Optimal. Leaf size=134 \[ \frac {22 \sqrt {1-2 x}}{390625}+\frac {2 (1-2 x)^{3/2}}{234375}-\frac {167115051 (1-2 x)^{5/2}}{2500000}+\frac {70752609 (1-2 x)^{7/2}}{700000}-\frac {665817 (1-2 x)^{9/2}}{10000}+\frac {507627 (1-2 x)^{11/2}}{22000}-\frac {43011 (1-2 x)^{13/2}}{10400}+\frac {243}{800} (1-2 x)^{15/2}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{390625} \]
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Rubi [A]
time = 0.03, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {90, 52, 65, 212}
\begin {gather*} \frac {243}{800} (1-2 x)^{15/2}-\frac {43011 (1-2 x)^{13/2}}{10400}+\frac {507627 (1-2 x)^{11/2}}{22000}-\frac {665817 (1-2 x)^{9/2}}{10000}+\frac {70752609 (1-2 x)^{7/2}}{700000}-\frac {167115051 (1-2 x)^{5/2}}{2500000}+\frac {2 (1-2 x)^{3/2}}{234375}+\frac {22 \sqrt {1-2 x}}{390625}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{390625} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 90
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (2+3 x)^6}{3+5 x} \, dx &=\int \left (\frac {167115051 (1-2 x)^{3/2}}{500000}-\frac {70752609 (1-2 x)^{5/2}}{100000}+\frac {5992353 (1-2 x)^{7/2}}{10000}-\frac {507627 (1-2 x)^{9/2}}{2000}+\frac {43011}{800} (1-2 x)^{11/2}-\frac {729}{160} (1-2 x)^{13/2}+\frac {(1-2 x)^{3/2}}{15625 (3+5 x)}\right ) \, dx\\ &=-\frac {167115051 (1-2 x)^{5/2}}{2500000}+\frac {70752609 (1-2 x)^{7/2}}{700000}-\frac {665817 (1-2 x)^{9/2}}{10000}+\frac {507627 (1-2 x)^{11/2}}{22000}-\frac {43011 (1-2 x)^{13/2}}{10400}+\frac {243}{800} (1-2 x)^{15/2}+\frac {\int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx}{15625}\\ &=\frac {2 (1-2 x)^{3/2}}{234375}-\frac {167115051 (1-2 x)^{5/2}}{2500000}+\frac {70752609 (1-2 x)^{7/2}}{700000}-\frac {665817 (1-2 x)^{9/2}}{10000}+\frac {507627 (1-2 x)^{11/2}}{22000}-\frac {43011 (1-2 x)^{13/2}}{10400}+\frac {243}{800} (1-2 x)^{15/2}+\frac {11 \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx}{78125}\\ &=\frac {22 \sqrt {1-2 x}}{390625}+\frac {2 (1-2 x)^{3/2}}{234375}-\frac {167115051 (1-2 x)^{5/2}}{2500000}+\frac {70752609 (1-2 x)^{7/2}}{700000}-\frac {665817 (1-2 x)^{9/2}}{10000}+\frac {507627 (1-2 x)^{11/2}}{22000}-\frac {43011 (1-2 x)^{13/2}}{10400}+\frac {243}{800} (1-2 x)^{15/2}+\frac {121 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{390625}\\ &=\frac {22 \sqrt {1-2 x}}{390625}+\frac {2 (1-2 x)^{3/2}}{234375}-\frac {167115051 (1-2 x)^{5/2}}{2500000}+\frac {70752609 (1-2 x)^{7/2}}{700000}-\frac {665817 (1-2 x)^{9/2}}{10000}+\frac {507627 (1-2 x)^{11/2}}{22000}-\frac {43011 (1-2 x)^{13/2}}{10400}+\frac {243}{800} (1-2 x)^{15/2}-\frac {121 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{390625}\\ &=\frac {22 \sqrt {1-2 x}}{390625}+\frac {2 (1-2 x)^{3/2}}{234375}-\frac {167115051 (1-2 x)^{5/2}}{2500000}+\frac {70752609 (1-2 x)^{7/2}}{700000}-\frac {665817 (1-2 x)^{9/2}}{10000}+\frac {507627 (1-2 x)^{11/2}}{22000}-\frac {43011 (1-2 x)^{13/2}}{10400}+\frac {243}{800} (1-2 x)^{15/2}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{390625}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 76, normalized size = 0.57 \begin {gather*} \frac {-5 \sqrt {1-2 x} \left (15379193944-9645684935 x-56176961670 x^2-61883481375 x^3+49094797500 x^4+174123928125 x^5+150857437500 x^6+45608062500 x^7\right )-66066 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5865234375} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 92, normalized size = 0.69
method | result | size |
risch | \(\frac {\left (45608062500 x^{7}+150857437500 x^{6}+174123928125 x^{5}+49094797500 x^{4}-61883481375 x^{3}-56176961670 x^{2}-9645684935 x +15379193944\right ) \left (-1+2 x \right )}{1173046875 \sqrt {1-2 x}}-\frac {22 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1953125}\) | \(69\) |
trager | \(\left (-\frac {972}{25} x^{7}-\frac {41796}{325} x^{6}-\frac {2653317}{17875} x^{5}-\frac {3740556}{89375} x^{4}+\frac {165022617}{3128125} x^{3}+\frac {3745130778}{78203125} x^{2}+\frac {1929136987}{234609375} x -\frac {15379193944}{1173046875}\right ) \sqrt {1-2 x}-\frac {11 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{1953125}\) | \(90\) |
