Optimal. Leaf size=65 \[ \frac{c \sqrt{c x^2} (a+b x)^{n+2}}{b^2 (n+2) x}-\frac{a c \sqrt{c x^2} (a+b x)^{n+1}}{b^2 (n+1) x} \]
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Rubi [A] time = 0.0179092, antiderivative size = 65, 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} \[ \frac{c \sqrt{c x^2} (a+b x)^{n+2}}{b^2 (n+2) x}-\frac{a c \sqrt{c x^2} (a+b x)^{n+1}}{b^2 (n+1) x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (c x^2\right )^{3/2} (a+b x)^n}{x^2} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int x (a+b x)^n \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (-\frac{a (a+b x)^n}{b}+\frac{(a+b x)^{1+n}}{b}\right ) \, dx}{x}\\ &=-\frac{a c \sqrt{c x^2} (a+b x)^{1+n}}{b^2 (1+n) x}+\frac{c \sqrt{c x^2} (a+b x)^{2+n}}{b^2 (2+n) x}\\ \end{align*}
Mathematica [A] time = 0.007752, size = 46, normalized size = 0.71 \[ \frac{c^2 x (a+b x)^{n+1} (b (n+1) x-a)}{b^2 (n+1) (n+2) \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 46, normalized size = 0.7 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{1+n} \left ( -bxn-bx+a \right ) }{{x}^{3}{b}^{2} \left ({n}^{2}+3\,n+2 \right ) } \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02159, size = 69, normalized size = 1.06 \begin{align*} \frac{{\left (b^{2} c^{\frac{3}{2}}{\left (n + 1\right )} x^{2} + a b c^{\frac{3}{2}} n x - a^{2} c^{\frac{3}{2}}\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 = 1.60575, size = 136, normalized size = 2.09 \begin{align*} \frac{{\left (a b c n x - a^{2} c +{\left (b^{2} c n + b^{2} c\right )} x^{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 [A] time = 1.08445, size = 161, normalized size = 2.48 \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 )} c^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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