Optimal. Leaf size=67 \[ -\frac{4 b \sqrt{c+d x} (b c-a d)}{d^3}-\frac{2 (b c-a d)^2}{d^3 \sqrt{c+d x}}+\frac{2 b^2 (c+d x)^{3/2}}{3 d^3} \]
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Rubi [A] time = 0.0206038, 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} \[ -\frac{4 b \sqrt{c+d x} (b c-a d)}{d^3}-\frac{2 (b c-a d)^2}{d^3 \sqrt{c+d x}}+\frac{2 b^2 (c+d x)^{3/2}}{3 d^3} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{(c+d x)^{3/2}} \, dx &=\int \left (\frac{(-b c+a d)^2}{d^2 (c+d x)^{3/2}}-\frac{2 b (b c-a d)}{d^2 \sqrt{c+d x}}+\frac{b^2 \sqrt{c+d x}}{d^2}\right ) \, dx\\ &=-\frac{2 (b c-a d)^2}{d^3 \sqrt{c+d x}}-\frac{4 b (b c-a d) \sqrt{c+d x}}{d^3}+\frac{2 b^2 (c+d x)^{3/2}}{3 d^3}\\ \end{align*}
Mathematica [A] time = 0.0340529, size = 59, normalized size = 0.88 \[ \frac{2 \left (-3 a^2 d^2+6 a b d (2 c+d x)+b^2 \left (-8 c^2-4 c d x+d^2 x^2\right )\right )}{3 d^3 \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 63, normalized size = 0.9 \begin{align*} -{\frac{-2\,{b}^{2}{x}^{2}{d}^{2}-12\,ab{d}^{2}x+8\,{b}^{2}cdx+6\,{a}^{2}{d}^{2}-24\,abcd+16\,{b}^{2}{c}^{2}}{3\,{d}^{3}}{\frac{1}{\sqrt{dx+c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.946276, size = 101, normalized size = 1.51 \begin{align*} \frac{2 \,{\left (\frac{{\left (d x + c\right )}^{\frac{3}{2}} b^{2} - 6 \,{\left (b^{2} c - a b d\right )} \sqrt{d x + c}}{d^{2}} - \frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}}{\sqrt{d x + c} d^{2}}\right )}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02985, size = 157, normalized size = 2.34 \begin{align*} \frac{2 \,{\left (b^{2} d^{2} x^{2} - 8 \, b^{2} c^{2} + 12 \, a b c d - 3 \, a^{2} d^{2} - 2 \,{\left (2 \, b^{2} c d - 3 \, a b d^{2}\right )} x\right )} \sqrt{d x + c}}{3 \,{\left (d^{4} x + c d^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.99504, size = 65, normalized size = 0.97 \begin{align*} \frac{2 b^{2} \left (c + d x\right )^{\frac{3}{2}}}{3 d^{3}} + \frac{\sqrt{c + d x} \left (4 a b d - 4 b^{2} c\right )}{d^{3}} - \frac{2 \left (a d - b c\right )^{2}}{d^{3} \sqrt{c + d x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05313, size = 113, normalized size = 1.69 \begin{align*} -\frac{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}}{\sqrt{d x + c} d^{3}} + \frac{2 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} b^{2} d^{6} - 6 \, \sqrt{d x + c} b^{2} c d^{6} + 6 \, \sqrt{d x + c} a b d^{7}\right )}}{3 \, d^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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