Optimal. Leaf size=67 \[ \frac{4 b (b c-a d)}{d^3 \sqrt{c+d x}}-\frac{2 (b c-a d)^2}{3 d^3 (c+d x)^{3/2}}+\frac{2 b^2 \sqrt{c+d x}}{d^3} \]
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Rubi [A] time = 0.0211512, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {43} \[ \frac{4 b (b c-a d)}{d^3 \sqrt{c+d x}}-\frac{2 (b c-a d)^2}{3 d^3 (c+d x)^{3/2}}+\frac{2 b^2 \sqrt{c+d x}}{d^3} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{(c+d x)^{5/2}} \, dx &=\int \left (\frac{(-b c+a d)^2}{d^2 (c+d x)^{5/2}}-\frac{2 b (b c-a d)}{d^2 (c+d x)^{3/2}}+\frac{b^2}{d^2 \sqrt{c+d x}}\right ) \, dx\\ &=-\frac{2 (b c-a d)^2}{3 d^3 (c+d x)^{3/2}}+\frac{4 b (b c-a d)}{d^3 \sqrt{c+d x}}+\frac{2 b^2 \sqrt{c+d x}}{d^3}\\ \end{align*}
Mathematica [A] time = 0.0321098, size = 62, normalized size = 0.93 \[ \frac{-2 a^2 d^2-4 a b d (2 c+3 d x)+2 b^2 \left (8 c^2+12 c d x+3 d^2 x^2\right )}{3 d^3 (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 62, normalized size = 0.9 \begin{align*} -{\frac{-6\,{b}^{2}{x}^{2}{d}^{2}+12\,ab{d}^{2}x-24\,{b}^{2}cdx+2\,{a}^{2}{d}^{2}+8\,abcd-16\,{b}^{2}{c}^{2}}{3\,{d}^{3}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.956515, size = 97, normalized size = 1.45 \begin{align*} \frac{2 \,{\left (\frac{3 \, \sqrt{d x + c} b^{2}}{d^{2}} - \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2} - 6 \,{\left (b^{2} c - a b d\right )}{\left (d x + c\right )}}{{\left (d x + c\right )}^{\frac{3}{2}} d^{2}}\right )}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.084, size = 174, normalized size = 2.6 \begin{align*} \frac{2 \,{\left (3 \, b^{2} d^{2} x^{2} + 8 \, b^{2} c^{2} - 4 \, a b c d - a^{2} d^{2} + 6 \,{\left (2 \, b^{2} c d - a b d^{2}\right )} x\right )} \sqrt{d x + c}}{3 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.20079, size = 265, normalized size = 3.96 \begin{align*} \begin{cases} - \frac{2 a^{2} d^{2}}{3 c d^{3} \sqrt{c + d x} + 3 d^{4} x \sqrt{c + d x}} - \frac{8 a b c d}{3 c d^{3} \sqrt{c + d x} + 3 d^{4} x \sqrt{c + d x}} - \frac{12 a b d^{2} x}{3 c d^{3} \sqrt{c + d x} + 3 d^{4} x \sqrt{c + d x}} + \frac{16 b^{2} c^{2}}{3 c d^{3} \sqrt{c + d x} + 3 d^{4} x \sqrt{c + d x}} + \frac{24 b^{2} c d x}{3 c d^{3} \sqrt{c + d x} + 3 d^{4} x \sqrt{c + d x}} + \frac{6 b^{2} d^{2} x^{2}}{3 c d^{3} \sqrt{c + d x} + 3 d^{4} x \sqrt{c + d x}} & \text{for}\: d \neq 0 \\\frac{a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3}}{c^{\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 = 1.08627, size = 97, normalized size = 1.45 \begin{align*} \frac{2 \, \sqrt{d x + c} b^{2}}{d^{3}} + \frac{2 \,{\left (6 \,{\left (d x + c\right )} b^{2} c - b^{2} c^{2} - 6 \,{\left (d x + c\right )} a b d + 2 \, a b c d - a^{2} d^{2}\right )}}{3 \,{\left (d x + c\right )}^{\frac{3}{2}} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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