Optimal. Leaf size=153 \[ \frac {34655 \sqrt {1-2 x}}{77 (5 x+3)}-\frac {1045 \sqrt {1-2 x}}{14 (5 x+3)^2}+\frac {139 \sqrt {1-2 x}}{14 (3 x+2) (5 x+3)^2}+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}+\frac {43467}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {66325}{11} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {99, 151, 156, 63, 206} \[ \frac {34655 \sqrt {1-2 x}}{77 (5 x+3)}-\frac {1045 \sqrt {1-2 x}}{14 (5 x+3)^2}+\frac {139 \sqrt {1-2 x}}{14 (3 x+2) (5 x+3)^2}+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}+\frac {43467}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {66325}{11} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 99
Rule 151
Rule 156
Rule 206
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx &=\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^2}-\frac {1}{2} \int \frac {-23+35 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx\\ &=\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^2}+\frac {139 \sqrt {1-2 x}}{14 (2+3 x) (3+5 x)^2}-\frac {1}{14} \int \frac {-2513+3475 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac {1045 \sqrt {1-2 x}}{14 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^2}+\frac {139 \sqrt {1-2 x}}{14 (2+3 x) (3+5 x)^2}+\frac {1}{308} \int \frac {-180818+206910 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {1045 \sqrt {1-2 x}}{14 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^2}+\frac {139 \sqrt {1-2 x}}{14 (2+3 x) (3+5 x)^2}+\frac {34655 \sqrt {1-2 x}}{77 (3+5 x)}-\frac {\int \frac {-7469374+4574460 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{3388}\\ &=-\frac {1045 \sqrt {1-2 x}}{14 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^2}+\frac {139 \sqrt {1-2 x}}{14 (2+3 x) (3+5 x)^2}+\frac {34655 \sqrt {1-2 x}}{77 (3+5 x)}-\frac {130401}{14} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {331625}{22} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {1045 \sqrt {1-2 x}}{14 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^2}+\frac {139 \sqrt {1-2 x}}{14 (2+3 x) (3+5 x)^2}+\frac {34655 \sqrt {1-2 x}}{77 (3+5 x)}+\frac {130401}{14} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {331625}{22} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {1045 \sqrt {1-2 x}}{14 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^2}+\frac {139 \sqrt {1-2 x}}{14 (2+3 x) (3+5 x)^2}+\frac {34655 \sqrt {1-2 x}}{77 (3+5 x)}+\frac {43467}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {66325}{11} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A] time = 0.13, size = 119, normalized size = 0.78 \[ \frac {10519014 \sqrt {21} \left (15 x^2+19 x+6\right )^2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-6499850 \sqrt {55} \left (15 x^2+19 x+6\right )^2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )+77 \sqrt {1-2 x} \left (3118950 x^3+5926515 x^2+3748007 x+788875\right )}{11858 (3 x+2)^2 (5 x+3)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 162, normalized size = 1.06 \[ \frac {3249925 \, \sqrt {11} \sqrt {5} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 5259507 \, \sqrt {7} \sqrt {3} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (3118950 \, x^{3} + 5926515 \, x^{2} + 3748007 \, x + 788875\right )} \sqrt {-2 \, x + 1}}{11858 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.25, size = 148, normalized size = 0.97 \[ \frac {66325}{242} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {43467}{98} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2 \, {\left (1559475 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 10604940 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 24027469 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 18137504 \, \sqrt {-2 \, x + 1}\right )}}{77 \, {\left (15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 94, normalized size = 0.61 \[ \frac {43467 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{49}-\frac {66325 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{121}+\frac {-\frac {24875 \left (-2 x +1\right )^{\frac {3}{2}}}{11}+4925 \sqrt {-2 x +1}}{\left (-10 x -6\right )^{2}}-\frac {972 \left (\frac {209 \left (-2 x +1\right )^{\frac {3}{2}}}{252}-\frac {211 \sqrt {-2 x +1}}{108}\right )}{\left (-6 x -4\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 146, normalized size = 0.95 \[ \frac {66325}{242} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {43467}{98} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (1559475 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 10604940 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 24027469 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 18137504 \, \sqrt {-2 \, x + 1}\right )}}{77 \, {\left (225 \, {\left (2 \, x - 1\right )}^{4} + 2040 \, {\left (2 \, x - 1\right )}^{3} + 6934 \, {\left (2 \, x - 1\right )}^{2} + 20944 \, x - 4543\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 107, normalized size = 0.70 \[ \frac {43467\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{49}-\frac {66325\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121}+\frac {\frac {471104\,\sqrt {1-2\,x}}{225}-\frac {48054938\,{\left (1-2\,x\right )}^{3/2}}{17325}+\frac {1413992\,{\left (1-2\,x\right )}^{5/2}}{1155}-\frac {13862\,{\left (1-2\,x\right )}^{7/2}}{77}}{\frac {20944\,x}{225}+\frac {6934\,{\left (2\,x-1\right )}^2}{225}+\frac {136\,{\left (2\,x-1\right )}^3}{15}+{\left (2\,x-1\right )}^4-\frac {4543}{225}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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