Optimal. Leaf size=130 \[ -\frac {\sqrt {1-2 x}}{315 (2+3 x)^5}+\frac {341 \sqrt {1-2 x}}{8820 (2+3 x)^4}-\frac {117 \sqrt {1-2 x}}{980 (2+3 x)^3}-\frac {117 \sqrt {1-2 x}}{2744 (2+3 x)^2}-\frac {351 \sqrt {1-2 x}}{19208 (2+3 x)}-\frac {117 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9604} \]
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Rubi [A]
time = 0.03, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {91, 79, 44, 65,
212} \begin {gather*} -\frac {351 \sqrt {1-2 x}}{19208 (3 x+2)}-\frac {117 \sqrt {1-2 x}}{2744 (3 x+2)^2}-\frac {117 \sqrt {1-2 x}}{980 (3 x+2)^3}+\frac {341 \sqrt {1-2 x}}{8820 (3 x+2)^4}-\frac {\sqrt {1-2 x}}{315 (3 x+2)^5}-\frac {117 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9604} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 79
Rule 91
Rule 212
Rubi steps
\begin {align*} \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^6} \, dx &=-\frac {\sqrt {1-2 x}}{315 (2+3 x)^5}+\frac {1}{315} \int \frac {1409+2625 x}{\sqrt {1-2 x} (2+3 x)^5} \, dx\\ &=-\frac {\sqrt {1-2 x}}{315 (2+3 x)^5}+\frac {341 \sqrt {1-2 x}}{8820 (2+3 x)^4}+\frac {351}{140} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=-\frac {\sqrt {1-2 x}}{315 (2+3 x)^5}+\frac {341 \sqrt {1-2 x}}{8820 (2+3 x)^4}-\frac {117 \sqrt {1-2 x}}{980 (2+3 x)^3}+\frac {117}{196} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {\sqrt {1-2 x}}{315 (2+3 x)^5}+\frac {341 \sqrt {1-2 x}}{8820 (2+3 x)^4}-\frac {117 \sqrt {1-2 x}}{980 (2+3 x)^3}-\frac {117 \sqrt {1-2 x}}{2744 (2+3 x)^2}+\frac {351 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{2744}\\ &=-\frac {\sqrt {1-2 x}}{315 (2+3 x)^5}+\frac {341 \sqrt {1-2 x}}{8820 (2+3 x)^4}-\frac {117 \sqrt {1-2 x}}{980 (2+3 x)^3}-\frac {117 \sqrt {1-2 x}}{2744 (2+3 x)^2}-\frac {351 \sqrt {1-2 x}}{19208 (2+3 x)}+\frac {351 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{19208}\\ &=-\frac {\sqrt {1-2 x}}{315 (2+3 x)^5}+\frac {341 \sqrt {1-2 x}}{8820 (2+3 x)^4}-\frac {117 \sqrt {1-2 x}}{980 (2+3 x)^3}-\frac {117 \sqrt {1-2 x}}{2744 (2+3 x)^2}-\frac {351 \sqrt {1-2 x}}{19208 (2+3 x)}-\frac {351 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{19208}\\ &=-\frac {\sqrt {1-2 x}}{315 (2+3 x)^5}+\frac {341 \sqrt {1-2 x}}{8820 (2+3 x)^4}-\frac {117 \sqrt {1-2 x}}{980 (2+3 x)^3}-\frac {117 \sqrt {1-2 x}}{2744 (2+3 x)^2}-\frac {351 \sqrt {1-2 x}}{19208 (2+3 x)}-\frac {117 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9604}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 70, normalized size = 0.54 \begin {gather*} \frac {-\frac {7 \sqrt {1-2 x} \left (298748+1327058 x+2110212 x^2+1468935 x^3+426465 x^4\right )}{2 (2+3 x)^5}-1755 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1008420} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 75, normalized size = 0.58
method | result | size |
risch | \(\frac {852930 x^{5}+2511405 x^{4}+2751489 x^{3}+543904 x^{2}-729562 x -298748}{288120 \left (2+3 x \right )^{5} \sqrt {1-2 x}}-\frac {117 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{67228}\) | \(61\) |
derivativedivides | \(-\frac {3888 \left (-\frac {117 \left (1-2 x \right )^{\frac {9}{2}}}{153664}+\frac {13 \left (1-2 x \right )^{\frac {7}{2}}}{1568}-\frac {26 \left (1-2 x \right )^{\frac {5}{2}}}{735}+\frac {77587 \left (1-2 x \right )^{\frac {3}{2}}}{1143072}-\frac {5287 \sqrt {1-2 x}}{108864}\right )}{\left (-4-6 x \right )^{5}}-\frac {117 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{67228}\) | \(75\) |
default | \(-\frac {3888 \left (-\frac {117 \left (1-2 x \right )^{\frac {9}{2}}}{153664}+\frac {13 \left (1-2 x \right )^{\frac {7}{2}}}{1568}-\frac {26 \left (1-2 x \right )^{\frac {5}{2}}}{735}+\frac {77587 \left (1-2 x \right )^{\frac {3}{2}}}{1143072}-\frac {5287 \sqrt {1-2 x}}{108864}\right )}{\left (-4-6 x \right )^{5}}-\frac {117 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{67228}\) | \(75\) |
trager | \(-\frac {\left (426465 x^{4}+1468935 x^{3}+2110212 x^{2}+1327058 x +298748\right ) \sqrt {1-2 x}}{288120 \left (2+3 x \right )^{5}}+\frac {117 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{134456}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 128, normalized size = 0.98 \begin {gather*} \frac {117}{134456} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {426465 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 4643730 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 19813248 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 38017630 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 27201615 \, \sqrt {-2 \, x + 1}}{144060 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.90, size = 120, normalized size = 0.92 \begin {gather*} \frac {1755 \, \sqrt {7} \sqrt {3} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 7 \, {\left (426465 \, x^{4} + 1468935 \, x^{3} + 2110212 \, x^{2} + 1327058 \, x + 298748\right )} \sqrt {-2 \, x + 1}}{2016840 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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 = 0.81, size = 116, normalized size = 0.89 \begin {gather*} \frac {117}{134456} \, \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 {426465 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 4643730 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 19813248 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 38017630 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 27201615 \, \sqrt {-2 \, x + 1}}{4609920 \, {\left (3 \, x + 2\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.20, size = 108, normalized size = 0.83 \begin {gather*} -\frac {117\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{67228}-\frac {\frac {5287\,\sqrt {1-2\,x}}{6804}-\frac {77587\,{\left (1-2\,x\right )}^{3/2}}{71442}+\frac {416\,{\left (1-2\,x\right )}^{5/2}}{735}-\frac {13\,{\left (1-2\,x\right )}^{7/2}}{98}+\frac {117\,{\left (1-2\,x\right )}^{9/2}}{9604}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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