\(\int (a+\frac {b}{x^3})^2 (c x)^m \, dx\) [2761]

Optimal result

Integrand size = 15, antiderivative size = 63 \[ \int \left (a+\frac {b}{x^3}\right )^2 (c x)^m \, dx=-\frac {b^2 c^5 (c x)^{-5+m}}{5-m}-\frac {2 a b c^2 (c x)^{-2+m}}{2-m}+\frac {a^2 (c x)^{1+m}}{c (1+m)} \]

[Out]

-b^2*c^5*(c*x)^(-5+m)/(5-m)-2*a*b*c^2*(c*x)^(-2+m)/(2-m)+a^2*(c*x)^(1+m)/c/(1+m)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {276} \[ \int \left (a+\frac {b}{x^3}\right )^2 (c x)^m \, dx=\frac {a^2 (c x)^{m+1}}{c (m+1)}-\frac {2 a b c^2 (c x)^{m-2}}{2-m}-\frac {b^2 c^5 (c x)^{m-5}}{5-m} \]

[In]

Int[(a + b/x^3)^2*(c*x)^m,x]

[Out]

-((b^2*c^5*(c*x)^(-5 + m))/(5 - m)) - (2*a*b*c^2*(c*x)^(-2 + m))/(2 - m) + (a^2*(c*x)^(1 + m))/(c*(1 + m))

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 (b^2 c^6 (c x)^{-6+m}+2 a b c^3 (c x)^{-3+m}+a^2 (c x)^m\right ) \, dx \\ & = -\frac {b^2 c^5 (c x)^{-5+m}}{5-m}-\frac {2 a b c^2 (c x)^{-2+m}}{2-m}+\frac {a^2 (c x)^{1+m}}{c (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.06 \[ \int \left (a+\frac {b}{x^3}\right )^2 (c x)^m \, dx=\frac {(c x)^m \left (b^2 \left (-2-m+m^2\right )+2 a b \left (-5-4 m+m^2\right ) x^3+a^2 \left (10-7 m+m^2\right ) x^6\right )}{(-5+m) (-2+m) (1+m) x^5} \]

[In]

Integrate[(a + b/x^3)^2*(c*x)^m,x]

[Out]

((c*x)^m*(b^2*(-2 - m + m^2) + 2*a*b*(-5 - 4*m + m^2)*x^3 + a^2*(10 - 7*m + m^2)*x^6))/((-5 + m)*(-2 + m)*(1 +
m)*x^5)

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.95

method	result	size
norman	\(\frac {\frac {b^{2} {\mathrm e}^{m \ln \left (c x \right )}}{-5+m}+\frac {a^{2} x^{6} {\mathrm e}^{m \ln \left (c x \right )}}{1+m}+\frac {2 a b \,x^{3} {\mathrm e}^{m \ln \left (c x \right )}}{-2+m}}{x^{5}}\)	\(60\)
gosper	\(\frac {\left (c x \right )^{m} \left (a^{2} m^{2} x^{6}-7 a^{2} m \,x^{6}+10 a^{2} x^{6}+2 a b \,m^{2} x^{3}-8 m \,x^{3} a b -10 a b \,x^{3}+b^{2} m^{2}-b^{2} m -2 b^{2}\right )}{x^{5} \left (1+m \right ) \left (-2+m \right ) \left (-5+m \right )}\)	\(96\)
risch	\(\frac {\left (c x \right )^{m} \left (a^{2} m^{2} x^{6}-7 a^{2} m \,x^{6}+10 a^{2} x^{6}+2 a b \,m^{2} x^{3}-8 m \,x^{3} a b -10 a b \,x^{3}+b^{2} m^{2}-b^{2} m -2 b^{2}\right )}{x^{5} \left (1+m \right ) \left (-2+m \right ) \left (-5+m \right )}\)	\(96\)
parallelrisch	\(\frac {x^{6} \left (c x \right )^{m} a^{2} m^{2}-7 x^{6} \left (c x \right )^{m} a^{2} m +10 x^{6} \left (c x \right )^{m} a^{2}+2 x^{3} \left (c x \right )^{m} a b \,m^{2}-8 x^{3} \left (c x \right )^{m} a b m -10 \left (c x \right )^{m} a b \,x^{3}+\left (c x \right )^{m} b^{2} m^{2}-\left (c x \right )^{m} b^{2} m -2 b^{2} \left (c x \right )^{m}}{x^{5} \left (1+m \right ) \left (-2+m \right ) \left (-5+m \right )}\)	\(136\)

[In]

int((a+b/x^3)^2*(c*x)^m,x,method=_RETURNVERBOSE)

[Out]

(b^2/(-5+m)*exp(m*ln(c*x))+a^2/(1+m)*x^6*exp(m*ln(c*x))+2*a*b/(-2+m)*x^3*exp(m*ln(c*x)))/x^5

Fricas [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.38 \[ \int \left (a+\frac {b}{x^3}\right )^2 (c x)^m \, dx=\frac {{\left ({\left (a^{2} m^{2} - 7 \, a^{2} m + 10 \, a^{2}\right )} x^{6} + b^{2} m^{2} + 2 \, {\left (a b m^{2} - 4 \, a b m - 5 \, a b\right )} x^{3} - b^{2} m - 2 \, b^{2}\right )} \left (c x\right )^{m}}{{\left (m^{3} - 6 \, m^{2} + 3 \, m + 10\right )} x^{5}} \]

[In]

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

[Out]

((a^2*m^2 - 7*a^2*m + 10*a^2)*x^6 + b^2*m^2 + 2*(a*b*m^2 - 4*a*b*m - 5*a*b)*x^3 - b^2*m - 2*b^2)*(c*x)^m/((m^3
- 6*m^2 + 3*m + 10)*x^5)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 449 vs. \(2 (49) = 98\).

