Optimal. Leaf size=38 \[ \frac {77}{12 (1-2 x)^{3/2}}-\frac {17}{\sqrt {1-2 x}}-\frac {15}{4} \sqrt {1-2 x} \]
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Rubi [A]
time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78}
\begin {gather*} -\frac {15}{4} \sqrt {1-2 x}-\frac {17}{\sqrt {1-2 x}}+\frac {77}{12 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rubi steps
\begin {align*} \int \frac {(2+3 x) (3+5 x)}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac {77}{4 (1-2 x)^{5/2}}-\frac {17}{(1-2 x)^{3/2}}+\frac {15}{4 \sqrt {1-2 x}}\right ) \, dx\\ &=\frac {77}{12 (1-2 x)^{3/2}}-\frac {17}{\sqrt {1-2 x}}-\frac {15}{4} \sqrt {1-2 x}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 23, normalized size = 0.61 \begin {gather*} -\frac {43-147 x+45 x^2}{3 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 29, normalized size = 0.76
method | result | size |
gosper | \(-\frac {45 x^{2}-147 x +43}{3 \left (1-2 x \right )^{\frac {3}{2}}}\) | \(20\) |
trager | \(-\frac {\left (45 x^{2}-147 x +43\right ) \sqrt {1-2 x}}{3 \left (-1+2 x \right )^{2}}\) | \(27\) |
risch | \(\frac {45 x^{2}-147 x +43}{3 \left (-1+2 x \right ) \sqrt {1-2 x}}\) | \(27\) |
derivativedivides | \(\frac {77}{12 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {17}{\sqrt {1-2 x}}-\frac {15 \sqrt {1-2 x}}{4}\) | \(29\) |
default | \(\frac {77}{12 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {17}{\sqrt {1-2 x}}-\frac {15 \sqrt {1-2 x}}{4}\) | \(29\) |
meijerg | \(-\frac {4 \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {\frac {19 \sqrt {\pi }}{3}-\frac {19 \sqrt {\pi }\, \left (-24 x +8\right )}{24 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {5 \left (-4 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (24 x^{2}-48 x +16\right )}{4 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{2 \sqrt {\pi }}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.39, size = 24, normalized size = 0.63 \begin {gather*} -\frac {15}{4} \, \sqrt {-2 \, x + 1} + \frac {408 \, x - 127}{12 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.77, size = 31, normalized size = 0.82 \begin {gather*} -\frac {{\left (45 \, x^{2} - 147 \, x + 43\right )} \sqrt {-2 \, x + 1}}{3 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs.
\(2 (32) = 64\).
time = 0.23, size = 65, normalized size = 1.71 \begin {gather*} - \frac {45 x^{2} \sqrt {1 - 2 x}}{12 x^{2} - 12 x + 3} + \frac {147 x \sqrt {1 - 2 x}}{12 x^{2} - 12 x + 3} - \frac {43 \sqrt {1 - 2 x}}{12 x^{2} - 12 x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.17, size = 31, normalized size = 0.82 \begin {gather*} -\frac {15}{4} \, \sqrt {-2 \, x + 1} - \frac {408 \, x - 127}{12 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.22, size = 29, normalized size = 0.76 \begin {gather*} \frac {45\,{\left (2\,x-1\right )}^2-408\,x+127}{\sqrt {1-2\,x}\,\left (24\,x-12\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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