Optimal. Leaf size=59 \[ \frac{a \sqrt{c x^2} (d x)^{m+2}}{d^2 (m+2) x}+\frac{b \sqrt{c x^2} (d x)^{m+3}}{d^3 (m+3) x} \]
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Rubi [A] time = 0.0267448, antiderivative size = 59, 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} \[ \frac{a \sqrt{c x^2} (d x)^{m+2}}{d^2 (m+2) x}+\frac{b \sqrt{c x^2} (d x)^{m+3}}{d^3 (m+3) x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 16
Rule 43
Rubi steps
\begin{align*} \int (d x)^m \sqrt{c x^2} (a+b x) \, dx &=\frac{\sqrt{c x^2} \int x (d x)^m (a+b x) \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int (d x)^{1+m} (a+b x) \, dx}{d x}\\ &=\frac{\sqrt{c x^2} \int \left (a (d x)^{1+m}+\frac{b (d x)^{2+m}}{d}\right ) \, dx}{d x}\\ &=\frac{a (d x)^{2+m} \sqrt{c x^2}}{d^2 (2+m) x}+\frac{b (d x)^{3+m} \sqrt{c x^2}}{d^3 (3+m) x}\\ \end{align*}
Mathematica [A] time = 0.0203458, size = 38, normalized size = 0.64 \[ \frac{x \sqrt{c x^2} (d x)^m (a (m+3)+b (m+2) x)}{(m+2) (m+3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 40, normalized size = 0.7 \begin{align*}{\frac{ \left ( bmx+am+2\,bx+3\,a \right ) x \left ( dx \right ) ^{m}}{ \left ( 3+m \right ) \left ( 2+m \right ) }\sqrt{c{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04417, size = 53, normalized size = 0.9 \begin{align*} \frac{b \sqrt{c} d^{m} x^{3} x^{m}}{m + 3} + \frac{a \sqrt{c} d^{m} x^{2} x^{m}}{m + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29818, size = 96, normalized size = 1.63 \begin{align*} \frac{{\left ({\left (b m + 2 \, b\right )} x^{2} +{\left (a m + 3 \, a\right )} x\right )} \sqrt{c x^{2}} \left (d x\right )^{m}}{m^{2} + 5 \, m + 6} \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(-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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