Optimal. Leaf size=28 \[ \frac{2 c (a+b x)^4 \sqrt{c (a+b x)^3}}{11 b} \]
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Rubi [A] time = 0.0116544, 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} \[ \frac{2 c (a+b x)^4 \sqrt{c (a+b x)^3}}{11 b} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 30
Rubi steps
\begin{align*} \int \left (c (a+b x)^3\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (c x^3\right )^{3/2} \, dx,x,a+b x\right )}{b}\\ &=\frac{\left (c \sqrt{c (a+b x)^3}\right ) \operatorname{Subst}\left (\int x^{9/2} \, dx,x,a+b x\right )}{b (a+b x)^{3/2}}\\ &=\frac{2 c (a+b x)^4 \sqrt{c (a+b x)^3}}{11 b}\\ \end{align*}
Mathematica [A] time = 0.0140574, size = 25, normalized size = 0.89 \[ \frac{2 (a+b x) \left (c (a+b x)^3\right )^{3/2}}{11 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 22, normalized size = 0.8 \begin{align*}{\frac{2\,bx+2\,a}{11\,b} \left ( c \left ( bx+a \right ) ^{3} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02487, size = 89, normalized size = 3.18 \begin{align*} \frac{2 \,{\left (b^{4} c^{\frac{3}{2}} x^{4} + 4 \, a b^{3} c^{\frac{3}{2}} x^{3} + 6 \, a^{2} b^{2} c^{\frac{3}{2}} x^{2} + 4 \, a^{3} b c^{\frac{3}{2}} x + a^{4} c^{\frac{3}{2}}\right )}{\left (b x + a\right )}^{\frac{3}{2}}}{11 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.2636, size = 176, normalized size = 6.29 \begin{align*} \frac{2 \,{\left (b^{4} c x^{4} + 4 \, a b^{3} c x^{3} + 6 \, a^{2} b^{2} c x^{2} + 4 \, a^{3} b c x + a^{4} c\right )} \sqrt{b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}}{11 \, b} \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 )^{3}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13111, size = 396, normalized size = 14.14 \begin{align*} \frac{2 \,{\left (1155 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{4} \mathrm{sgn}\left (b x + a\right ) - \frac{924 \,{\left (5 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a c - 3 \,{\left (b c x + a c\right )}^{\frac{5}{2}}\right )} a^{3} \mathrm{sgn}\left (b x + a\right )}{c} + \frac{198 \,{\left (35 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{2} c^{2} - 42 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a c + 15 \,{\left (b c x + a c\right )}^{\frac{7}{2}}\right )} a^{2} \mathrm{sgn}\left (b x + a\right )}{c^{2}} - \frac{44 \,{\left (105 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{3} c^{3} - 189 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a^{2} c^{2} + 135 \,{\left (b c x + a c\right )}^{\frac{7}{2}} a c - 35 \,{\left (b c x + a c\right )}^{\frac{9}{2}}\right )} a \mathrm{sgn}\left (b x + a\right )}{c^{3}} + \frac{{\left (1155 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{4} c^{4} - 2772 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a^{3} c^{3} + 2970 \,{\left (b c x + a c\right )}^{\frac{7}{2}} a^{2} c^{2} - 1540 \,{\left (b c x + a c\right )}^{\frac{9}{2}} a c + 315 \,{\left (b c x + a c\right )}^{\frac{11}{2}}\right )} \mathrm{sgn}\left (b x + a\right )}{c^{4}}\right )}}{3465 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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