Optimal. Leaf size=120 \[ \frac {10836 \sqrt {1-2 x} (2+3 x)^2}{15125}-\frac {36 \sqrt {1-2 x} (2+3 x)^3}{605 (3+5 x)}+\frac {7 (2+3 x)^4}{11 \sqrt {1-2 x} (3+5 x)}+\frac {504 \sqrt {1-2 x} (4499+1500 x)}{75625}-\frac {336 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{75625 \sqrt {55}} \]
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Rubi [A]
time = 0.03, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {100, 154, 158,
152, 65, 212} \begin {gather*} \frac {7 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)}-\frac {36 \sqrt {1-2 x} (3 x+2)^3}{605 (5 x+3)}+\frac {10836 \sqrt {1-2 x} (3 x+2)^2}{15125}+\frac {504 \sqrt {1-2 x} (1500 x+4499)}{75625}-\frac {336 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{75625 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 152
Rule 154
Rule 158
Rule 212
Rubi steps
\begin {align*} \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^2} \, dx &=\frac {7 (2+3 x)^4}{11 \sqrt {1-2 x} (3+5 x)}-\frac {1}{11} \int \frac {(2+3 x)^3 (180+312 x)}{\sqrt {1-2 x} (3+5 x)^2} \, dx\\ &=-\frac {36 \sqrt {1-2 x} (2+3 x)^3}{605 (3+5 x)}+\frac {7 (2+3 x)^4}{11 \sqrt {1-2 x} (3+5 x)}-\frac {1}{605} \int \frac {(2+3 x)^2 (6468+10836 x)}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {10836 \sqrt {1-2 x} (2+3 x)^2}{15125}-\frac {36 \sqrt {1-2 x} (2+3 x)^3}{605 (3+5 x)}+\frac {7 (2+3 x)^4}{11 \sqrt {1-2 x} (3+5 x)}+\frac {\int \frac {(-453432-756000 x) (2+3 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{15125}\\ &=\frac {10836 \sqrt {1-2 x} (2+3 x)^2}{15125}-\frac {36 \sqrt {1-2 x} (2+3 x)^3}{605 (3+5 x)}+\frac {7 (2+3 x)^4}{11 \sqrt {1-2 x} (3+5 x)}+\frac {504 \sqrt {1-2 x} (4499+1500 x)}{75625}+\frac {168 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{75625}\\ &=\frac {10836 \sqrt {1-2 x} (2+3 x)^2}{15125}-\frac {36 \sqrt {1-2 x} (2+3 x)^3}{605 (3+5 x)}+\frac {7 (2+3 x)^4}{11 \sqrt {1-2 x} (3+5 x)}+\frac {504 \sqrt {1-2 x} (4499+1500 x)}{75625}-\frac {168 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{75625}\\ &=\frac {10836 \sqrt {1-2 x} (2+3 x)^2}{15125}-\frac {36 \sqrt {1-2 x} (2+3 x)^3}{605 (3+5 x)}+\frac {7 (2+3 x)^4}{11 \sqrt {1-2 x} (3+5 x)}+\frac {504 \sqrt {1-2 x} (4499+1500 x)}{75625}-\frac {336 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{75625 \sqrt {55}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 68, normalized size = 0.57 \begin {gather*} \frac {-\frac {55 \left (-8186648-6264264 x+14309460 x^2+3789720 x^3+735075 x^4\right )}{\sqrt {1-2 x} (3+5 x)}-336 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{4159375} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 72, normalized size = 0.60
method | result | size |
risch | \(-\frac {735075 x^{4}+3789720 x^{3}+14309460 x^{2}-6264264 x -8186648}{75625 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {336 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{4159375}\) | \(56\) |
derivativedivides | \(\frac {243 \left (1-2 x \right )^{\frac {5}{2}}}{1000}-\frac {2943 \left (1-2 x \right )^{\frac {3}{2}}}{1000}+\frac {107109 \sqrt {1-2 x}}{5000}+\frac {2 \sqrt {1-2 x}}{378125 \left (-\frac {6}{5}-2 x \right )}-\frac {336 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{4159375}+\frac {16807}{968 \sqrt {1-2 x}}\) | \(72\) |
default | \(\frac {243 \left (1-2 x \right )^{\frac {5}{2}}}{1000}-\frac {2943 \left (1-2 x \right )^{\frac {3}{2}}}{1000}+\frac {107109 \sqrt {1-2 x}}{5000}+\frac {2 \sqrt {1-2 x}}{378125 \left (-\frac {6}{5}-2 x \right )}-\frac {336 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{4159375}+\frac {16807}{968 \sqrt {1-2 x}}\) | \(72\) |
trager | \(\frac {\left (735075 x^{4}+3789720 x^{3}+14309460 x^{2}-6264264 x -8186648\right ) \sqrt {1-2 x}}{756250 x^{2}+75625 x -226875}-\frac {168 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{4159375}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 92, normalized size = 0.77 \begin {gather*} \frac {243}{1000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {2943}{1000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {168}{4159375} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {107109}{5000} \, \sqrt {-2 \, x + 1} - \frac {52521891 \, x + 31513117}{302500 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.58, size = 80, normalized size = 0.67 \begin {gather*} \frac {168 \, \sqrt {55} {\left (10 \, x^{2} + x - 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (735075 \, x^{4} + 3789720 \, x^{3} + 14309460 \, x^{2} - 6264264 \, x - 8186648\right )} \sqrt {-2 \, x + 1}}{4159375 \, {\left (10 \, x^{2} + x - 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [A]
time = 1.72, size = 102, normalized size = 0.85 \begin {gather*} \frac {243}{1000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {2943}{1000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {168}{4159375} \, \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 {107109}{5000} \, \sqrt {-2 \, x + 1} - \frac {52521891 \, x + 31513117}{302500 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 75, normalized size = 0.62 \begin {gather*} \frac {\frac {52521891\,x}{1512500}+\frac {31513117}{1512500}}{\frac {11\,\sqrt {1-2\,x}}{5}-{\left (1-2\,x\right )}^{3/2}}+\frac {107109\,\sqrt {1-2\,x}}{5000}-\frac {2943\,{\left (1-2\,x\right )}^{3/2}}{1000}+\frac {243\,{\left (1-2\,x\right )}^{5/2}}{1000}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,336{}\mathrm {i}}{4159375} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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