b**5*x**12/12

________________________________________________________________________________________