Time = 0.39 (sec) , antiderivative size = 449, normalized size of antiderivative = 7.13 \[ \int \left (a+\frac {b}{x^3}\right )^2 (c x)^m \, dx=\begin {cases} \frac {a^{2} \log {\left (x \right )} - \frac {2 a b}{3 x^{3}} - \frac {b^{2}}{6 x^{6}}}{c} & \text {for}\: m = -1 \\c^{2} \left (\frac {a^{2} x^{3}}{3} + 2 a b \log {\left (x \right )} - \frac {b^{2}}{3 x^{3}}\right ) & \text {for}\: m = 2 \\c^{5} \left (\frac {a^{2} x^{6}}{6} + \frac {2 a b x^{3}}{3} + b^{2} \log {\left (x \right )}\right ) & \text {for}\: m = 5 \\\frac {a^{2} m^{2} x^{6} \left (c x\right )^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} - \frac {7 a^{2} m x^{6} \left (c x\right )^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} + \frac {10 a^{2} x^{6} \left (c x\right )^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} + \frac {2 a b m^{2} x^{3} \left (c x\right )^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} - \frac {8 a b m x^{3} \left (c x\right )^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} - \frac {10 a b x^{3} \left (c x\right )^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} + \frac {b^{2} m^{2} \left (c x\right )^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} - \frac {b^{2} m \left (c x\right )^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} - \frac {2 b^{2} \left (c x\right )^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise(((a**2*log(x) - 2*a*b/(3*x**3) - b**2/(6*x**6))/c, Eq(m, -1)), (c**2*(a**2*x**3/3 + 2*a*b*log(x) - b
**2/(3*x**3)), Eq(m, 2)), (c**5*(a**2*x**6/6 + 2*a*b*x**3/3 + b**2*log(x)), Eq(m, 5)), (a**2*m**2*x**6*(c*x)**
m/(m**3*x**5 - 6*m**2*x**5 + 3*m*x**5 + 10*x**5) - 7*a**2*m*x**6*(c*x)**m/(m**3*x**5 - 6*m**2*x**5 + 3*m*x**5
+ 10*x**5) + 10*a**2*x**6*(c*x)**m/(m**3*x**5 - 6*m**2*x**5 + 3*m*x**5 + 10*x**5) + 2*a*b*m**2*x**3*(c*x)**m/(
m**3*x**5 - 6*m**2*x**5 + 3*m*x**5 + 10*x**5) - 8*a*b*m*x**3*(c*x)**m/(m**3*x**5 - 6*m**2*x**5 + 3*m*x**5 + 10
*x**5) - 10*a*b*x**3*(c*x)**m/(m**3*x**5 - 6*m**2*x**5 + 3*m*x**5 + 10*x**5) + b**2*m**2*(c*x)**m/(m**3*x**5 -
6*m**2*x**5 + 3*m*x**5 + 10*x**5) - b**2*m*(c*x)**m/(m**3*x**5 - 6*m**2*x**5 + 3*m*x**5 + 10*x**5) - 2*b**2*(
c*x)**m/(m**3*x**5 - 6*m**2*x**5 + 3*m*x**5 + 10*x**5), True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int \left (a+\frac {b}{x^3}\right )^2 (c x)^m \, dx=\frac {\left (c x\right )^{m + 1} a^{2}}{c {\left (m + 1\right )}} + \frac {2 \, a b c^{m} x^{m}}{{\left (m - 2\right )} x^{2}} + \frac {b^{2} c^{m} x^{m}}{{\left (m - 5\right )} x^{5}} \]

[In]

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

[Out]

(c*x)^(m + 1)*a^2/(c*(m + 1)) + 2*a*b*c^m*x^m/((m - 2)*x^2) + b^2*c^m*x^m/((m - 5)*x^5)

Giac [F]

\[ \int \left (a+\frac {b}{x^3}\right )^2 (c x)^m \, dx=\int { \left (c x\right )^{m} {\left (a + \frac {b}{x^{3}}\right )}^{2} \,d x } \]

[In]

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

[Out]

integrate((c*x)^m*(a + b/x^3)^2, x)

Mupad [B] (verification not implemented)

Time = 5.72 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.54 \[ \int \left (a+\frac {b}{x^3}\right )^2 (c x)^m \, dx=-\frac {{\left (c\,x\right )}^m\,\left (-a^2\,m^2\,x^6+7\,a^2\,m\,x^6-10\,a^2\,x^6-2\,a\,b\,m^2\,x^3+8\,a\,b\,m\,x^3+10\,a\,b\,x^3-b^2\,m^2+b^2\,m+2\,b^2\right )}{x^5\,\left (m^3-6\,m^2+3\,m+10\right )} \]

[In]

int((c*x)^m*(a + b/x^3)^2,x)

[Out]

-((c*x)^m*(b^2*m + 2*b^2 - b^2*m^2 - 10*a^2*x^6 + 7*a^2*m*x^6 - a^2*m^2*x^6 + 10*a*b*x^3 + 8*a*b*m*x^3 - 2*a*b
*m^2*x^3))/(x^5*(3*m - 6*m^2 + m^3 + 10))