Optimal. Leaf size=22 \[ \frac{2 (a c+b c x)^{5/2}}{5 b c^6} \]
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Rubi [A] time = 0.0042234, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {21, 32} \[ \frac{2 (a c+b c x)^{5/2}}{5 b c^6} \]
Antiderivative was successfully verified.
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Rule 21
Rule 32
Rubi steps
\begin{align*} \int \frac{(a+b x)^5}{(a c+b c x)^{7/2}} \, dx &=\frac{\int (a c+b c x)^{3/2} \, dx}{c^5}\\ &=\frac{2 (a c+b c x)^{5/2}}{5 b c^6}\\ \end{align*}
Mathematica [A] time = 0.0126118, size = 25, normalized size = 1.14 \[ \frac{2 (a+b x)^6}{5 b (c (a+b x))^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 23, normalized size = 1.1 \begin{align*}{\frac{2\, \left ( bx+a \right ) ^{6}}{5\,b} \left ( bcx+ac \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.979997, size = 24, normalized size = 1.09 \begin{align*} \frac{2 \,{\left (b c x + a c\right )}^{\frac{5}{2}}}{5 \, b c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00158, size = 77, normalized size = 3.5 \begin{align*} \frac{2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b c x + a c}}{5 \, b c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.49026, size = 80, normalized size = 3.64 \begin{align*} \begin{cases} \frac{2 a^{2} \sqrt{a c + b c x}}{5 b c^{4}} + \frac{4 a x \sqrt{a c + b c x}}{5 c^{4}} + \frac{2 b x^{2} \sqrt{a c + b c x}}{5 c^{4}} & \text{for}\: b \neq 0 \\\frac{a^{5} x}{\left (a c\right )^{\frac{7}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.07611, size = 143, normalized size = 6.5 \begin{align*} \frac{2 \,{\left (15 \, \sqrt{b c x + a c} a^{2} - \frac{10 \,{\left (3 \, \sqrt{b c x + a c} a c -{\left (b c x + a c\right )}^{\frac{3}{2}}\right )} a}{c} + \frac{15 \, \sqrt{b c x + a c} a^{2} c^{2} - 10 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a c + 3 \,{\left (b c x + a c\right )}^{\frac{5}{2}}}{c^{2}}\right )}}{15 \, b c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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