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

Optimal. Leaf size=61 \[ \frac{3 a^2 b x^{m+4}}{m+4}+\frac{a^3 x^{m+1}}{m+1}+\frac{3 a b^2 x^{m+7}}{m+7}+\frac{b^3 x^{m+10}}{m+10} \]

[Out]

(a^3*x^(1 + m))/(1 + m) + (3*a^2*b*x^(4 + m))/(4 + m) + (3*a*b^2*x^(7 + m))/(7 + m) + (b^3*x^(10 + m))/(10 + m
)

________________________________________________________________________________________

Rubi [A]  time = 0.0218334, antiderivative size = 61, 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} \[ \frac{3 a^2 b x^{m+4}}{m+4}+\frac{a^3 x^{m+1}}{m+1}+\frac{3 a b^2 x^{m+7}}{m+7}+\frac{b^3 x^{m+10}}{m+10} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*x^3)^3,x]

[Out]

(a^3*x^(1 + m))/(1 + m) + (3*a^2*b*x^(4 + m))/(4 + m) + (3*a*b^2*x^(7 + m))/(7 + m) + (b^3*x^(10 + m))/(10 + 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 )^3 \, dx &=\int \left (a^3 x^m+3 a^2 b x^{3+m}+3 a b^2 x^{6+m}+b^3 x^{9+m}\right ) \, dx\\ &=\frac{a^3 x^{1+m}}{1+m}+\frac{3 a^2 b x^{4+m}}{4+m}+\frac{3 a b^2 x^{7+m}}{7+m}+\frac{b^3 x^{10+m}}{10+m}\\ \end{align*}

Mathematica [A]  time = 0.0302107, size = 56, normalized size = 0.92 \[ x^{m+1} \left (\frac{3 a^2 b x^3}{m+4}+\frac{a^3}{m+1}+\frac{3 a b^2 x^6}{m+7}+\frac{b^3 x^9}{m+10}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a + b*x^3)^3,x]

[Out]

x^(1 + m)*(a^3/(1 + m) + (3*a^2*b*x^3)/(4 + m) + (3*a*b^2*x^6)/(7 + m) + (b^3*x^9)/(10 + m))

________________________________________________________________________________________

Maple [B]  time = 0.006, size = 178, normalized size = 2.9 \begin{align*}{\frac{{x}^{1+m} \left ({b}^{3}{m}^{3}{x}^{9}+12\,{b}^{3}{m}^{2}{x}^{9}+39\,{b}^{3}m{x}^{9}+3\,a{b}^{2}{m}^{3}{x}^{6}+28\,{b}^{3}{x}^{9}+45\,a{b}^{2}{m}^{2}{x}^{6}+162\,a{b}^{2}m{x}^{6}+3\,{a}^{2}b{m}^{3}{x}^{3}+120\,a{b}^{2}{x}^{6}+54\,{a}^{2}b{m}^{2}{x}^{3}+261\,{a}^{2}bm{x}^{3}+{a}^{3}{m}^{3}+210\,{a}^{2}b{x}^{3}+21\,{a}^{3}{m}^{2}+138\,{a}^{3}m+280\,{a}^{3} \right ) }{ \left ( 10+m \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)^3,x)

[Out]

x^(1+m)*(b^3*m^3*x^9+12*b^3*m^2*x^9+39*b^3*m*x^9+3*a*b^2*m^3*x^6+28*b^3*x^9+45*a*b^2*m^2*x^6+162*a*b^2*m*x^6+3
*a^2*b*m^3*x^3+120*a*b^2*x^6+54*a^2*b*m^2*x^3+261*a^2*b*m*x^3+a^3*m^3+210*a^2*b*x^3+21*a^3*m^2+138*a^3*m+280*a
^3)/(10+m)/(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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.56619, size = 352, normalized size = 5.77 \begin{align*} \frac{{\left ({\left (b^{3} m^{3} + 12 \, b^{3} m^{2} + 39 \, b^{3} m + 28 \, b^{3}\right )} x^{10} + 3 \,{\left (a b^{2} m^{3} + 15 \, a b^{2} m^{2} + 54 \, a b^{2} m + 40 \, a b^{2}\right )} x^{7} + 3 \,{\left (a^{2} b m^{3} + 18 \, a^{2} b m^{2} + 87 \, a^{2} b m + 70 \, a^{2} b\right )} x^{4} +{\left (a^{3} m^{3} + 21 \, a^{3} m^{2} + 138 \, a^{3} m + 280 \, a^{3}\right )} x\right )} x^{m}}{m^{4} + 22 \, m^{3} + 159 \, m^{2} + 418 \, m + 280} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b^3*m^3 + 12*b^3*m^2 + 39*b^3*m + 28*b^3)*x^10 + 3*(a*b^2*m^3 + 15*a*b^2*m^2 + 54*a*b^2*m + 40*a*b^2)*x^7 +
3*(a^2*b*m^3 + 18*a^2*b*m^2 + 87*a^2*b*m + 70*a^2*b)*x^4 + (a^3*m^3 + 21*a^3*m^2 + 138*a^3*m + 280*a^3)*x)*x^m
/(m^4 + 22*m^3 + 159*m^2 + 418*m + 280)

