Optimal. Leaf size=28 \[ -\frac{2 c \sqrt{\frac{c}{(a+b x)^3}}}{7 b (a+b x)^2} \]
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Rubi [A] time = 0.0087339, 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} \[ -\frac{2 c \sqrt{\frac{c}{(a+b x)^3}}}{7 b (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 30
Rubi steps
\begin{align*} \int \left (\frac{c}{(a+b x)^3}\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{c}{x^3}\right )^{3/2} \, dx,x,a+b x\right )}{b}\\ &=\frac{\left (c \sqrt{\frac{c}{(a+b x)^3}} (a+b x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{9/2}} \, dx,x,a+b x\right )}{b}\\ &=-\frac{2 c \sqrt{\frac{c}{(a+b x)^3}}}{7 b (a+b x)^2}\\ \end{align*}
Mathematica [A] time = 0.009371, size = 25, normalized size = 0.89 \[ -\frac{2 (a+b x) \left (\frac{c}{(a+b x)^3}\right )^{3/2}}{7 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 22, normalized size = 0.8 \begin{align*} -{\frac{2\,bx+2\,a}{7\,b} \left ({\frac{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 [A] time = 1.37796, size = 32, normalized size = 1.14 \begin{align*} -\frac{2 \,{\left (b c^{\frac{3}{2}} x + a c^{\frac{3}{2}}\right )}}{7 \,{\left (b x + a\right )}^{\frac{9}{2}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.20741, size = 119, normalized size = 4.25 \begin{align*} -\frac{2 \, c \sqrt{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{7 \,{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.7178, size = 357, normalized size = 12.75 \begin{align*} \begin{cases} - \frac{67 a^{3} c^{\frac{3}{2}} \left (\frac{1}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac{3}{2}}}{14 a^{2} b + 28 a b^{2} x + 14 b^{3} x^{2}} - \frac{201 a^{2} b c^{\frac{3}{2}} x \left (\frac{1}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac{3}{2}}}{14 a^{2} b + 28 a b^{2} x + 14 b^{3} x^{2}} - \frac{201 a b^{2} c^{\frac{3}{2}} x^{2} \left (\frac{1}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac{3}{2}}}{14 a^{2} b + 28 a b^{2} x + 14 b^{3} x^{2}} - \frac{67 b^{3} c^{\frac{3}{2}} x^{3} \left (\frac{1}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac{3}{2}}}{14 a^{2} b + 28 a b^{2} x + 14 b^{3} x^{2}} + \frac{63 c^{\frac{3}{2}} \sqrt{\frac{1}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}}}{14 a^{2} b + 28 a b^{2} x + 14 b^{3} x^{2}} & \text{for}\: b \neq 0 \\x \left (\frac{c}{a^{3}}\right )^{\frac{3}{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11565, size = 70, normalized size = 2.5 \begin{align*} -\frac{2 \, c^{5} \mathrm{sgn}\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right ) \mathrm{sgn}\left (b x + a\right )}{7 \,{\left (b c x + a c\right )}^{\frac{7}{2}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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