derivativedivides | \(\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{234375}-\frac {167115051 \left (1-2 x \right )^{\frac {5}{2}}}{2500000}+\frac {70752609 \left (1-2 x \right )^{\frac {7}{2}}}{700000}-\frac {665817 \left (1-2 x \right )^{\frac {9}{2}}}{10000}+\frac {507627 \left (1-2 x \right )^{\frac {11}{2}}}{22000}-\frac {43011 \left (1-2 x \right )^{\frac {13}{2}}}{10400}+\frac {243 \left (1-2 x \right )^{\frac {15}{2}}}{800}-\frac {22 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1953125}+\frac {22 \sqrt {1-2 x}}{390625}\) | \(92\) |
default | \(\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{234375}-\frac {167115051 \left (1-2 x \right )^{\frac {5}{2}}}{2500000}+\frac {70752609 \left (1-2 x \right )^{\frac {7}{2}}}{700000}-\frac {665817 \left (1-2 x \right )^{\frac {9}{2}}}{10000}+\frac {507627 \left (1-2 x \right )^{\frac {11}{2}}}{22000}-\frac {43011 \left (1-2 x \right )^{\frac {13}{2}}}{10400}+\frac {243 \left (1-2 x \right )^{\frac {15}{2}}}{800}-\frac {22 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1953125}+\frac {22 \sqrt {1-2 x}}{390625}\) | \(92\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 109, normalized size = 0.81 \begin {gather*} \frac {243}{800} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} - \frac {43011}{10400} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {507627}{22000} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {665817}{10000} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {70752609}{700000} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {167115051}{2500000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2}{234375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{1953125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {22}{390625} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.03, size = 81, normalized size = 0.60 \begin {gather*} \frac {11}{1953125} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - \frac {1}{1173046875} \, {\left (45608062500 \, x^{7} + 150857437500 \, x^{6} + 174123928125 \, x^{5} + 49094797500 \, x^{4} - 61883481375 \, x^{3} - 56176961670 \, x^{2} - 9645684935 \, x + 15379193944\right )} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 77.88, size = 155, normalized size = 1.16 \begin {gather*} \frac {243 \left (1 - 2 x\right )^{\frac {15}{2}}}{800} - \frac {43011 \left (1 - 2 x\right )^{\frac {13}{2}}}{10400} + \frac {507627 \left (1 - 2 x\right )^{\frac {11}{2}}}{22000} - \frac {665817 \left (1 - 2 x\right )^{\frac {9}{2}}}{10000} + \frac {70752609 \left (1 - 2 x\right )^{\frac {7}{2}}}{700000} - \frac {167115051 \left (1 - 2 x\right )^{\frac {5}{2}}}{2500000} + \frac {2 \left (1 - 2 x\right )^{\frac {3}{2}}}{234375} + \frac {22 \sqrt {1 - 2 x}}{390625} + \frac {242 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x < - \frac {3}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x > - \frac {3}{5} \end {cases}\right )}{390625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.59, size = 154, normalized size = 1.15 \begin {gather*} -\frac {243}{800} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} - \frac {43011}{10400} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {507627}{22000} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {665817}{10000} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {70752609}{700000} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {167115051}{2500000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2}{234375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{1953125} \, \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 {22}{390625} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.18, size = 93, normalized size = 0.69 \begin {gather*} \frac {22\,\sqrt {1-2\,x}}{390625}+\frac {2\,{\left (1-2\,x\right )}^{3/2}}{234375}-\frac {167115051\,{\left (1-2\,x\right )}^{5/2}}{2500000}+\frac {70752609\,{\left (1-2\,x\right )}^{7/2}}{700000}-\frac {665817\,{\left (1-2\,x\right )}^{9/2}}{10000}+\frac {507627\,{\left (1-2\,x\right )}^{11/2}}{22000}-\frac {43011\,{\left (1-2\,x\right )}^{13/2}}{10400}+\frac {243\,{\left (1-2\,x\right )}^{15/2}}{800}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,22{}\mathrm {i}}{1953125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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