________________________________________________________________________________________

Sympy [A]  time = 2.9404, size = 666, normalized size = 10.92 \begin{align*} \begin{cases} - \frac{a^{3}}{9 x^{9}} - \frac{a^{2} b}{2 x^{6}} - \frac{a b^{2}}{x^{3}} + b^{3} \log{\left (x \right )} & \text{for}\: m = -10 \\- \frac{a^{3}}{6 x^{6}} - \frac{a^{2} b}{x^{3}} + 3 a b^{2} \log{\left (x \right )} + \frac{b^{3} x^{3}}{3} & \text{for}\: m = -7 \\- \frac{a^{3}}{3 x^{3}} + 3 a^{2} b \log{\left (x \right )} + a b^{2} x^{3} + \frac{b^{3} x^{6}}{6} & \text{for}\: m = -4 \\a^{3} \log{\left (x \right )} + a^{2} b x^{3} + \frac{a b^{2} x^{6}}{2} + \frac{b^{3} x^{9}}{9} & \text{for}\: m = -1 \\\frac{a^{3} m^{3} x x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac{21 a^{3} m^{2} x x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac{138 a^{3} m x x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac{280 a^{3} x x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac{3 a^{2} b m^{3} x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac{54 a^{2} b m^{2} x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac{261 a^{2} b m x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac{210 a^{2} b x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac{3 a b^{2} m^{3} x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac{45 a b^{2} m^{2} x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac{162 a b^{2} m x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac{120 a b^{2} x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac{b^{3} m^{3} x^{10} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac{12 b^{3} m^{2} x^{10} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac{39 b^{3} m x^{10} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac{28 b^{3} x^{10} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**3/(9*x**9) - a**2*b/(2*x**6) - a*b**2/x**3 + b**3*log(x), Eq(m, -10)), (-a**3/(6*x**6) - a**2*b
/x**3 + 3*a*b**2*log(x) + b**3*x**3/3, Eq(m, -7)), (-a**3/(3*x**3) + 3*a**2*b*log(x) + a*b**2*x**3 + b**3*x**6
/6, Eq(m, -4)), (a**3*log(x) + a**2*b*x**3 + a*b**2*x**6/2 + b**3*x**9/9, Eq(m, -1)), (a**3*m**3*x*x**m/(m**4
+ 22*m**3 + 159*m**2 + 418*m + 280) + 21*a**3*m**2*x*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 138*a**3
*m*x*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 280*a**3*x*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280
) + 3*a**2*b*m**3*x**4*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 54*a**2*b*m**2*x**4*x**m/(m**4 + 22*m*
*3 + 159*m**2 + 418*m + 280) + 261*a**2*b*m*x**4*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 210*a**2*b*x
**4*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 3*a*b**2*m**3*x**7*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*
m + 280) + 45*a*b**2*m**2*x**7*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 162*a*b**2*m*x**7*x**m/(m**4 +
22*m**3 + 159*m**2 + 418*m + 280) + 120*a*b**2*x**7*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + b**3*m**
3*x**10*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 12*b**3*m**2*x**10*x**m/(m**4 + 22*m**3 + 159*m**2 +
418*m + 280) + 39*b**3*m*x**10*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 28*b**3*x**10*x**m/(m**4 + 22*
m**3 + 159*m**2 + 418*m + 280), True))

________________________________________________________________________________________

Giac [B]  time = 1.12349, size = 302, normalized size = 4.95 \begin{align*} \frac{b^{3} m^{3} x^{10} x^{m} + 12 \, b^{3} m^{2} x^{10} x^{m} + 39 \, b^{3} m x^{10} x^{m} + 3 \, a b^{2} m^{3} x^{7} x^{m} + 28 \, b^{3} x^{10} x^{m} + 45 \, a b^{2} m^{2} x^{7} x^{m} + 162 \, a b^{2} m x^{7} x^{m} + 3 \, a^{2} b m^{3} x^{4} x^{m} + 120 \, a b^{2} x^{7} x^{m} + 54 \, a^{2} b m^{2} x^{4} x^{m} + 261 \, a^{2} b m x^{4} x^{m} + a^{3} m^{3} x x^{m} + 210 \, a^{2} b x^{4} x^{m} + 21 \, a^{3} m^{2} x x^{m} + 138 \, a^{3} m x x^{m} + 280 \, a^{3} x x^{m}}{m^{4} + 22 \, m^{3} + 159 \, m^{2} + 418 \, m + 280} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^3*m^3*x^10*x^m + 12*b^3*m^2*x^10*x^m + 39*b^3*m*x^10*x^m + 3*a*b^2*m^3*x^7*x^m + 28*b^3*x^10*x^m + 45*a*b^2
*m^2*x^7*x^m + 162*a*b^2*m*x^7*x^m + 3*a^2*b*m^3*x^4*x^m + 120*a*b^2*x^7*x^m + 54*a^2*b*m^2*x^4*x^m + 261*a^2*
b*m*x^4*x^m + a^3*m^3*x*x^m + 210*a^2*b*x^4*x^m + 21*a^3*m^2*x*x^m + 138*a^3*m*x*x^m + 280*a^3*x*x^m)/(m^4 + 2
2*m^3 + 159*m^2 + 418*m + 280)