Optimal. Leaf size=153 \[ \frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac {584179 \sqrt {1-2 x}}{196 (3 x+2)}+\frac {25159 \sqrt {1-2 x}}{84 (3 x+2)^2}+\frac {1201 \sqrt {1-2 x}}{30 (3 x+2)^3}+\frac {63 \sqrt {1-2 x}}{10 (3 x+2)^4}+\frac {20149879 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{98 \sqrt {21}}-6050 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {98, 149, 151, 156, 63, 206} \begin {gather*} \frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac {584179 \sqrt {1-2 x}}{196 (3 x+2)}+\frac {25159 \sqrt {1-2 x}}{84 (3 x+2)^2}+\frac {1201 \sqrt {1-2 x}}{30 (3 x+2)^3}+\frac {63 \sqrt {1-2 x}}{10 (3 x+2)^4}+\frac {20149879 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{98 \sqrt {21}}-6050 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 149
Rule 151
Rule 156
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)} \, dx &=\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac {1}{15} \int \frac {(228-225 x) \sqrt {1-2 x}}{(2+3 x)^5 (3+5 x)} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac {63 \sqrt {1-2 x}}{10 (2+3 x)^4}-\frac {1}{180} \int \frac {-25182+37890 x}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac {63 \sqrt {1-2 x}}{10 (2+3 x)^4}+\frac {1201 \sqrt {1-2 x}}{30 (2+3 x)^3}-\frac {\int \frac {-2761290+3783150 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)} \, dx}{3780}\\ &=\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac {63 \sqrt {1-2 x}}{10 (2+3 x)^4}+\frac {1201 \sqrt {1-2 x}}{30 (2+3 x)^3}+\frac {25159 \sqrt {1-2 x}}{84 (2+3 x)^2}-\frac {\int \frac {-209531070+237752550 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)} \, dx}{52920}\\ &=\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac {63 \sqrt {1-2 x}}{10 (2+3 x)^4}+\frac {1201 \sqrt {1-2 x}}{30 (2+3 x)^3}+\frac {25159 \sqrt {1-2 x}}{84 (2+3 x)^2}+\frac {584179 \sqrt {1-2 x}}{196 (2+3 x)}-\frac {\int \frac {-9014096070+5520491550 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{370440}\\ &=\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac {63 \sqrt {1-2 x}}{10 (2+3 x)^4}+\frac {1201 \sqrt {1-2 x}}{30 (2+3 x)^3}+\frac {25159 \sqrt {1-2 x}}{84 (2+3 x)^2}+\frac {584179 \sqrt {1-2 x}}{196 (2+3 x)}-\frac {20149879}{196} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+166375 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac {63 \sqrt {1-2 x}}{10 (2+3 x)^4}+\frac {1201 \sqrt {1-2 x}}{30 (2+3 x)^3}+\frac {25159 \sqrt {1-2 x}}{84 (2+3 x)^2}+\frac {584179 \sqrt {1-2 x}}{196 (2+3 x)}+\frac {20149879}{196} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-166375 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac {63 \sqrt {1-2 x}}{10 (2+3 x)^4}+\frac {1201 \sqrt {1-2 x}}{30 (2+3 x)^3}+\frac {25159 \sqrt {1-2 x}}{84 (2+3 x)^2}+\frac {584179 \sqrt {1-2 x}}{196 (2+3 x)}+\frac {20149879 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{98 \sqrt {21}}-6050 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A] time = 0.17, size = 93, normalized size = 0.61 \begin {gather*} \frac {\sqrt {1-2 x} \left (709777485 x^4+1916515215 x^3+1941349752 x^2+874383298 x+147756688\right )}{2940 (3 x+2)^5}+\frac {20149879 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{98 \sqrt {21}}-6050 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.71, size = 113, normalized size = 0.74 \begin {gather*} -\frac {\sqrt {1-2 x} \left (709777485 (1-2 x)^4-6672140370 (1-2 x)^3+23523155208 (1-2 x)^2-36864065630 (1-2 x)+21667380315\right )}{1470 (3 (1-2 x)-7)^5}+\frac {20149879 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{98 \sqrt {21}}-6050 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.54, size = 170, normalized size = 1.11 \begin {gather*} \frac {62254500 \, \sqrt {55} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 100749395 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 7 \, {\left (709777485 \, x^{4} + 1916515215 \, x^{3} + 1941349752 \, x^{2} + 874383298 \, x + 147756688\right )} \sqrt {-2 \, x + 1}}{20580 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.09, size = 155, normalized size = 1.01 \begin {gather*} 3025 \, \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 {20149879}{4116} \, \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 {709777485 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 6672140370 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 23523155208 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 36864065630 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 21667380315 \, \sqrt {-2 \, x + 1}}{47040 \, {\left (3 \, x + 2\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 93, normalized size = 0.61 \begin {gather*} \frac {20149879 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{2058}-6050 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )-\frac {486 \left (\frac {584179 \left (-2 x +1\right )^{\frac {9}{2}}}{588}-\frac {504319 \left (-2 x +1\right )^{\frac {7}{2}}}{54}+\frac {13335122 \left (-2 x +1\right )^{\frac {5}{2}}}{405}-\frac {75232787 \left (-2 x +1\right )^{\frac {3}{2}}}{1458}+\frac {29479429 \sqrt {-2 x +1}}{972}\right )}{\left (-6 x -4\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.18, size = 164, normalized size = 1.07 \begin {gather*} 3025 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {20149879}{4116} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {709777485 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 6672140370 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 23523155208 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 36864065630 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 21667380315 \, \sqrt {-2 \, x + 1}}{1470 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 125, normalized size = 0.82 \begin {gather*} \frac {20149879\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2058}-6050\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {29479429\,\sqrt {1-2\,x}}{486}-\frac {75232787\,{\left (1-2\,x\right )}^{3/2}}{729}+\frac {26670244\,{\left (1-2\,x\right )}^{5/2}}{405}-\frac {504319\,{\left (1-2\,x\right )}^{7/2}}{27}+\frac {584179\,{\left (1-2\,x\right )}^{9/2}}{294}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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