Optimal. Leaf size=30 \[ \frac{(a+b x)^5}{6 b c^2 \sqrt{\frac{c}{(a+b x)^2}}} \]
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Rubi [A] time = 0.0090109, 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} \[ \frac{(a+b x)^5}{6 b c^2 \sqrt{\frac{c}{(a+b x)^2}}} \]
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)^2}\right )^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (\frac{c}{x^2}\right )^{5/2}} \, dx,x,a+b x\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int x^5 \, dx,x,a+b x\right )}{b c^2 \sqrt{\frac{c}{(a+b x)^2}} (a+b x)}\\ &=\frac{(a+b x)^5}{6 b c^2 \sqrt{\frac{c}{(a+b x)^2}}}\\ \end{align*}
Mathematica [A] time = 0.0167713, size = 25, normalized size = 0.83 \[ \frac{a+b x}{6 b \left (\frac{c}{(a+b x)^2}\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 73, normalized size = 2.4 \begin{align*}{\frac{x \left ({b}^{5}{x}^{5}+6\,a{b}^{4}{x}^{4}+15\,{a}^{2}{b}^{3}{x}^{3}+20\,{a}^{3}{b}^{2}{x}^{2}+15\,{a}^{4}bx+6\,{a}^{5} \right ) }{6\, \left ( bx+a \right ) ^{5}} \left ({\frac{c}{ \left ( bx+a \right ) ^{2}}} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.28996, size = 80, normalized size = 2.67 \begin{align*} \frac{b^{5} x^{6} + 6 \, a b^{4} x^{5} + 15 \, a^{2} b^{3} x^{4} + 20 \, a^{3} b^{2} x^{3} + 15 \, a^{4} b x^{2} + 6 \, a^{5} x}{6 \, c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.20083, size = 193, normalized size = 6.43 \begin{align*} \frac{{\left (b^{6} x^{7} + 7 \, a b^{5} x^{6} + 21 \, a^{2} b^{4} x^{5} + 35 \, a^{3} b^{3} x^{4} + 35 \, a^{4} b^{2} x^{3} + 21 \, a^{5} b x^{2} + 6 \, a^{6} x\right )} \sqrt{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{6 \, 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 )^{2}}\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 )}^{2}}\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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