Optimal. Leaf size=31 \[ \frac{c \sqrt{c x^2} (a+b x)^{n+1}}{b (n+1) x} \]
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Rubi [A] time = 0.0059132, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 32} \[ \frac{c \sqrt{c x^2} (a+b x)^{n+1}}{b (n+1) x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 32
Rubi steps
\begin{align*} \int \frac{\left (c x^2\right )^{3/2} (a+b x)^n}{x^3} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int (a+b x)^n \, dx}{x}\\ &=\frac{c \sqrt{c x^2} (a+b x)^{1+n}}{b (1+n) x}\\ \end{align*}
Mathematica [A] time = 0.0138324, size = 30, normalized size = 0.97 \[ \frac{\left (c x^2\right )^{3/2} (a+b x)^{n+1}}{b (n+1) x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 29, normalized size = 0.9 \begin{align*}{\frac{ \left ( bx+a \right ) ^{1+n}}{b \left ( 1+n \right ){x}^{3}} \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 = 0.991608, size = 38, normalized size = 1.23 \begin{align*} \frac{{\left (b c^{\frac{3}{2}} x + a c^{\frac{3}{2}}\right )}{\left (b x + a\right )}^{n}}{b{\left (n + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61113, size = 72, normalized size = 2.32 \begin{align*} \frac{{\left (b c x + a c\right )} \sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{{\left (b n + b\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 = 1.05204, size = 57, normalized size = 1.84 \begin{align*} -c^{\frac{3}{2}}{\left (\frac{a^{n + 1} \mathrm{sgn}\left (x\right )}{b n + b} - \frac{{\left (b x + a\right )}^{n + 1} \mathrm{sgn}\left (x\right )}{b{\left (n + 1\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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