Optimal. Leaf size=28 \[ \frac{c (a+b x)^3 \sqrt{c (a+b x)^2}}{4 b} \]
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Rubi [A] time = 0.0106663, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {247, 15, 30} \[ \frac{c (a+b x)^3 \sqrt{c (a+b x)^2}}{4 b} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 30
Rubi steps
\begin{align*} \int \left (c (a+b x)^2\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (c x^2\right )^{3/2} \, dx,x,a+b x\right )}{b}\\ &=\frac{\left (c \sqrt{c (a+b x)^2}\right ) \operatorname{Subst}\left (\int x^3 \, dx,x,a+b x\right )}{b (a+b x)}\\ &=\frac{c (a+b x)^3 \sqrt{c (a+b x)^2}}{4 b}\\ \end{align*}
Mathematica [A] time = 0.0090135, size = 25, normalized size = 0.89 \[ \frac{(a+b x) \left (c (a+b x)^2\right )^{3/2}}{4 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 51, normalized size = 1.8 \begin{align*}{\frac{x \left ({b}^{3}{x}^{3}+4\,a{b}^{2}{x}^{2}+6\,{a}^{2}bx+4\,{a}^{3} \right ) }{4\, \left ( bx+a \right ) ^{3}} \left ( c \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.22641, size = 144, normalized size = 5.14 \begin{align*} \frac{{\left (b^{3} c x^{4} + 4 \, a b^{2} c x^{3} + 6 \, a^{2} b c x^{2} + 4 \, a^{3} c x\right )} \sqrt{b^{2} c x^{2} + 2 \, a b c x + a^{2} c}}{4 \,{\left (b x + a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09758, size = 28, normalized size = 1. \begin{align*} \frac{{\left (b x + a\right )}^{4} c^{\frac{3}{2}} \mathrm{sgn}\left (b x + a\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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