Optimal. Leaf size=93 \[ -\frac {11 \sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{2 (2+3 x)^2}-\frac {121 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{28 \sqrt {7}} \]
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Rubi [A]
time = 0.01, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {96, 95, 210}
\begin {gather*} -\frac {121 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{28 \sqrt {7}}+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{2 (3 x+2)^2}-\frac {11 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 96
Rule 210
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^3} \, dx &=\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{2 (2+3 x)^2}+\frac {11}{4} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {11 \sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{2 (2+3 x)^2}+\frac {121}{56} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {11 \sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{2 (2+3 x)^2}+\frac {121}{28} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {11 \sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{2 (2+3 x)^2}-\frac {121 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{28 \sqrt {7}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 69, normalized size = 0.74 \begin {gather*} \frac {1}{196} \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (20+37 x)}{(2+3 x)^2}-121 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right ) \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. \(153\) vs.
\(2(72)=144\).
time = 0.12, size = 154, normalized size = 1.66
method | result | size |
risch | \(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (37 x +20\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{28 \left (2+3 x \right )^{2} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {121 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{392 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(119\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (1089 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+1452 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +484 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+518 x \sqrt {-10 x^{2}-x +3}+280 \sqrt {-10 x^{2}-x +3}\right )}{392 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{2}}\) | \(154\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 90, normalized size = 0.97 \begin {gather*} \frac {121}{392} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {5}{21} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{14 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {37 \, \sqrt {-10 \, x^{2} - x + 3}}{84 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.84, size = 86, normalized size = 0.92 \begin {gather*} -\frac {121 \, \sqrt {7} {\left (9 \, x^{2} + 12 \, x + 4\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 ) - 14 \, {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{392 \, {\left (9 \, x^{2} + 12 \, x + 4\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 {\sqrt {1 - 2 x} \sqrt {5 x + 3}}{\left (3 x + 2\right )^{3}}\, 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. 250 vs.
\(2 (72) = 144\).
time = 0.60, size = 250, normalized size = 2.69 \begin {gather*} \frac {121}{3920} \, \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 {121 \, \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 {280 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {1120 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{14 \, {\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 )}^{2} + 280\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 12.62, size = 1037, normalized size = 11.15 \begin {gather*} \frac {\frac {2129\,{\left (\sqrt {1-2\,x}-1\right )}^5}{875\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}-\frac {4258\,{\left (\sqrt {1-2\,x}-1\right )}^3}{4375\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {158\,\left (\sqrt {1-2\,x}-1\right )}{4375\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {79\,{\left (\sqrt {1-2\,x}-1\right )}^7}{140\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}+\frac {991\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{4375\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}-\frac {376\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{875\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {991\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{700\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}}{\frac {544\,{\left (\sqrt {1-2\,x}-1\right )}^2}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}-\frac {1764\,{\left (\sqrt {1-2\,x}-1\right )}^4}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {136\,{\left (\sqrt {1-2\,x}-1\right )}^6}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^8}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}-\frac {96\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^3}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}+\frac {48\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^5}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}+\frac {12\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^7}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}-\frac {96\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{625\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {16}{625}}-\frac {121\,\sqrt {7}\,\mathrm {atan}\left (\frac {\frac {121\,\sqrt {7}\,\left (\frac {726\,\sqrt {3}}{875}+\frac {363\,\left (\sqrt {1-2\,x}-1\right )}{875\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {363\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{175\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}-\frac {\sqrt {7}\,\left (\frac {212\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {888\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {536}{125}\right )\,121{}\mathrm {i}}{392}\right )}{392}+\frac {121\,\sqrt {7}\,\left (\frac {726\,\sqrt {3}}{875}+\frac {363\,\left (\sqrt {1-2\,x}-1\right )}{875\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {363\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{175\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {\sqrt {7}\,\left (\frac {212\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {888\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {536}{125}\right )\,121{}\mathrm {i}}{392}\right )}{392}}{\frac {14641\,{\left (\sqrt {1-2\,x}-1\right )}^2}{9800\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {14641}{24500}+\frac {\sqrt {7}\,\left (\frac {726\,\sqrt {3}}{875}+\frac {363\,\left (\sqrt {1-2\,x}-1\right )}{875\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {363\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{175\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}-\frac {\sqrt {7}\,\left (\frac {212\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {888\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {536}{125}\right )\,121{}\mathrm {i}}{392}\right )\,121{}\mathrm {i}}{392}-\frac {\sqrt {7}\,\left (\frac {726\,\sqrt {3}}{875}+\frac {363\,\left (\sqrt {1-2\,x}-1\right )}{875\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {363\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{175\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {\sqrt {7}\,\left (\frac {212\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {888\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {536}{125}\right )\,121{}\mathrm {i}}{392}\right )\,121{}\mathrm {i}}{392}}\right )}{196} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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