Optimal. Leaf size=30 \[ -\frac{c^2 \sqrt{\frac{c}{(a+b x)^2}}}{4 b (a+b x)^3} \]
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Rubi [A] time = 0.0087175, 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} \[ -\frac{c^2 \sqrt{\frac{c}{(a+b x)^2}}}{4 b (a+b x)^3} \]
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)^2}\right )^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{c}{x^2}\right )^{5/2} \, dx,x,a+b x\right )}{b}\\ &=\frac{\left (c^2 \sqrt{\frac{c}{(a+b x)^2}} (a+b x)\right ) \operatorname{Subst}\left (\int \frac{1}{x^5} \, dx,x,a+b x\right )}{b}\\ &=-\frac{c^2 \sqrt{\frac{c}{(a+b x)^2}}}{4 b (a+b x)^3}\\ \end{align*}
Mathematica [A] time = 0.0104442, size = 25, normalized size = 0.83 \[ -\frac{(a+b x) \left (\frac{c}{(a+b x)^2}\right )^{5/2}}{4 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 22, normalized size = 0.7 \begin{align*} -{\frac{bx+a}{4\,b} \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 [A] time = 1.19904, size = 66, normalized size = 2.2 \begin{align*} -\frac{c^{\frac{5}{2}}}{4 \,{\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.31098, size = 122, normalized size = 4.07 \begin{align*} -\frac{c^{2} \sqrt{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{4 \,{\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\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 \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 [A] time = 1.11585, size = 1, normalized size = 0.03 \begin{align*} +\infty \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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