Optimal. Leaf size=32 \[ -\frac{2 (c+d x)^{5/2}}{5 (a+b x)^{5/2} (b c-a d)} \]
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Rubi [A] time = 0.0030631, 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} \[ -\frac{2 (c+d x)^{5/2}}{5 (a+b x)^{5/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int \frac{(c+d x)^{3/2}}{(a+b x)^{7/2}} \, dx &=-\frac{2 (c+d x)^{5/2}}{5 (b c-a d) (a+b x)^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0140612, size = 32, normalized size = 1. \[ -\frac{2 (c+d x)^{5/2}}{5 (a+b x)^{5/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 27, normalized size = 0.8 \begin{align*}{\frac{2}{5\,ad-5\,bc} \left ( dx+c \right ) ^{{\frac{5}{2}}} \left ( bx+a \right ) ^{-{\frac{5}{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 = 4.3366, size = 216, normalized size = 6.75 \begin{align*} -\frac{2 \,{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \sqrt{b x + a} \sqrt{d x + c}}{5 \,{\left (a^{3} b c - a^{4} d +{\left (b^{4} c - a b^{3} d\right )} x^{3} + 3 \,{\left (a b^{3} c - a^{2} b^{2} d\right )} x^{2} + 3 \,{\left (a^{2} b^{2} c - a^{3} b d\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.53208, size = 505, normalized size = 15.78 \begin{align*} -\frac{4 \,{\left (\sqrt{b d} b^{8} c^{4} d^{2}{\left | b \right |} - 4 \, \sqrt{b d} a b^{7} c^{3} d^{3}{\left | b \right |} + 6 \, \sqrt{b d} a^{2} b^{6} c^{2} d^{4}{\left | b \right |} - 4 \, \sqrt{b d} a^{3} b^{5} c d^{5}{\left | b \right |} + \sqrt{b d} a^{4} b^{4} d^{6}{\left | b \right |} + 10 \, \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} b^{4} c^{2} d^{2}{\left | b \right |} - 20 \, \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} a b^{3} c d^{3}{\left | b \right |} + 10 \, \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} a^{2} b^{2} d^{4}{\left | b \right |} + 5 \, \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 )}^{8} d^{2}{\left | b \right |}\right )}}{5 \,{\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 )}^{5} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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