Optimal. Leaf size=30 \[ -\frac{2 c^2 \sqrt{\frac{c}{a+b x}}}{3 b (a+b x)} \]
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Rubi [A] time = 0.0079889, 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} \[ -\frac{2 c^2 \sqrt{\frac{c}{a+b x}}}{3 b (a+b x)} \]
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}\right )^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{c}{x}\right )^{5/2} \, dx,x,a+b x\right )}{b}\\ &=\frac{\left (c^2 \sqrt{\frac{c}{a+b x}} \sqrt{a+b x}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{5/2}} \, dx,x,a+b x\right )}{b}\\ &=-\frac{2 c^2 \sqrt{\frac{c}{a+b x}}}{3 b (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0115844, size = 21, normalized size = 0.7 \[ -\frac{2 c \left (\frac{c}{a+b x}\right )^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 22, normalized size = 0.7 \begin{align*} -{\frac{2\,bx+2\,a}{3\,b} \left ({\frac{c}{bx+a}} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.970897, size = 23, normalized size = 0.77 \begin{align*} -\frac{2 \, c \left (\frac{c}{b x + a}\right )^{\frac{3}{2}}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3314, size = 57, normalized size = 1.9 \begin{align*} -\frac{2 \, c^{2} \sqrt{\frac{c}{b x + a}}}{3 \,{\left (b^{2} x + a b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.70138, size = 51, normalized size = 1.7 \begin{align*} \begin{cases} - \frac{2 a c^{\frac{5}{2}} \left (\frac{1}{a + b x}\right )^{\frac{5}{2}}}{3 b} - \frac{2 c^{\frac{5}{2}} x \left (\frac{1}{a + b x}\right )^{\frac{5}{2}}}{3} & \text{for}\: b \neq 0 \\x \left (\frac{c}{a}\right )^{\frac{5}{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15108, size = 32, normalized size = 1.07 \begin{align*} -\frac{2 \, c^{4} \mathrm{sgn}\left (b x + a\right )}{3 \,{\left (b c x + a c\right )}^{\frac{3}{2}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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