Optimal. Leaf size=93 \[ -\frac{a^2 d^2 x (d x)^{m-2}}{c (2-m) \sqrt{c x^2}}-\frac{2 a b d x (d x)^{m-1}}{c (1-m) \sqrt{c x^2}}+\frac{b^2 x (d x)^m}{c m \sqrt{c x^2}} \]
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Rubi [A] time = 0.0450581, antiderivative size = 93, 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} \[ -\frac{a^2 d^2 x (d x)^{m-2}}{c (2-m) \sqrt{c x^2}}-\frac{2 a b d x (d x)^{m-1}}{c (1-m) \sqrt{c x^2}}+\frac{b^2 x (d x)^m}{c m \sqrt{c x^2}} \]
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 )^{3/2}} \, dx &=\frac{x \int \frac{(d x)^m (a+b x)^2}{x^3} \, dx}{c \sqrt{c x^2}}\\ &=\frac{\left (d^3 x\right ) \int (d x)^{-3+m} (a+b x)^2 \, dx}{c \sqrt{c x^2}}\\ &=\frac{\left (d^3 x\right ) \int \left (a^2 (d x)^{-3+m}+\frac{2 a b (d x)^{-2+m}}{d}+\frac{b^2 (d x)^{-1+m}}{d^2}\right ) \, dx}{c \sqrt{c x^2}}\\ &=-\frac{a^2 d^2 x (d x)^{-2+m}}{c (2-m) \sqrt{c x^2}}-\frac{2 a b d x (d x)^{-1+m}}{c (1-m) \sqrt{c x^2}}+\frac{b^2 x (d x)^m}{c m \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0803281, size = 62, normalized size = 0.67 \[ \frac{x (d x)^m \left (a^2 (m-1) m+2 a b (m-2) m x+b^2 \left (m^2-3 m+2\right ) x^2\right )}{(m-2) (m-1) m \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 83, normalized size = 0.9 \begin{align*}{\frac{ \left ({b}^{2}{m}^{2}{x}^{2}+2\,abx{m}^{2}-3\,{b}^{2}m{x}^{2}+{a}^{2}{m}^{2}-4\,abxm+2\,{b}^{2}{x}^{2}-{a}^{2}m \right ) x \left ( dx \right ) ^{m}}{m \left ( -1+m \right ) \left ( -2+m \right ) } \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09372, size = 80, normalized size = 0.86 \begin{align*} \frac{b^{2} d^{m} x^{m}}{c^{\frac{3}{2}} m} + \frac{2 \, a b d^{m} x^{m}}{c^{\frac{3}{2}}{\left (m - 1\right )} x} + \frac{a^{2} d^{m} x^{m}}{c^{\frac{3}{2}}{\left (m - 2\right )} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30778, size = 185, normalized size = 1.99 \begin{align*} \frac{{\left (a^{2} m^{2} - a^{2} m +{\left (b^{2} m^{2} - 3 \, b^{2} m + 2 \, b^{2}\right )} x^{2} + 2 \,{\left (a b m^{2} - 2 \, a b m\right )} x\right )} \sqrt{c x^{2}} \left (d x\right )^{m}}{{\left (c^{2} m^{3} - 3 \, c^{2} m^{2} + 2 \, c^{2} m\right )} x^{3}} \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{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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