Optimal. Leaf size=63 \[ \frac {a^2 (c x)^{m+1}}{c (m+1)}-\frac {2 a b c^2 (c x)^{m-2}}{2-m}-\frac {b^2 c^5 (c x)^{m-5}}{5-m} \]
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Rubi [A] time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270} \[ \frac {a^2 (c x)^{m+1}}{c (m+1)}-\frac {2 a b c^2 (c x)^{m-2}}{2-m}-\frac {b^2 c^5 (c x)^{m-5}}{5-m} \]
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^3}\right )^2 (c x)^m \, dx &=\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*}
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Mathematica [A] time = 0.04, size = 67, normalized size = 1.06 \[ \frac {(c x)^m \left (a^2 \left (m^2-7 m+10\right ) x^6+2 a b \left (m^2-4 m-5\right ) x^3+b^2 \left (m^2-m-2\right )\right )}{(m-5) (m-2) (m+1) x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 87, normalized size = 1.38 \[ \frac {{\left ({\left (a^{2} m^{2} - 7 \, a^{2} m + 10 \, a^{2}\right )} x^{6} + b^{2} m^{2} + 2 \, {\left (a b m^{2} - 4 \, a b m - 5 \, a b\right )} x^{3} - b^{2} m - 2 \, b^{2}\right )} \left (c x\right )^{m}}{{\left (m^{3} - 6 \, m^{2} + 3 \, m + 10\right )} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c x\right )^{m} {\left (a + \frac {b}{x^{3}}\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 96, normalized size = 1.52 \[ \frac {\left (a^{2} m^{2} x^{6}-7 a^{2} m \,x^{6}+10 a^{2} x^{6}+2 a b \,m^{2} x^{3}-8 a b m \,x^{3}-10 a b \,x^{3}+b^{2} m^{2}-b^{2} m -2 b^{2}\right ) \left (c x \right )^{m}}{\left (m +1\right ) \left (m -2\right ) \left (m -5\right ) x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 56, normalized size = 0.89 \[ \frac {\left (c x\right )^{m + 1} a^{2}}{c {\left (m + 1\right )}} + \frac {2 \, a b c^{m} x^{m}}{{\left (m - 2\right )} x^{2}} + \frac {b^{2} c^{m} x^{m}}{{\left (m - 5\right )} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 97, normalized size = 1.54 \[ -\frac {{\left (c\,x\right )}^m\,\left (-a^2\,m^2\,x^6+7\,a^2\,m\,x^6-10\,a^2\,x^6-2\,a\,b\,m^2\,x^3+8\,a\,b\,m\,x^3+10\,a\,b\,x^3-b^2\,m^2+b^2\,m+2\,b^2\right )}{x^5\,\left (m^3-6\,m^2+3\,m+10\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.61, size = 464, normalized size = 7.37 \[ \begin {cases} \frac {a^{2} \log {\relax (x )} - \frac {2 a b}{3 x^{3}} - \frac {b^{2}}{6 x^{6}}}{c} & \text {for}\: m = -1 \\c^{2} \left (\frac {a^{2} x^{3}}{3} + 2 a b \log {\relax (x )} - \frac {b^{2}}{3 x^{3}}\right ) & \text {for}\: m = 2 \\c^{5} \left (\frac {a^{2} x^{6}}{6} + \frac {2 a b x^{3}}{3} + b^{2} \log {\relax (x )}\right ) & \text {for}\: m = 5 \\\frac {a^{2} c^{m} m^{2} x^{6} x^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} - \frac {7 a^{2} c^{m} m x^{6} x^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} + \frac {10 a^{2} c^{m} x^{6} x^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} + \frac {2 a b c^{m} m^{2} x^{3} x^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} - \frac {8 a b c^{m} m x^{3} x^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} - \frac {10 a b c^{m} x^{3} x^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} + \frac {b^{2} c^{m} m^{2} x^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} - \frac {b^{2} c^{m} m x^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} - \frac {2 b^{2} c^{m} x^{m}}{m^{3} x^{5} - 6 m^{2} x^{5} + 3 m x^{5} + 10 x^{5}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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