Optimal. Leaf size=59 \[ \frac{x (a+b x)^{n+2}}{b^2 (n+2) \sqrt{c x^2}}-\frac{a x (a+b x)^{n+1}}{b^2 (n+1) \sqrt{c x^2}} \]
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Rubi [A] time = 0.0157513, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 43} \[ \frac{x (a+b x)^{n+2}}{b^2 (n+2) \sqrt{c x^2}}-\frac{a x (a+b x)^{n+1}}{b^2 (n+1) \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2 (a+b x)^n}{\sqrt{c x^2}} \, dx &=\frac{x \int x (a+b x)^n \, dx}{\sqrt{c x^2}}\\ &=\frac{x \int \left (-\frac{a (a+b x)^n}{b}+\frac{(a+b x)^{1+n}}{b}\right ) \, dx}{\sqrt{c x^2}}\\ &=-\frac{a x (a+b x)^{1+n}}{b^2 (1+n) \sqrt{c x^2}}+\frac{x (a+b x)^{2+n}}{b^2 (2+n) \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0206006, size = 43, normalized size = 0.73 \[ \frac{x (a+b x)^{n+1} (b (n+1) x-a)}{b^2 (n+1) (n+2) \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 44, normalized size = 0.8 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{1+n}x \left ( -bxn-bx+a \right ) }{{b}^{2} \left ({n}^{2}+3\,n+2 \right ) }{\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 = 1.02931, size = 61, normalized size = 1.03 \begin{align*} \frac{{\left (b^{2}{\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )}{\left (b x + a\right )}^{n}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2} \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70888, size = 134, normalized size = 2.27 \begin{align*} \frac{{\left (a b n x +{\left (b^{2} n + b^{2}\right )} x^{2} - a^{2}\right )} \sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{{\left (b^{2} c n^{2} + 3 \, b^{2} c n + 2 \, b^{2} c\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 )}^{n} x^{2}}{\sqrt{c x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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