3.2754 \(\int \left (a+\frac{b}{x^3}\right )^2 (c x)^m \, dx\)

Optimal. Leaf size=63 \[ \frac{a^2 (c x)^{m+1}}{c (m+1)}-\frac{2 a b c^2 (c x)^{m-2}}{2-m}-\frac{b^2 c^5 (c x)^{m-5}}{5-m} \]

[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))

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Rubi [A]  time = 0.0797327, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{a^2 (c x)^{m+1}}{c (m+1)}-\frac{2 a b c^2 (c x)^{m-2}}{2-m}-\frac{b^2 c^5 (c x)^{m-5}}{5-m} \]

Antiderivative was successfully verified.

[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))

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Rubi in Sympy [A]  time = 14.0617, size = 49, normalized size = 0.78 \[ \frac{a^{2} \left (c x\right )^{m + 1}}{c \left (m + 1\right )} - \frac{2 a b c^{2} \left (c x\right )^{m - 2}}{- m + 2} - \frac{b^{2} c^{5} \left (c x\right )^{m - 5}}{- m + 5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**3)**2*(c*x)**m,x)

[Out]

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

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Mathematica [A]  time = 0.0545363, size = 62, normalized size = 0.98 \[ \frac{x^6 \left (a+\frac{b}{x^3}\right )^2 (c x)^m \left (\frac{a^2 x}{m+1}+\frac{2 a b}{(m-2) x^2}+\frac{b^2}{(m-5) x^5}\right )}{\left (a x^3+b\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^3)^2*(c*x)^m,x]

[Out]

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

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Maple [A]  time = 0.007, size = 96, normalized size = 1.5 \[{\frac{ \left ( cx \right ) ^{m} \left ({a}^{2}{m}^{2}{x}^{6}-7\,{a}^{2}m{x}^{6}+10\,{a}^{2}{x}^{6}+2\,ab{m}^{2}{x}^{3}-8\,abm{x}^{3}-10\,ab{x}^{3}+{b}^{2}{m}^{2}-{b}^{2}m-2\,{b}^{2} \right ) }{{x}^{5} \left ( 1+m \right ) \left ( -2+m \right ) \left ( -5+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^3)^2*(c*x)^m,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229839, size = 117, normalized size = 1.86 \[ \frac{{\left ({\left (a^{2} m^{2} - 7 \, a^{2} m + 10 \, a^{2}\right )} x^{6} + b^{2} m^{2} + 2 \,{\left (a b m^{2} - 4 \, a b m - 5 \, a b\right )} x^{3} - b^{2} m - 2 \, b^{2}\right )} \left (c x\right )^{m}}{{\left (m^{3} - 6 \, m^{2} + 3 \, m + 10\right )} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^m*(a + b/x^3)^2,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)

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Sympy [A]  time = 5.44974, size = 464, normalized size = 7.37 \[ \begin{cases} \frac{a^{2} \log{\left (x \right )} - \frac{2 a b}{3 x^{3}} - \frac{b^{2}}{6 x^{6}}}{c} & \text{for}\: m = -1 \\c^{2} \left (\frac{a^{2} x^{3}}{3} + 2 a b \log{\left (x \right )} - \frac{b^{2}}{3 x^{3}}\right ) & \text{for}\: m = 2 \\c^{5} \left (\frac{a^{2} x^{6}}{6} + \frac{2 a b x^{3}}{3} + b^{2} \log{\left (x \right )}\right ) & \text{for}\: m = 5 \\\frac{a^{2} c^{m} m^{2} x^{6} x^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} - \frac{7 a^{2} c^{m} m x^{6} x^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} + \frac{10 a^{2} c^{m} x^{6} x^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} + \frac{2 a b c^{m} m^{2} x^{3} x^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} - \frac{8 a b c^{m} m x^{3} x^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} - \frac{10 a b c^{m} x^{3} x^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} + \frac{b^{2} c^{m} m^{2} x^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} - \frac{b^{2} c^{m} m x^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} - \frac{2 b^{2} c^{m} x^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (c x\right )^{m}{\left (a + \frac{b}{x^{3}}\right )}^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x)^m*(a + b/x^3)^2, x)