Optimal. Leaf size=48 \[ \frac{a x (d x)^m}{m \sqrt{c x^2}}+\frac{b x (d x)^{m+1}}{d (m+1) \sqrt{c x^2}} \]
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Rubi [A] time = 0.0193592, antiderivative size = 48, 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} \[ \frac{a x (d x)^m}{m \sqrt{c x^2}}+\frac{b x (d x)^{m+1}}{d (m+1) \sqrt{c x^2}} \]
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)}{\sqrt{c x^2}} \, dx &=\frac{x \int \frac{(d x)^m (a+b x)}{x} \, dx}{\sqrt{c x^2}}\\ &=\frac{(d x) \int (d x)^{-1+m} (a+b x) \, dx}{\sqrt{c x^2}}\\ &=\frac{(d x) \int \left (a (d x)^{-1+m}+\frac{b (d x)^m}{d}\right ) \, dx}{\sqrt{c x^2}}\\ &=\frac{a x (d x)^m}{m \sqrt{c x^2}}+\frac{b x (d x)^{1+m}}{d (1+m) \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0156799, size = 33, normalized size = 0.69 \[ \frac{x (d x)^m (a m+a+b m x)}{m (m+1) \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 32, normalized size = 0.7 \begin{align*}{\frac{ \left ( bmx+am+a \right ) x \left ( dx \right ) ^{m}}{ \left ( 1+m \right ) m}{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05922, size = 43, normalized size = 0.9 \begin{align*} \frac{b d^{m} x x^{m}}{\sqrt{c}{\left (m + 1\right )}} + \frac{a d^{m} x^{m}}{\sqrt{c} m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35777, size = 77, normalized size = 1.6 \begin{align*} \frac{{\left (b m x + a m + a\right )} \sqrt{c x^{2}} \left (d x\right )^{m}}{{\left (c m^{2} + c m\right )} x} \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}}{\sqrt{c x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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