Optimal. Leaf size=32 \[ -\frac{2 (c+d x)^{3/2}}{3 (a+b x)^{3/2} (b c-a d)} \]
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Rubi [A] time = 0.0029979, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {37} \[ -\frac{2 (c+d x)^{3/2}}{3 (a+b x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int \frac{\sqrt{c+d x}}{(a+b x)^{5/2}} \, dx &=-\frac{2 (c+d x)^{3/2}}{3 (b c-a d) (a+b x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.011371, size = 32, normalized size = 1. \[ -\frac{2 (c+d x)^{3/2}}{3 (a+b x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 27, normalized size = 0.8 \begin{align*}{\frac{2}{3\,ad-3\,bc} \left ( dx+c \right ) ^{{\frac{3}{2}}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.3436, size = 140, normalized size = 4.38 \begin{align*} -\frac{2 \, \sqrt{b x + a}{\left (d x + c\right )}^{\frac{3}{2}}}{3 \,{\left (a^{2} b c - a^{3} d +{\left (b^{3} c - a b^{2} d\right )} x^{2} + 2 \,{\left (a b^{2} c - a^{2} b d\right )} x\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 \frac{\sqrt{c + d x}}{\left (a + b x\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13717, size = 205, normalized size = 6.41 \begin{align*} -\frac{4 \,{\left (\sqrt{b d} b^{4} c^{2} d - 2 \, \sqrt{b d} a b^{3} c d^{2} + \sqrt{b d} a^{2} b^{2} d^{3} + 3 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} d\right )}{\left | b \right |}}{3 \,{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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