Optimal. Leaf size=105 \[ -\frac{a^2 d^4 x (d x)^{m-4}}{c^2 (4-m) \sqrt{c x^2}}-\frac{2 a b d^3 x (d x)^{m-3}}{c^2 (3-m) \sqrt{c x^2}}-\frac{b^2 d^2 x (d x)^{m-2}}{c^2 (2-m) \sqrt{c x^2}} \]
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Rubi [A] time = 0.0542726, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {15, 16, 43} \[ -\frac{a^2 d^4 x (d x)^{m-4}}{c^2 (4-m) \sqrt{c x^2}}-\frac{2 a b d^3 x (d x)^{m-3}}{c^2 (3-m) \sqrt{c x^2}}-\frac{b^2 d^2 x (d x)^{m-2}}{c^2 (2-m) \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 16
Rule 43
Rubi steps
\begin{align*} \int \frac{(d x)^m (a+b x)^2}{\left (c x^2\right )^{5/2}} \, dx &=\frac{x \int \frac{(d x)^m (a+b x)^2}{x^5} \, dx}{c^2 \sqrt{c x^2}}\\ &=\frac{\left (d^5 x\right ) \int (d x)^{-5+m} (a+b x)^2 \, dx}{c^2 \sqrt{c x^2}}\\ &=\frac{\left (d^5 x\right ) \int \left (a^2 (d x)^{-5+m}+\frac{2 a b (d x)^{-4+m}}{d}+\frac{b^2 (d x)^{-3+m}}{d^2}\right ) \, dx}{c^2 \sqrt{c x^2}}\\ &=-\frac{a^2 d^4 x (d x)^{-4+m}}{c^2 (4-m) \sqrt{c x^2}}-\frac{2 a b d^3 x (d x)^{-3+m}}{c^2 (3-m) \sqrt{c x^2}}-\frac{b^2 d^2 x (d x)^{-2+m}}{c^2 (2-m) \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0553227, size = 72, normalized size = 0.69 \[ \frac{x (d x)^m \left (a^2 \left (m^2-5 m+6\right )+2 a b \left (m^2-6 m+8\right ) x+b^2 \left (m^2-7 m+12\right ) x^2\right )}{(m-4) (m-3) (m-2) \left (c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 95, normalized size = 0.9 \begin{align*}{\frac{ \left ({b}^{2}{m}^{2}{x}^{2}+2\,ab{m}^{2}x-7\,{b}^{2}m{x}^{2}+{a}^{2}{m}^{2}-12\,abmx+12\,{b}^{2}{x}^{2}-5\,{a}^{2}m+16\,abx+6\,{a}^{2} \right ) x \left ( dx \right ) ^{m}}{ \left ( -2+m \right ) \left ( -3+m \right ) \left ( -4+m \right ) } \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09451, size = 86, normalized size = 0.82 \begin{align*} \frac{b^{2} d^{m} x^{m}}{c^{\frac{5}{2}}{\left (m - 2\right )} x^{2}} + \frac{2 \, a b d^{m} x^{m}}{c^{\frac{5}{2}}{\left (m - 3\right )} x^{3}} + \frac{a^{2} d^{m} x^{m}}{c^{\frac{5}{2}}{\left (m - 4\right )} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34616, size = 224, normalized size = 2.13 \begin{align*} \frac{{\left (a^{2} m^{2} - 5 \, a^{2} m +{\left (b^{2} m^{2} - 7 \, b^{2} m + 12 \, b^{2}\right )} x^{2} + 6 \, a^{2} + 2 \,{\left (a b m^{2} - 6 \, a b m + 8 \, a b\right )} x\right )} \sqrt{c x^{2}} \left (d x\right )^{m}}{{\left (c^{3} m^{3} - 9 \, c^{3} m^{2} + 26 \, c^{3} m - 24 \, c^{3}\right )} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{2} \left (d x\right )^{m}}{\left (c x^{2}\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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