Optimal. Leaf size=158 \[ -\frac {48645 \sqrt {1-2 x}}{98 (5 x+3)}+\frac {7261 \sqrt {1-2 x}}{147 (3 x+2) (5 x+3)}+\frac {139 \sqrt {1-2 x}}{42 (3 x+2)^2 (5 x+3)}+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}-\frac {335579}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+6650 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 158, 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} \[ -\frac {48645 \sqrt {1-2 x}}{98 (5 x+3)}+\frac {7261 \sqrt {1-2 x}}{147 (3 x+2) (5 x+3)}+\frac {139 \sqrt {1-2 x}}{42 (3 x+2)^2 (5 x+3)}+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}-\frac {335579}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+6650 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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)^4 (3+5 x)^2} \, dx &=\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)}-\frac {1}{3} \int \frac {-23+35 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx\\ &=\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)}+\frac {139 \sqrt {1-2 x}}{42 (2+3 x)^2 (3+5 x)}-\frac {1}{42} \int \frac {-2524+3475 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx\\ &=\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)}+\frac {139 \sqrt {1-2 x}}{42 (2+3 x)^2 (3+5 x)}+\frac {7261 \sqrt {1-2 x}}{147 (2+3 x) (3+5 x)}-\frac {1}{294} \int \frac {-190359+217830 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {48645 \sqrt {1-2 x}}{98 (3+5 x)}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)}+\frac {139 \sqrt {1-2 x}}{42 (2+3 x)^2 (3+5 x)}+\frac {7261 \sqrt {1-2 x}}{147 (2+3 x) (3+5 x)}+\frac {\int \frac {-7863537+4815855 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{3234}\\ &=-\frac {48645 \sqrt {1-2 x}}{98 (3+5 x)}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)}+\frac {139 \sqrt {1-2 x}}{42 (2+3 x)^2 (3+5 x)}+\frac {7261 \sqrt {1-2 x}}{147 (2+3 x) (3+5 x)}+\frac {1006737}{98} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-16625 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {48645 \sqrt {1-2 x}}{98 (3+5 x)}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)}+\frac {139 \sqrt {1-2 x}}{42 (2+3 x)^2 (3+5 x)}+\frac {7261 \sqrt {1-2 x}}{147 (2+3 x) (3+5 x)}-\frac {1006737}{98} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+16625 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {48645 \sqrt {1-2 x}}{98 (3+5 x)}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)}+\frac {139 \sqrt {1-2 x}}{42 (2+3 x)^2 (3+5 x)}+\frac {7261 \sqrt {1-2 x}}{147 (2+3 x) (3+5 x)}-\frac {335579}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+6650 \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 = 0.16, size = 99, normalized size = 0.63 \[ -\frac {\sqrt {1-2 x} \left (1313415 x^3+2583264 x^2+1692159 x+369116\right )}{98 (3 x+2)^3 (5 x+3)}-\frac {335579}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+6650 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 162, normalized size = 1.03 \[ \frac {2280950 \, \sqrt {11} \sqrt {5} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 3691369 \, \sqrt {7} \sqrt {3} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (1313415 \, x^{3} + 2583264 \, x^{2} + 1692159 \, x + 369116\right )} \sqrt {-2 \, x + 1}}{7546 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.29, size = 139, normalized size = 0.88 \[ -\frac {3325}{11} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {335579}{686} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {125 \, \sqrt {-2 \, x + 1}}{5 \, x + 3} - \frac {3 \, {\left (65511 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 308644 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 363629 \, \sqrt {-2 \, x + 1}\right )}}{392 \, {\left (3 \, x + 2\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 91, normalized size = 0.58 \[ -\frac {335579 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{343}+\frac {6650 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{11}+\frac {50 \sqrt {-2 x +1}}{-2 x -\frac {6}{5}}+\frac {\frac {196533 \left (-2 x +1\right )^{\frac {5}{2}}}{49}-\frac {132276 \left (-2 x +1\right )^{\frac {3}{2}}}{7}+22263 \sqrt {-2 x +1}}{\left (-6 x -4\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 146, normalized size = 0.92 \[ -\frac {3325}{11} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {335579}{686} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1313415 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 9106773 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 21041937 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 16201507 \, \sqrt {-2 \, x + 1}}{49 \, {\left (135 \, {\left (2 \, x - 1\right )}^{4} + 1242 \, {\left (2 \, x - 1\right )}^{3} + 4284 \, {\left (2 \, x - 1\right )}^{2} + 13132 \, x - 2793\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 108, normalized size = 0.68 \[ \frac {6650\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{11}-\frac {335579\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {\frac {330643\,\sqrt {1-2\,x}}{135}-\frac {111333\,{\left (1-2\,x\right )}^{3/2}}{35}+\frac {3035591\,{\left (1-2\,x\right )}^{5/2}}{2205}-\frac {9729\,{\left (1-2\,x\right )}^{7/2}}{49}}{\frac {13132\,x}{135}+\frac {476\,{\left (2\,x-1\right )}^2}{15}+\frac {46\,{\left (2\,x-1\right )}^3}{5}+{\left (2\,x-1\right )}^4-\frac {931}{45}} \]
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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