Optimal. Leaf size=42 \[ \frac {2 b (c+d x)^{5/2}}{5 d^2}-\frac {2 (c+d x)^{3/2} (b c-a d)}{3 d^2} \]
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Rubi [A] time = 0.01, antiderivative size = 42, 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} \[ \frac {2 b (c+d x)^{5/2}}{5 d^2}-\frac {2 (c+d x)^{3/2} (b c-a d)}{3 d^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int (a+b x) \sqrt {c+d x} \, dx &=\int \left (\frac {(-b c+a d) \sqrt {c+d x}}{d}+\frac {b (c+d x)^{3/2}}{d}\right ) \, dx\\ &=-\frac {2 (b c-a d) (c+d x)^{3/2}}{3 d^2}+\frac {2 b (c+d x)^{5/2}}{5 d^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 30, normalized size = 0.71 \[ \frac {2 (c+d x)^{3/2} (5 a d-2 b c+3 b d x)}{15 d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 46, normalized size = 1.10 \[ \frac {2 \, {\left (3 \, b d^{2} x^{2} - 2 \, b c^{2} + 5 \, a c d + {\left (b c d + 5 \, a d^{2}\right )} x\right )} \sqrt {d x + c}}{15 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.37, size = 100, normalized size = 2.38 \[ \frac {2 \, {\left (15 \, \sqrt {d x + c} a c + 5 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a + \frac {5 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} b c}{d} + \frac {{\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} b}{d}\right )}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 27, normalized size = 0.64 \[ \frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (3 b d x +5 a d -2 b c \right )}{15 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 33, normalized size = 0.79 \[ \frac {2 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} b - 5 \, {\left (b c - a d\right )} {\left (d x + c\right )}^{\frac {3}{2}}\right )}}{15 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 29, normalized size = 0.69 \[ \frac {2\,{\left (c+d\,x\right )}^{3/2}\,\left (5\,a\,d-5\,b\,c+3\,b\,\left (c+d\,x\right )\right )}{15\,d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.12, size = 36, normalized size = 0.86 \[ \frac {2 \left (\frac {b \left (c + d x\right )^{\frac {5}{2}}}{5 d} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (a d - b c\right )}{3 d}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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