Optimal. Leaf size=40 \[ \frac {2 (b c-a d)}{3 d^2 (c+d x)^{3/2}}-\frac {2 b}{d^2 \sqrt {c+d x}} \]
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Rubi [A] time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {2 (b c-a d)}{3 d^2 (c+d x)^{3/2}}-\frac {2 b}{d^2 \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {a+b x}{(c+d x)^{5/2}} \, dx &=\int \left (\frac {-b c+a d}{d (c+d x)^{5/2}}+\frac {b}{d (c+d x)^{3/2}}\right ) \, dx\\ &=\frac {2 (b c-a d)}{3 d^2 (c+d x)^{3/2}}-\frac {2 b}{d^2 \sqrt {c+d x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 29, normalized size = 0.72 \begin {gather*} -\frac {2 (a d+2 b c+3 b d x)}{3 d^2 (c+d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 32, normalized size = 0.80 \begin {gather*} -\frac {2 (a d+3 b (c+d x)-b c)}{3 d^2 (c+d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 46, normalized size = 1.15 \begin {gather*} -\frac {2 \, {\left (3 \, b d x + 2 \, b c + a d\right )} \sqrt {d x + c}}{3 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 28, normalized size = 0.70 \begin {gather*} -\frac {2 \, {\left (3 \, {\left (d x + c\right )} b - b c + a d\right )}}{3 \, {\left (d x + c\right )}^{\frac {3}{2}} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 0.65 \begin {gather*} -\frac {2 \left (3 b d x +a d +2 b c \right )}{3 \left (d x +c \right )^{\frac {3}{2}} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 28, normalized size = 0.70 \begin {gather*} -\frac {2 \, {\left (3 \, {\left (d x + c\right )} b - b c + a d\right )}}{3 \, {\left (d x + c\right )}^{\frac {3}{2}} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 29, normalized size = 0.72 \begin {gather*} -\frac {2\,a\,d-2\,b\,c+6\,b\,\left (c+d\,x\right )}{3\,d^2\,{\left (c+d\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.12, size = 124, normalized size = 3.10 \begin {gather*} \begin {cases} - \frac {2 a d}{3 c d^{2} \sqrt {c + d x} + 3 d^{3} x \sqrt {c + d x}} - \frac {4 b c}{3 c d^{2} \sqrt {c + d x} + 3 d^{3} x \sqrt {c + d x}} - \frac {6 b d x}{3 c d^{2} \sqrt {c + d x} + 3 d^{3} x \sqrt {c + d x}} & \text {for}\: d \neq 0 \\\frac {a x + \frac {b x^{2}}{2}}{c^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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