Optimal. Leaf size=30 \[ -\frac{2}{7 b c (a+b x)^2 \sqrt{c (a+b x)^3}} \]
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Rubi [A] time = 0.0126426, antiderivative size = 30, 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} \[ -\frac{2}{7 b c (a+b x)^2 \sqrt{c (a+b x)^3}} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 30
Rubi steps
\begin{align*} \int \frac{1}{\left (c (a+b x)^3\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (c x^3\right )^{3/2}} \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x)^{3/2} \operatorname{Subst}\left (\int \frac{1}{x^{9/2}} \, dx,x,a+b x\right )}{b c \sqrt{c (a+b x)^3}}\\ &=-\frac{2}{7 b c (a+b x)^2 \sqrt{c (a+b x)^3}}\\ \end{align*}
Mathematica [A] time = 0.0108384, size = 25, normalized size = 0.83 \[ -\frac{2 (a+b x)}{7 b \left (c (a+b x)^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 22, normalized size = 0.7 \begin{align*} -{\frac{2\,bx+2\,a}{7\,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 [A] time = 1.02268, size = 58, normalized size = 1.93 \begin{align*} -\frac{2 \, \sqrt{c}}{7 \,{\left (b^{3} c^{2} x^{2} + 2 \, a b^{2} c^{2} x + a^{2} b c^{2}\right )}{\left (b x + a\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.25945, size = 221, normalized size = 7.37 \begin{align*} -\frac{2 \, \sqrt{b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}}{7 \,{\left (b^{6} c^{2} x^{5} + 5 \, a b^{5} c^{2} x^{4} + 10 \, a^{2} b^{4} c^{2} x^{3} + 10 \, a^{3} b^{3} c^{2} x^{2} + 5 \, a^{4} b^{2} c^{2} x + a^{5} b c^{2}\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 \frac{1}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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