Optimal. Leaf size=141 \[ \frac {2794 \sqrt {1-2 x}}{78125}+\frac {254 (1-2 x)^{3/2}}{46875}-\frac {32 (1-2 x)^{5/2} (2+3 x)^2}{4125}+\frac {39}{275} (1-2 x)^{5/2} (2+3 x)^3-\frac {(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{5/2} (1347116+1110975 x)}{3609375}-\frac {2794 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125} \]
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Rubi [A]
time = 0.03, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {99, 158, 152,
52, 65, 212} \begin {gather*} -\frac {(1-2 x)^{5/2} (3 x+2)^4}{5 (5 x+3)}+\frac {39}{275} (1-2 x)^{5/2} (3 x+2)^3-\frac {32 (1-2 x)^{5/2} (3 x+2)^2}{4125}+\frac {254 (1-2 x)^{3/2}}{46875}-\frac {(1-2 x)^{5/2} (1110975 x+1347116)}{3609375}+\frac {2794 \sqrt {1-2 x}}{78125}-\frac {2794 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 99
Rule 152
Rule 158
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^2} \, dx &=-\frac {(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}+\frac {1}{5} \int \frac {(2-39 x) (1-2 x)^{3/2} (2+3 x)^3}{3+5 x} \, dx\\ &=\frac {39}{275} (1-2 x)^{5/2} (2+3 x)^3-\frac {(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}-\frac {1}{275} \int \frac {(-337-96 x) (1-2 x)^{3/2} (2+3 x)^2}{3+5 x} \, dx\\ &=-\frac {32 (1-2 x)^{5/2} (2+3 x)^2}{4125}+\frac {39}{275} (1-2 x)^{5/2} (2+3 x)^3-\frac {(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}+\frac {\int \frac {(1-2 x)^{3/2} (2+3 x) (29178+44439 x)}{3+5 x} \, dx}{12375}\\ &=-\frac {32 (1-2 x)^{5/2} (2+3 x)^2}{4125}+\frac {39}{275} (1-2 x)^{5/2} (2+3 x)^3-\frac {(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{5/2} (1347116+1110975 x)}{3609375}+\frac {127 \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx}{3125}\\ &=\frac {254 (1-2 x)^{3/2}}{46875}-\frac {32 (1-2 x)^{5/2} (2+3 x)^2}{4125}+\frac {39}{275} (1-2 x)^{5/2} (2+3 x)^3-\frac {(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{5/2} (1347116+1110975 x)}{3609375}+\frac {1397 \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx}{15625}\\ &=\frac {2794 \sqrt {1-2 x}}{78125}+\frac {254 (1-2 x)^{3/2}}{46875}-\frac {32 (1-2 x)^{5/2} (2+3 x)^2}{4125}+\frac {39}{275} (1-2 x)^{5/2} (2+3 x)^3-\frac {(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{5/2} (1347116+1110975 x)}{3609375}+\frac {15367 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{78125}\\ &=\frac {2794 \sqrt {1-2 x}}{78125}+\frac {254 (1-2 x)^{3/2}}{46875}-\frac {32 (1-2 x)^{5/2} (2+3 x)^2}{4125}+\frac {39}{275} (1-2 x)^{5/2} (2+3 x)^3-\frac {(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{5/2} (1347116+1110975 x)}{3609375}-\frac {15367 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{78125}\\ &=\frac {2794 \sqrt {1-2 x}}{78125}+\frac {254 (1-2 x)^{3/2}}{46875}-\frac {32 (1-2 x)^{5/2} (2+3 x)^2}{4125}+\frac {39}{275} (1-2 x)^{5/2} (2+3 x)^3-\frac {(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{5/2} (1347116+1110975 x)}{3609375}-\frac {2794 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 78, normalized size = 0.55 \begin {gather*} \frac {\frac {5 \sqrt {1-2 x} \left (-15982128+50081215 x+85482115 x^2-214071975 x^3-173598750 x^4+237037500 x^5+212625000 x^6\right )}{3+5 x}-645414 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{90234375} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 90, normalized size = 0.64
method | result | size |
risch | \(-\frac {425250000 x^{7}+261450000 x^{6}-584235000 x^{5}-254545200 x^{4}+385036205 x^{3}+14680315 x^{2}-82045471 x +15982128}{18046875 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {2794 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{390625}\) | \(71\) |
derivativedivides | \(-\frac {81 \left (1-2 x \right )^{\frac {11}{2}}}{1100}+\frac {111 \left (1-2 x \right )^{\frac {9}{2}}}{250}-\frac {12393 \left (1-2 x \right )^{\frac {7}{2}}}{17500}+\frac {24 \left (1-2 x \right )^{\frac {5}{2}}}{15625}+\frac {52 \left (1-2 x \right )^{\frac {3}{2}}}{9375}+\frac {2816 \sqrt {1-2 x}}{78125}+\frac {242 \sqrt {1-2 x}}{390625 \left (-\frac {6}{5}-2 x \right )}-\frac {2794 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{390625}\) | \(90\) |
default | \(-\frac {81 \left (1-2 x \right )^{\frac {11}{2}}}{1100}+\frac {111 \left (1-2 x \right )^{\frac {9}{2}}}{250}-\frac {12393 \left (1-2 x \right )^{\frac {7}{2}}}{17500}+\frac {24 \left (1-2 x \right )^{\frac {5}{2}}}{15625}+\frac {52 \left (1-2 x \right )^{\frac {3}{2}}}{9375}+\frac {2816 \sqrt {1-2 x}}{78125}+\frac {242 \sqrt {1-2 x}}{390625 \left (-\frac {6}{5}-2 x \right )}-\frac {2794 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{390625}\) | \(90\) |
trager | \(\frac {\left (212625000 x^{6}+237037500 x^{5}-173598750 x^{4}-214071975 x^{3}+85482115 x^{2}+50081215 x -15982128\right ) \sqrt {1-2 x}}{54140625+90234375 x}-\frac {1397 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{390625}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 107, normalized size = 0.76 \begin {gather*} -\frac {81}{1100} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {111}{250} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {12393}{17500} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {24}{15625} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {52}{9375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1397}{390625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2816}{78125} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{78125 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.04, size = 90, normalized size = 0.64 \begin {gather*} \frac {322707 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 5 \, {\left (212625000 \, x^{6} + 237037500 \, x^{5} - 173598750 \, x^{4} - 214071975 \, x^{3} + 85482115 \, x^{2} + 50081215 \, x - 15982128\right )} \sqrt {-2 \, x + 1}}{90234375 \, {\left (5 \, 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 = 0.75, size = 138, normalized size = 0.98 \begin {gather*} \frac {81}{1100} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {111}{250} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {12393}{17500} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {24}{15625} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {52}{9375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1397}{390625} \, \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 {2816}{78125} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{78125 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 91, normalized size = 0.65 \begin {gather*} \frac {2816\,\sqrt {1-2\,x}}{78125}-\frac {242\,\sqrt {1-2\,x}}{390625\,\left (2\,x+\frac {6}{5}\right )}+\frac {52\,{\left (1-2\,x\right )}^{3/2}}{9375}+\frac {24\,{\left (1-2\,x\right )}^{5/2}}{15625}-\frac {12393\,{\left (1-2\,x\right )}^{7/2}}{17500}+\frac {111\,{\left (1-2\,x\right )}^{9/2}}{250}-\frac {81\,{\left (1-2\,x\right )}^{11/2}}{1100}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2794{}\mathrm {i}}{390625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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