Optimal. Leaf size=63 \[ \frac{\sqrt{c x^2} (a+b x)^{n+2}}{b^2 (n+2) x}-\frac{a \sqrt{c x^2} (a+b x)^{n+1}}{b^2 (n+1) x} \]
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Rubi [A] time = 0.0169286, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {15, 43} \[ \frac{\sqrt{c x^2} (a+b x)^{n+2}}{b^2 (n+2) x}-\frac{a \sqrt{c x^2} (a+b x)^{n+1}}{b^2 (n+1) x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \sqrt{c x^2} (a+b x)^n \, dx &=\frac{\sqrt{c x^2} \int x (a+b x)^n \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int \left (-\frac{a (a+b x)^n}{b}+\frac{(a+b x)^{1+n}}{b}\right ) \, dx}{x}\\ &=-\frac{a \sqrt{c x^2} (a+b x)^{1+n}}{b^2 (1+n) x}+\frac{\sqrt{c x^2} (a+b x)^{2+n}}{b^2 (2+n) x}\\ \end{align*}
Mathematica [A] time = 0.0304787, size = 44, normalized size = 0.7 \[ \frac{c x (a+b x)^{n+1} (b (n+1) x-a)}{b^2 (n+1) (n+2) \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 46, normalized size = 0.7 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{1+n} \left ( -bxn-bx+a \right ) }{x{b}^{2} \left ({n}^{2}+3\,n+2 \right ) }\sqrt{c{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03805, size = 69, normalized size = 1.1 \begin{align*} \frac{{\left (b^{2} \sqrt{c}{\left (n + 1\right )} x^{2} + a b \sqrt{c} n x - a^{2} \sqrt{c}\right )}{\left (b x + a\right )}^{n}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.21974, size = 126, normalized size = 2. \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} n^{2} + 3 \, b^{2} n + 2 \, b^{2}\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 [B] time = 1.06232, size = 161, normalized size = 2.56 \begin{align*}{\left (\frac{a^{2} a^{n} \mathrm{sgn}\left (x\right )}{b^{2} n^{2} + 3 \, b^{2} n + 2 \, b^{2}} + \frac{{\left (b x + a\right )}^{n} b^{2} n x^{2} \mathrm{sgn}\left (x\right ) +{\left (b x + a\right )}^{n} a b n x \mathrm{sgn}\left (x\right ) +{\left (b x + a\right )}^{n} b^{2} x^{2} \mathrm{sgn}\left (x\right ) -{\left (b x + a\right )}^{n} a^{2} \mathrm{sgn}\left (x\right )}{b^{2} n^{2} + 3 \, b^{2} n + 2 \, b^{2}}\right )} \sqrt{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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