3.1438 \(\int x^m (a+b x^7)^2 \, dx\)

Optimal. Leaf size=43 \[ \frac{a^2 x^{m+1}}{m+1}+\frac{2 a b x^{m+8}}{m+8}+\frac{b^2 x^{m+15}}{m+15} \]

[Out]

(a^2*x^(1 + m))/(1 + m) + (2*a*b*x^(8 + m))/(8 + m) + (b^2*x^(15 + m))/(15 + m)

________________________________________________________________________________________

Rubi [A]  time = 0.0170874, 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} \[ \frac{a^2 x^{m+1}}{m+1}+\frac{2 a b x^{m+8}}{m+8}+\frac{b^2 x^{m+15}}{m+15} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*x^7)^2,x]

[Out]

(a^2*x^(1 + m))/(1 + m) + (2*a*b*x^(8 + m))/(8 + m) + (b^2*x^(15 + m))/(15 + 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^7\right )^2 \, dx &=\int \left (a^2 x^m+2 a b x^{7+m}+b^2 x^{14+m}\right ) \, dx\\ &=\frac{a^2 x^{1+m}}{1+m}+\frac{2 a b x^{8+m}}{8+m}+\frac{b^2 x^{15+m}}{15+m}\\ \end{align*}

Mathematica [A]  time = 0.0213593, size = 40, normalized size = 0.93 \[ x^{m+1} \left (\frac{a^2}{m+1}+\frac{2 a b x^7}{m+8}+\frac{b^2 x^{14}}{m+15}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a + b*x^7)^2,x]

[Out]

x^(1 + m)*(a^2/(1 + m) + (2*a*b*x^7)/(8 + m) + (b^2*x^14)/(15 + m))

________________________________________________________________________________________

Maple [B]  time = 0.006, size = 93, normalized size = 2.2 \begin{align*}{\frac{{x}^{1+m} \left ({b}^{2}{m}^{2}{x}^{14}+9\,{b}^{2}m{x}^{14}+8\,{b}^{2}{x}^{14}+2\,ab{m}^{2}{x}^{7}+32\,abm{x}^{7}+30\,ab{x}^{7}+{a}^{2}{m}^{2}+23\,{a}^{2}m+120\,{a}^{2} \right ) }{ \left ( 1+m \right ) \left ( 8+m \right ) \left ( 15+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(b*x^7+a)^2,x)

[Out]

x^(1+m)*(b^2*m^2*x^14+9*b^2*m*x^14+8*b^2*x^14+2*a*b*m^2*x^7+32*a*b*m*x^7+30*a*b*x^7+a^2*m^2+23*a^2*m+120*a^2)/
(1+m)/(8+m)/(15+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^7+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.76554, size = 192, normalized size = 4.47 \begin{align*} \frac{{\left ({\left (b^{2} m^{2} + 9 \, b^{2} m + 8 \, b^{2}\right )} x^{15} + 2 \,{\left (a b m^{2} + 16 \, a b m + 15 \, a b\right )} x^{8} +{\left (a^{2} m^{2} + 23 \, a^{2} m + 120 \, a^{2}\right )} x\right )} x^{m}}{m^{3} + 24 \, m^{2} + 143 \, m + 120} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^7+a)^2,x, algorithm="fricas")

[Out]

((b^2*m^2 + 9*b^2*m + 8*b^2)*x^15 + 2*(a*b*m^2 + 16*a*b*m + 15*a*b)*x^8 + (a^2*m^2 + 23*a^2*m + 120*a^2)*x)*x^
m/(m^3 + 24*m^2 + 143*m + 120)

________________________________________________________________________________________

Sympy [A]  time = 7.08843, size = 313, normalized size = 7.28 \begin{align*} \begin{cases} - \frac{a^{2}}{14 x^{14}} - \frac{2 a b}{7 x^{7}} + b^{2} \log{\left (x \right )} & \text{for}\: m = -15 \\- \frac{a^{2}}{7 x^{7}} + 2 a b \log{\left (x \right )} + \frac{b^{2} x^{7}}{7} & \text{for}\: m = -8 \\a^{2} \log{\left (x \right )} + \frac{2 a b x^{7}}{7} + \frac{b^{2} x^{14}}{14} & \text{for}\: m = -1 \\\frac{a^{2} m^{2} x x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac{23 a^{2} m x x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac{120 a^{2} x x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac{2 a b m^{2} x^{8} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac{32 a b m x^{8} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac{30 a b x^{8} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac{b^{2} m^{2} x^{15} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac{9 b^{2} m x^{15} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac{8 b^{2} x^{15} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(b*x**7+a)**2,x)

[Out]

Piecewise((-a**2/(14*x**14) - 2*a*b/(7*x**7) + b**2*log(x), Eq(m, -15)), (-a**2/(7*x**7) + 2*a*b*log(x) + b**2
*x**7/7, Eq(m, -8)), (a**2*log(x) + 2*a*b*x**7/7 + b**2*x**14/14, Eq(m, -1)), (a**2*m**2*x*x**m/(m**3 + 24*m**
2 + 143*m + 120) + 23*a**2*m*x*x**m/(m**3 + 24*m**2 + 143*m + 120) + 120*a**2*x*x**m/(m**3 + 24*m**2 + 143*m +
120) + 2*a*b*m**2*x**8*x**m/(m**3 + 24*m**2 + 143*m + 120) + 32*a*b*m*x**8*x**m/(m**3 + 24*m**2 + 143*m + 120
) + 30*a*b*x**8*x**m/(m**3 + 24*m**2 + 143*m + 120) + b**2*m**2*x**15*x**m/(m**3 + 24*m**2 + 143*m + 120) + 9*
b**2*m*x**15*x**m/(m**3 + 24*m**2 + 143*m + 120) + 8*b**2*x**15*x**m/(m**3 + 24*m**2 + 143*m + 120), True))

________________________________________________________________________________________

Giac [B]  time = 1.24667, size = 158, normalized size = 3.67 \begin{align*} \frac{b^{2} m^{2} x^{15} x^{m} + 9 \, b^{2} m x^{15} x^{m} + 8 \, b^{2} x^{15} x^{m} + 2 \, a b m^{2} x^{8} x^{m} + 32 \, a b m x^{8} x^{m} + 30 \, a b x^{8} x^{m} + a^{2} m^{2} x x^{m} + 23 \, a^{2} m x x^{m} + 120 \, a^{2} x x^{m}}{m^{3} + 24 \, m^{2} + 143 \, m + 120} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^7+a)^2,x, algorithm="giac")

[Out]

(b^2*m^2*x^15*x^m + 9*b^2*m*x^15*x^m + 8*b^2*x^15*x^m + 2*a*b*m^2*x^8*x^m + 32*a*b*m*x^8*x^m + 30*a*b*x^8*x^m
+ a^2*m^2*x*x^m + 23*a^2*m*x*x^m + 120*a^2*x*x^m)/(m^3 + 24*m^2 + 143*m + 120)