Optimal. Leaf size=154 \[ \frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)}+\frac {6649 \sqrt {1-2 x}}{27 (3 x+2) (5 x+3)}+\frac {917 \sqrt {1-2 x}}{54 (3 x+2)^2 (5 x+3)}-\frac {44545 \sqrt {1-2 x}}{18 (5 x+3)}-\frac {307295 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3 \sqrt {21}}+3014 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 154, 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 = {98, 149, 151, 156, 63, 206} \[ \frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)}+\frac {6649 \sqrt {1-2 x}}{27 (3 x+2) (5 x+3)}+\frac {917 \sqrt {1-2 x}}{54 (3 x+2)^2 (5 x+3)}-\frac {44545 \sqrt {1-2 x}}{18 (5 x+3)}-\frac {307295 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3 \sqrt {21}}+3014 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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)^4 (3+5 x)^2} \, dx &=\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)}+\frac {1}{9} \int \frac {(197-163 x) \sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^2} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)}+\frac {917 \sqrt {1-2 x}}{54 (2+3 x)^2 (3+5 x)}-\frac {1}{54} \int \frac {-16180+22273 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)}+\frac {917 \sqrt {1-2 x}}{54 (2+3 x)^2 (3+5 x)}+\frac {6649 \sqrt {1-2 x}}{27 (2+3 x) (3+5 x)}-\frac {1}{378} \int \frac {-1220205+1396290 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {44545 \sqrt {1-2 x}}{18 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)}+\frac {917 \sqrt {1-2 x}}{54 (2+3 x)^2 (3+5 x)}+\frac {6649 \sqrt {1-2 x}}{27 (2+3 x) (3+5 x)}+\frac {\int \frac {-50405355+30869685 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{4158}\\ &=-\frac {44545 \sqrt {1-2 x}}{18 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)}+\frac {917 \sqrt {1-2 x}}{54 (2+3 x)^2 (3+5 x)}+\frac {6649 \sqrt {1-2 x}}{27 (2+3 x) (3+5 x)}+\frac {307295}{6} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-82885 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {44545 \sqrt {1-2 x}}{18 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)}+\frac {917 \sqrt {1-2 x}}{54 (2+3 x)^2 (3+5 x)}+\frac {6649 \sqrt {1-2 x}}{27 (2+3 x) (3+5 x)}-\frac {307295}{6} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+82885 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {44545 \sqrt {1-2 x}}{18 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)}+\frac {917 \sqrt {1-2 x}}{54 (2+3 x)^2 (3+5 x)}+\frac {6649 \sqrt {1-2 x}}{27 (2+3 x) (3+5 x)}-\frac {307295 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3 \sqrt {21}}+3014 \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 = 95, normalized size = 0.62 \[ -\frac {\sqrt {1-2 x} \left (400905 x^3+788512 x^2+516513 x+112668\right )}{6 (3 x+2)^3 (5 x+3)}-\frac {307295 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3 \sqrt {21}}+3014 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 150, normalized size = 0.97 \[ \frac {189882 \, \sqrt {55} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac {5 \, x - \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 307295 \, \sqrt {21} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (400905 \, x^{3} + 788512 \, x^{2} + 516513 \, x + 112668\right )} \sqrt {-2 \, x + 1}}{126 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 139, normalized size = 0.90 \[ -1507 \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {307295}{126} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {605 \, \sqrt {-2 \, x + 1}}{5 \, x + 3} - \frac {60579 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 285460 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 336385 \, \sqrt {-2 \, x + 1}}{24 \, {\left (3 \, x + 2\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 91, normalized size = 0.59 \[ -\frac {307295 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{63}+3014 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )+\frac {242 \sqrt {-2 x +1}}{-2 x -\frac {6}{5}}+\frac {20193 \left (-2 x +1\right )^{\frac {5}{2}}-\frac {285460 \left (-2 x +1\right )^{\frac {3}{2}}}{3}+\frac {336385 \sqrt {-2 x +1}}{3}}{\left (-6 x -4\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 146, normalized size = 0.95 \[ -1507 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {307295}{126} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {400905 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 2779739 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 6422815 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 4945325 \, \sqrt {-2 \, x + 1}}{3 \, {\left (135 \, {\left (2 \, x - 1\right )}^{4} + 1242 \, {\left (2 \, x - 1\right )}^{3} + 4284 \, {\left (2 \, x - 1\right )}^{2} + 13132 \, x - 2793\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 108, normalized size = 0.70 \[ 3014\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )-\frac {307295\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{63}-\frac {\frac {989065\,\sqrt {1-2\,x}}{81}-\frac {1284563\,{\left (1-2\,x\right )}^{3/2}}{81}+\frac {2779739\,{\left (1-2\,x\right )}^{5/2}}{405}-\frac {8909\,{\left (1-2\,x\right )}^{7/2}}{9}}{\frac {13132\,x}{135}+\frac {476\,{\left (2\,x-1\right )}^2}{15}+\frac {46\,{\left (2\,x-1\right )}^3}{5}+{\left (2\,x-1\right )}^4-\frac {931}{45}} \]
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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