Optimal. Leaf size=81 \[ \frac{a^2 x (d x)^m}{m \sqrt{c x^2}}+\frac{2 a b x (d x)^{m+1}}{d (m+1) \sqrt{c x^2}}+\frac{b^2 x (d x)^{m+2}}{d^2 (m+2) \sqrt{c x^2}} \]
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Rubi [A] time = 0.0355124, antiderivative size = 81, 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} \[ \frac{a^2 x (d x)^m}{m \sqrt{c x^2}}+\frac{2 a b x (d x)^{m+1}}{d (m+1) \sqrt{c x^2}}+\frac{b^2 x (d x)^{m+2}}{d^2 (m+2) \sqrt{c x^2}} \]
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}{\sqrt{c x^2}} \, dx &=\frac{x \int \frac{(d x)^m (a+b x)^2}{x} \, dx}{\sqrt{c x^2}}\\ &=\frac{(d x) \int (d x)^{-1+m} (a+b x)^2 \, dx}{\sqrt{c x^2}}\\ &=\frac{(d x) \int \left (a^2 (d x)^{-1+m}+\frac{2 a b (d x)^m}{d}+\frac{b^2 (d x)^{1+m}}{d^2}\right ) \, dx}{\sqrt{c x^2}}\\ &=\frac{a^2 x (d x)^m}{m \sqrt{c x^2}}+\frac{2 a b x (d x)^{1+m}}{d (1+m) \sqrt{c x^2}}+\frac{b^2 x (d x)^{2+m}}{d^2 (2+m) \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.032731, size = 62, normalized size = 0.77 \[ \frac{x (d x)^m \left (a^2 \left (m^2+3 m+2\right )+2 a b m (m+2) x+b^2 m (m+1) x^2\right )}{m (m+1) (m+2) \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 79, normalized size = 1. \begin{align*}{\frac{ \left ({b}^{2}{x}^{2}{m}^{2}+2\,abx{m}^{2}+{b}^{2}{x}^{2}m+{a}^{2}{m}^{2}+4\,abxm+3\,{a}^{2}m+2\,{a}^{2} \right ) x \left ( dx \right ) ^{m}}{ \left ( 2+m \right ) \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 = 1.08843, size = 77, normalized size = 0.95 \begin{align*} \frac{b^{2} d^{m} x^{2} x^{m}}{\sqrt{c}{\left (m + 2\right )}} + \frac{2 \, a b d^{m} x x^{m}}{\sqrt{c}{\left (m + 1\right )}} + \frac{a^{2} 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 = 1.42444, size = 174, normalized size = 2.15 \begin{align*} \frac{{\left (a^{2} m^{2} + 3 \, a^{2} m +{\left (b^{2} m^{2} + b^{2} m\right )} x^{2} + 2 \, a^{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 m^{3} + 3 \, c m^{2} + 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 )}^{2} \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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