Optimal. Leaf size=125 \[ -\frac {12295}{41503 \sqrt {1-2 x}}+\frac {33}{14 (1-2 x)^{3/2} (3 x+2)}-\frac {1115}{1617 (1-2 x)^{3/2}}+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2}+\frac {3645}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {1250}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {103, 151, 152, 156, 63, 206} \[ -\frac {12295}{41503 \sqrt {1-2 x}}+\frac {33}{14 (1-2 x)^{3/2} (3 x+2)}-\frac {1115}{1617 (1-2 x)^{3/2}}+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2}+\frac {3645}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {1250}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 103
Rule 151
Rule 152
Rule 156
Rule 206
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)} \, dx &=\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac {1}{14} \int \frac {7-105 x}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)} \, dx\\ &=\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac {33}{14 (1-2 x)^{3/2} (2+3 x)}+\frac {1}{98} \int \frac {-1015-5775 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {1115}{1617 (1-2 x)^{3/2}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac {33}{14 (1-2 x)^{3/2} (2+3 x)}-\frac {\int \frac {-\frac {46515}{2}+\frac {351225 x}{2}}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{11319}\\ &=-\frac {1115}{1617 (1-2 x)^{3/2}}-\frac {12295}{41503 \sqrt {1-2 x}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac {33}{14 (1-2 x)^{3/2} (2+3 x)}+\frac {2 \int \frac {\frac {6679995}{4}-\frac {3872925 x}{4}}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{871563}\\ &=-\frac {1115}{1617 (1-2 x)^{3/2}}-\frac {12295}{41503 \sqrt {1-2 x}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac {33}{14 (1-2 x)^{3/2} (2+3 x)}-\frac {10935}{686} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {3125}{121} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {1115}{1617 (1-2 x)^{3/2}}-\frac {12295}{41503 \sqrt {1-2 x}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac {33}{14 (1-2 x)^{3/2} (2+3 x)}+\frac {10935}{686} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {3125}{121} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {1115}{1617 (1-2 x)^{3/2}}-\frac {12295}{41503 \sqrt {1-2 x}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac {33}{14 (1-2 x)^{3/2} (2+3 x)}+\frac {3645}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {1250}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [C] time = 0.05, size = 83, normalized size = 0.66 \[ -\frac {26730 (3 x+2)^2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {3}{7}-\frac {6 x}{7}\right )-7 \left (3500 (3 x+2)^2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\frac {5}{11} (2 x-1)\right )+99 (33 x+23)\right )}{3234 (1-2 x)^{3/2} (3 x+2)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 162, normalized size = 1.30 \[ \frac {9003750 \, \sqrt {11} \sqrt {5} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 14554485 \, \sqrt {7} \sqrt {3} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (1327860 \, x^{3} - 438840 \, x^{2} - 594687 \, x + 245383\right )} \sqrt {-2 \, x + 1}}{19174386 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.26, size = 128, normalized size = 1.02 \[ \frac {625}{1331} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {3645}{4802} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {16 \, {\left (804 \, x - 479\right )}}{871563 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {27 \, {\left (243 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 581 \, \sqrt {-2 \, x + 1}\right )}}{9604 \, {\left (3 \, x + 2\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 84, normalized size = 0.67 \[ \frac {3645 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{2401}-\frac {1250 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{1331}+\frac {16}{11319 \left (-2 x +1\right )^{\frac {3}{2}}}+\frac {2144}{290521 \sqrt {-2 x +1}}-\frac {486 \left (\frac {27 \left (-2 x +1\right )^{\frac {3}{2}}}{2}-\frac {581 \sqrt {-2 x +1}}{18}\right )}{2401 \left (-6 x -4\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 128, normalized size = 1.02 \[ \frac {625}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {3645}{4802} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {331965 \, {\left (2 \, x - 1\right )}^{3} + 776475 \, {\left (2 \, x - 1\right )}^{2} - 75264 \, x + 46256}{124509 \, {\left (9 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 42 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 49 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 89, normalized size = 0.71 \[ \frac {3645\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {1250\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331}+\frac {\frac {12325\,{\left (2\,x-1\right )}^2}{17787}-\frac {512\,x}{7623}+\frac {12295\,{\left (2\,x-1\right )}^3}{41503}+\frac {944}{22869}}{\frac {49\,{\left (1-2\,x\right )}^{3/2}}{9}-\frac {14\,{\left (1-2\,x\right )}^{5/2}}{3}+{\left (1-2\,x\right )}^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: MellinTransformStripError} \]
Verification of antiderivative is not currently implemented for this CAS.
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