Optimal. Leaf size=32 \[ \frac{2 (a+b x)^{3/2}}{3 (c+d x)^{3/2} (b c-a d)} \]
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Rubi [A] time = 0.0027823, 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} \[ \frac{2 (a+b x)^{3/2}}{3 (c+d x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x}}{(c+d x)^{5/2}} \, dx &=\frac{2 (a+b x)^{3/2}}{3 (b c-a d) (c+d x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0094239, size = 32, normalized size = 1. \[ \frac{2 (a+b x)^{3/2}}{3 (c+d x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 27, normalized size = 0.8 \begin{align*} -{\frac{2}{3\,ad-3\,bc} \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( dx+c \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 = 3.19569, size = 139, normalized size = 4.34 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}} \sqrt{d x + c}}{3 \,{\left (b c^{3} - a c^{2} d +{\left (b c d^{2} - a d^{3}\right )} x^{2} + 2 \,{\left (b c^{2} d - a c d^{2}\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{a + b x}}{\left (c + d 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 = 1.14782, size = 90, normalized size = 2.81 \begin{align*} -\frac{{\left (b x + a\right )}^{\frac{3}{2}} b^{4} d}{24 \,{\left (b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}\right )}{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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