Optimal. Leaf size=30 \[ \frac{2 (a+b x)^4}{11 b c \sqrt{\frac{c}{(a+b x)^3}}} \]
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Rubi [A] time = 0.0088129, 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} \[ \frac{2 (a+b x)^4}{11 b c \sqrt{\frac{c}{(a+b x)^3}}} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 30
Rubi steps
\begin{align*} \int \frac{1}{\left (\frac{c}{(a+b x)^3}\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (\frac{c}{x^3}\right )^{3/2}} \, dx,x,a+b x\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int x^{9/2} \, dx,x,a+b x\right )}{b c \sqrt{\frac{c}{(a+b x)^3}} (a+b x)^{3/2}}\\ &=\frac{2 (a+b x)^4}{11 b c \sqrt{\frac{c}{(a+b x)^3}}}\\ \end{align*}
Mathematica [A] time = 0.0204846, size = 25, normalized size = 0.83 \[ \frac{2 (a+b x)}{11 b \left (\frac{c}{(a+b x)^3}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 22, normalized size = 0.7 \begin{align*}{\frac{2\,bx+2\,a}{11\,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 = 1.04503, size = 36, normalized size = 1.2 \begin{align*} \frac{2 \,{\left (b \sqrt{c} x + a \sqrt{c}\right )}{\left (b x + a\right )}^{\frac{9}{2}}}{11 \, b c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.32665, size = 235, normalized size = 7.83 \begin{align*} \frac{2 \,{\left (b^{7} x^{7} + 7 \, a b^{6} x^{6} + 21 \, a^{2} b^{5} x^{5} + 35 \, a^{3} b^{4} x^{4} + 35 \, a^{4} b^{3} x^{3} + 21 \, a^{5} b^{2} x^{2} + 7 \, a^{6} b x + a^{7}\right )} \sqrt{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{11 \, b c^{2}} \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 (\frac{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*} \int \frac{1}{\left (\frac{c}{{\left (b x + a\right )}^{3}}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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