Optimal. Leaf size=65 \[ -\frac{a d^2 x (d x)^{m-2}}{c (2-m) \sqrt{c x^2}}-\frac{b d x (d x)^{m-1}}{c (1-m) \sqrt{c x^2}} \]
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Rubi [A] time = 0.0358064, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {15, 16, 43} \[ -\frac{a d^2 x (d x)^{m-2}}{c (2-m) \sqrt{c x^2}}-\frac{b d x (d x)^{m-1}}{c (1-m) \sqrt{c x^2}} \]
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)}{\left (c x^2\right )^{3/2}} \, dx &=\frac{x \int \frac{(d x)^m (a+b x)}{x^3} \, dx}{c \sqrt{c x^2}}\\ &=\frac{\left (d^3 x\right ) \int (d x)^{-3+m} (a+b x) \, dx}{c \sqrt{c x^2}}\\ &=\frac{\left (d^3 x\right ) \int \left (a (d x)^{-3+m}+\frac{b (d x)^{-2+m}}{d}\right ) \, dx}{c \sqrt{c x^2}}\\ &=-\frac{a d^2 x (d x)^{-2+m}}{c (2-m) \sqrt{c x^2}}-\frac{b d x (d x)^{-1+m}}{c (1-m) \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0261452, size = 38, normalized size = 0.58 \[ \frac{x (d x)^m (a (m-1)+b (m-2) x)}{(m-2) (m-1) \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 40, normalized size = 0.6 \begin{align*}{\frac{ \left ( bmx+am-2\,bx-a \right ) x \left ( dx \right ) ^{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 = 1.06085, size = 53, normalized size = 0.82 \begin{align*} \frac{b d^{m} x^{m}}{c^{\frac{3}{2}}{\left (m - 1\right )} x} + \frac{a 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 = 1.3613, size = 109, normalized size = 1.68 \begin{align*} \frac{\sqrt{c x^{2}}{\left (a m +{\left (b m - 2 \, b\right )} x - a\right )} \left (d x\right )^{m}}{{\left (c^{2} m^{2} - 3 \, c^{2} m + 2 \, c^{2}\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 )} \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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