Optimal. Leaf size=77 \[ -\frac {10 \sqrt {1-2 x}}{33 (3+5 x)^{3/2}}+\frac {950 \sqrt {1-2 x}}{363 \sqrt {3+5 x}}-\frac {18 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{\sqrt {7}} \]
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Rubi [A]
time = 0.02, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {106, 157, 12,
95, 210} \begin {gather*} -\frac {18 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}}+\frac {950 \sqrt {1-2 x}}{363 \sqrt {5 x+3}}-\frac {10 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 106
Rule 157
Rule 210
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx &=-\frac {10 \sqrt {1-2 x}}{33 (3+5 x)^{3/2}}-\frac {2}{33} \int \frac {\frac {59}{2}-30 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {10 \sqrt {1-2 x}}{33 (3+5 x)^{3/2}}+\frac {950 \sqrt {1-2 x}}{363 \sqrt {3+5 x}}+\frac {4}{363} \int \frac {3267}{4 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {10 \sqrt {1-2 x}}{33 (3+5 x)^{3/2}}+\frac {950 \sqrt {1-2 x}}{363 \sqrt {3+5 x}}+9 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {10 \sqrt {1-2 x}}{33 (3+5 x)^{3/2}}+\frac {950 \sqrt {1-2 x}}{363 \sqrt {3+5 x}}+18 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {10 \sqrt {1-2 x}}{33 (3+5 x)^{3/2}}+\frac {950 \sqrt {1-2 x}}{363 \sqrt {3+5 x}}-\frac {18 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{\sqrt {7}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 60, normalized size = 0.78 \begin {gather*} \frac {10 \sqrt {1-2 x} (274+475 x)}{363 (3+5 x)^{3/2}}-\frac {18 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{\sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(146\) vs.
\(2(58)=116\).
time = 0.08, size = 147, normalized size = 1.91
method | result | size |
default | \(\frac {\left (81675 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+98010 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +29403 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+33250 x \sqrt {-10 x^{2}-x +3}+19180 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{2541 \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(147\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.49, size = 86, normalized size = 1.12 \begin {gather*} -\frac {3267 \, \sqrt {7} {\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 70 \, {\left (475 \, x + 274\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2541 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {1 - 2 x} \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 190 vs.
\(2 (58) = 116\).
time = 1.21, size = 190, normalized size = 2.47 \begin {gather*} \frac {9}{70} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1}{5808} \, \sqrt {10} {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {744 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {2976 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {1-2\,x}\,\left (3\,x+2\right )\,{\left (5\,x+3\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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