3.586 \(\int x^m (a+b x^3)^2 \, dx\)

Optimal. Leaf size=43 \[ \frac{a^2 x^{m+1}}{m+1}+\frac{2 a b x^{m+4}}{m+4}+\frac{b^2 x^{m+7}}{m+7} \]

[Out]

(a^2*x^(1 + m))/(1 + m) + (2*a*b*x^(4 + m))/(4 + m) + (b^2*x^(7 + m))/(7 + m)

________________________________________________________________________________________

Rubi [A]  time = 0.0144864, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {270} \[ \frac{a^2 x^{m+1}}{m+1}+\frac{2 a b x^{m+4}}{m+4}+\frac{b^2 x^{m+7}}{m+7} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*x^3)^2,x]

[Out]

(a^2*x^(1 + m))/(1 + m) + (2*a*b*x^(4 + m))/(4 + m) + (b^2*x^(7 + m))/(7 + m)

Rule 270

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*} \int x^m \left (a+b x^3\right )^2 \, dx &=\int \left (a^2 x^m+2 a b x^{3+m}+b^2 x^{6+m}\right ) \, dx\\ &=\frac{a^2 x^{1+m}}{1+m}+\frac{2 a b x^{4+m}}{4+m}+\frac{b^2 x^{7+m}}{7+m}\\ \end{align*}

Mathematica [A]  time = 0.0194526, size = 40, normalized size = 0.93 \[ x^{m+1} \left (\frac{a^2}{m+1}+\frac{2 a b x^3}{m+4}+\frac{b^2 x^6}{m+7}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a + b*x^3)^2,x]

[Out]

x^(1 + m)*(a^2/(1 + m) + (2*a*b*x^3)/(4 + m) + (b^2*x^6)/(7 + m))

________________________________________________________________________________________

Maple [B]  time = 0.005, size = 93, normalized size = 2.2 \begin{align*}{\frac{{x}^{1+m} \left ({b}^{2}{m}^{2}{x}^{6}+5\,{b}^{2}m{x}^{6}+4\,{b}^{2}{x}^{6}+2\,ab{m}^{2}{x}^{3}+16\,abm{x}^{3}+14\,{x}^{3}ab+{a}^{2}{m}^{2}+11\,{a}^{2}m+28\,{a}^{2} \right ) }{ \left ( 7+m \right ) \left ( 4+m \right ) \left ( 1+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(b*x^3+a)^2,x)

[Out]

x^(1+m)*(b^2*m^2*x^6+5*b^2*m*x^6+4*b^2*x^6+2*a*b*m^2*x^3+16*a*b*m*x^3+14*a*b*x^3+a^2*m^2+11*a^2*m+28*a^2)/(7+m
)/(4+m)/(1+m)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^3+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.57324, size = 184, normalized size = 4.28 \begin{align*} \frac{{\left ({\left (b^{2} m^{2} + 5 \, b^{2} m + 4 \, b^{2}\right )} x^{7} + 2 \,{\left (a b m^{2} + 8 \, a b m + 7 \, a b\right )} x^{4} +{\left (a^{2} m^{2} + 11 \, a^{2} m + 28 \, a^{2}\right )} x\right )} x^{m}}{m^{3} + 12 \, m^{2} + 39 \, m + 28} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^3+a)^2,x, algorithm="fricas")

[Out]

((b^2*m^2 + 5*b^2*m + 4*b^2)*x^7 + 2*(a*b*m^2 + 8*a*b*m + 7*a*b)*x^4 + (a^2*m^2 + 11*a^2*m + 28*a^2)*x)*x^m/(m
^3 + 12*m^2 + 39*m + 28)

________________________________________________________________________________________

Sympy [A]  time = 1.39388, size = 313, normalized size = 7.28 \begin{align*} \begin{cases} - \frac{a^{2}}{6 x^{6}} - \frac{2 a b}{3 x^{3}} + b^{2} \log{\left (x \right )} & \text{for}\: m = -7 \\- \frac{a^{2}}{3 x^{3}} + 2 a b \log{\left (x \right )} + \frac{b^{2} x^{3}}{3} & \text{for}\: m = -4 \\a^{2} \log{\left (x \right )} + \frac{2 a b x^{3}}{3} + \frac{b^{2} x^{6}}{6} & \text{for}\: m = -1 \\\frac{a^{2} m^{2} x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{11 a^{2} m x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{28 a^{2} x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{2 a b m^{2} x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{16 a b m x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{14 a b x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{b^{2} m^{2} x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{5 b^{2} m x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{4 b^{2} x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(b*x**3+a)**2,x)

[Out]

Piecewise((-a**2/(6*x**6) - 2*a*b/(3*x**3) + b**2*log(x), Eq(m, -7)), (-a**2/(3*x**3) + 2*a*b*log(x) + b**2*x*
*3/3, Eq(m, -4)), (a**2*log(x) + 2*a*b*x**3/3 + b**2*x**6/6, Eq(m, -1)), (a**2*m**2*x*x**m/(m**3 + 12*m**2 + 3
9*m + 28) + 11*a**2*m*x*x**m/(m**3 + 12*m**2 + 39*m + 28) + 28*a**2*x*x**m/(m**3 + 12*m**2 + 39*m + 28) + 2*a*
b*m**2*x**4*x**m/(m**3 + 12*m**2 + 39*m + 28) + 16*a*b*m*x**4*x**m/(m**3 + 12*m**2 + 39*m + 28) + 14*a*b*x**4*
x**m/(m**3 + 12*m**2 + 39*m + 28) + b**2*m**2*x**7*x**m/(m**3 + 12*m**2 + 39*m + 28) + 5*b**2*m*x**7*x**m/(m**
3 + 12*m**2 + 39*m + 28) + 4*b**2*x**7*x**m/(m**3 + 12*m**2 + 39*m + 28), True))

________________________________________________________________________________________

Giac [B]  time = 1.10532, size = 158, normalized size = 3.67 \begin{align*} \frac{b^{2} m^{2} x^{7} x^{m} + 5 \, b^{2} m x^{7} x^{m} + 4 \, b^{2} x^{7} x^{m} + 2 \, a b m^{2} x^{4} x^{m} + 16 \, a b m x^{4} x^{m} + 14 \, a b x^{4} x^{m} + a^{2} m^{2} x x^{m} + 11 \, a^{2} m x x^{m} + 28 \, a^{2} x x^{m}}{m^{3} + 12 \, m^{2} + 39 \, m + 28} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^3+a)^2,x, algorithm="giac")

[Out]

(b^2*m^2*x^7*x^m + 5*b^2*m*x^7*x^m + 4*b^2*x^7*x^m + 2*a*b*m^2*x^4*x^m + 16*a*b*m*x^4*x^m + 14*a*b*x^4*x^m + a
^2*m^2*x*x^m + 11*a^2*m*x*x^m + 28*a^2*x*x^m)/(m^3 + 12*m^2 + 39*m + 28)