Optimal. Leaf size=30 \[ \frac{2 (a+b x)^7}{17 b c^2 \sqrt{\frac{c}{(a+b x)^3}}} \]
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Rubi [A] time = 0.0095543, 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} \[ \frac{2 (a+b x)^7}{17 b c^2 \sqrt{\frac{c}{(a+b x)^3}}} \]
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 )^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (\frac{c}{x^3}\right )^{5/2}} \, dx,x,a+b x\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int x^{15/2} \, dx,x,a+b x\right )}{b c^2 \sqrt{\frac{c}{(a+b x)^3}} (a+b x)^{3/2}}\\ &=\frac{2 (a+b x)^7}{17 b c^2 \sqrt{\frac{c}{(a+b x)^3}}}\\ \end{align*}
Mathematica [A] time = 0.0144372, size = 25, normalized size = 0.83 \[ \frac{2 (a+b x)}{17 b \left (\frac{c}{(a+b x)^3}\right )^{5/2}} \]
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}{17\,b} \left ({\frac{c}{ \left ( bx+a \right ) ^{3}}} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.26531, 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{15}{2}}}{17 \, b c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.21817, size = 317, normalized size = 10.57 \begin{align*} \frac{2 \,{\left (b^{10} x^{10} + 10 \, a b^{9} x^{9} + 45 \, a^{2} b^{8} x^{8} + 120 \, a^{3} b^{7} x^{7} + 210 \, a^{4} b^{6} x^{6} + 252 \, a^{5} b^{5} x^{5} + 210 \, a^{6} b^{4} x^{4} + 120 \, a^{7} b^{3} x^{3} + 45 \, a^{8} b^{2} x^{2} + 10 \, a^{9} b x + a^{10}\right )} \sqrt{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{17 \, b c^{3}} \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{5}{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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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