Optimal. Leaf size=95 \[ -\frac {125}{108} (1-2 x)^{9/2}+\frac {400}{63} (1-2 x)^{7/2}-\frac {1027}{108} (1-2 x)^{5/2}-\frac {2}{243} (1-2 x)^{3/2}-\frac {14}{243} \sqrt {1-2 x}+\frac {14}{243} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 95, 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 = {88, 50, 63, 206} \[ -\frac {125}{108} (1-2 x)^{9/2}+\frac {400}{63} (1-2 x)^{7/2}-\frac {1027}{108} (1-2 x)^{5/2}-\frac {2}{243} (1-2 x)^{3/2}-\frac {14}{243} \sqrt {1-2 x}+\frac {14}{243} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 88
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{2+3 x} \, dx &=\int \left (\frac {5135}{108} (1-2 x)^{3/2}-\frac {400}{9} (1-2 x)^{5/2}+\frac {125}{12} (1-2 x)^{7/2}-\frac {(1-2 x)^{3/2}}{27 (2+3 x)}\right ) \, dx\\ &=-\frac {1027}{108} (1-2 x)^{5/2}+\frac {400}{63} (1-2 x)^{7/2}-\frac {125}{108} (1-2 x)^{9/2}-\frac {1}{27} \int \frac {(1-2 x)^{3/2}}{2+3 x} \, dx\\ &=-\frac {2}{243} (1-2 x)^{3/2}-\frac {1027}{108} (1-2 x)^{5/2}+\frac {400}{63} (1-2 x)^{7/2}-\frac {125}{108} (1-2 x)^{9/2}-\frac {7}{81} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=-\frac {14}{243} \sqrt {1-2 x}-\frac {2}{243} (1-2 x)^{3/2}-\frac {1027}{108} (1-2 x)^{5/2}+\frac {400}{63} (1-2 x)^{7/2}-\frac {125}{108} (1-2 x)^{9/2}-\frac {49}{243} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {14}{243} \sqrt {1-2 x}-\frac {2}{243} (1-2 x)^{3/2}-\frac {1027}{108} (1-2 x)^{5/2}+\frac {400}{63} (1-2 x)^{7/2}-\frac {125}{108} (1-2 x)^{9/2}+\frac {49}{243} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {14}{243} \sqrt {1-2 x}-\frac {2}{243} (1-2 x)^{3/2}-\frac {1027}{108} (1-2 x)^{5/2}+\frac {400}{63} (1-2 x)^{7/2}-\frac {125}{108} (1-2 x)^{9/2}+\frac {14}{243} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 61, normalized size = 0.64 \[ \frac {98 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-3 \sqrt {1-2 x} \left (31500 x^4+23400 x^3-17649 x^2-15679 x+7456\right )}{5103} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 67, normalized size = 0.71 \[ \frac {7}{729} \, \sqrt {7} \sqrt {3} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) - \frac {1}{1701} \, {\left (31500 \, x^{4} + 23400 \, x^{3} - 17649 \, x^{2} - 15679 \, x + 7456\right )} \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 106, normalized size = 1.12 \[ -\frac {125}{108} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {400}{63} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {1027}{108} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {2}{243} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {7}{729} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {14}{243} \, \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 65, normalized size = 0.68 \[ \frac {14 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{729}-\frac {2 \left (-2 x +1\right )^{\frac {3}{2}}}{243}-\frac {1027 \left (-2 x +1\right )^{\frac {5}{2}}}{108}+\frac {400 \left (-2 x +1\right )^{\frac {7}{2}}}{63}-\frac {125 \left (-2 x +1\right )^{\frac {9}{2}}}{108}-\frac {14 \sqrt {-2 x +1}}{243} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 82, normalized size = 0.86 \[ -\frac {125}{108} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {400}{63} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {1027}{108} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {2}{243} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {7}{729} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {14}{243} \, \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 66, normalized size = 0.69 \[ \frac {400\,{\left (1-2\,x\right )}^{7/2}}{63}-\frac {2\,{\left (1-2\,x\right )}^{3/2}}{243}-\frac {1027\,{\left (1-2\,x\right )}^{5/2}}{108}-\frac {14\,\sqrt {1-2\,x}}{243}-\frac {125\,{\left (1-2\,x\right )}^{9/2}}{108}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,14{}\mathrm {i}}{729} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 55.28, size = 126, normalized size = 1.33 \[ - \frac {125 \left (1 - 2 x\right )^{\frac {9}{2}}}{108} + \frac {400 \left (1 - 2 x\right )^{\frac {7}{2}}}{63} - \frac {1027 \left (1 - 2 x\right )^{\frac {5}{2}}}{108} - \frac {2 \left (1 - 2 x\right )^{\frac {3}{2}}}{243} - \frac {14 \sqrt {1 - 2 x}}{243} - \frac {98 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 < - \frac {7}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 > - \frac {7}{3} \end {cases}\right )}{243} \]
Verification of antiderivative is not currently implemented for this CAS.
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