Optimal. Leaf size=42 \[ \frac{2 b (c+d x)^{7/2}}{7 d^2}-\frac{2 (c+d x)^{5/2} (b c-a d)}{5 d^2} \]
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Rubi [A] time = 0.0139126, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ \frac{2 b (c+d x)^{7/2}}{7 d^2}-\frac{2 (c+d x)^{5/2} (b c-a d)}{5 d^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x) (c+d x)^{3/2} \, dx &=\int \left (\frac{(-b c+a d) (c+d x)^{3/2}}{d}+\frac{b (c+d x)^{5/2}}{d}\right ) \, dx\\ &=-\frac{2 (b c-a d) (c+d x)^{5/2}}{5 d^2}+\frac{2 b (c+d x)^{7/2}}{7 d^2}\\ \end{align*}
Mathematica [A] time = 0.0202293, size = 30, normalized size = 0.71 \[ \frac{2 (c+d x)^{5/2} (7 a d-2 b c+5 b d x)}{35 d^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 27, normalized size = 0.6 \begin{align*}{\frac{10\,bdx+14\,ad-4\,bc}{35\,{d}^{2}} \left ( dx+c \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.948696, size = 45, normalized size = 1.07 \begin{align*} \frac{2 \,{\left (5 \,{\left (d x + c\right )}^{\frac{7}{2}} b - 7 \,{\left (b c - a d\right )}{\left (d x + c\right )}^{\frac{5}{2}}\right )}}{35 \, d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.94714, size = 155, normalized size = 3.69 \begin{align*} \frac{2 \,{\left (5 \, b d^{3} x^{3} - 2 \, b c^{3} + 7 \, a c^{2} d +{\left (8 \, b c d^{2} + 7 \, a d^{3}\right )} x^{2} +{\left (b c^{2} d + 14 \, a c d^{2}\right )} x\right )} \sqrt{d x + c}}{35 \, d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.681699, size = 146, normalized size = 3.48 \begin{align*} \begin{cases} \frac{2 a c^{2} \sqrt{c + d x}}{5 d} + \frac{4 a c x \sqrt{c + d x}}{5} + \frac{2 a d x^{2} \sqrt{c + d x}}{5} - \frac{4 b c^{3} \sqrt{c + d x}}{35 d^{2}} + \frac{2 b c^{2} x \sqrt{c + d x}}{35 d} + \frac{16 b c x^{2} \sqrt{c + d x}}{35} + \frac{2 b d x^{3} \sqrt{c + d x}}{7} & \text{for}\: d \neq 0 \\c^{\frac{3}{2}} \left (a x + \frac{b x^{2}}{2}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.04931, size = 140, normalized size = 3.33 \begin{align*} \frac{2 \,{\left (35 \,{\left (d x + c\right )}^{\frac{3}{2}} a c + 7 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} c\right )} a + \frac{7 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} c\right )} b c}{d} + \frac{{\left (15 \,{\left (d x + c\right )}^{\frac{7}{2}} - 42 \,{\left (d x + c\right )}^{\frac{5}{2}} c + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2}\right )} b}{d}\right )}}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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