Optimal. Leaf size=249 \[ -\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {3+5 x}}{524126295 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}-\frac {1305025844 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}}-\frac {37904696 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}} \]
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Rubi [A]
time = 0.07, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 157,
164, 114, 120} \begin {gather*} -\frac {37904696 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}}-\frac {1305025844 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}}-\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac {118 \sqrt {1-2 x} (5 x+3)^{3/2}}{2079 (3 x+2)^{9/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {5 x+3}}{524126295 \sqrt {3 x+2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {5 x+3}}{74875185 (3 x+2)^{3/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {5 x+3}}{10696455 (3 x+2)^{5/2}}-\frac {13022 \sqrt {1-2 x} \sqrt {5 x+3}}{305613 (3 x+2)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 99
Rule 114
Rule 120
Rule 155
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {2}{33} \int \frac {\left (\frac {19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{11/2}} \, dx\\ &=-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {4 \int \frac {\left (-\frac {189}{4}-\frac {5025 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx}{6237}\\ &=-\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {8 \int \frac {-\frac {676497}{8}-185700 x}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{916839}\\ &=-\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {16 \int \frac {\frac {1286433}{2}-\frac {14125635 x}{8}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{32089365}\\ &=-\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {32 \int \frac {\frac {687512943}{16}-\frac {109221165 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{673876665}\\ &=-\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {3+5 x}}{524126295 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {64 \int \frac {\frac {2319498765}{4}+\frac {14681540745 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4717136655}\\ &=-\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {3+5 x}}{524126295 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {18952348 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{47647845}+\frac {1305025844 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{524126295}\\ &=-\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {3+5 x}}{524126295 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}-\frac {1305025844 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}}-\frac {37904696 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 5.23, size = 112, normalized size = 0.45 \begin {gather*} \frac {\frac {48 \sqrt {2-4 x} \sqrt {3+5 x} \left (21813966691+162787885893 x+484598540169 x^2+719808574005 x^3+534040213536 x^4+158560640046 x^5\right )}{(2+3 x)^{11/2}}+20880413504 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-10873573760 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{12579031080 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(586\) vs.
\(2(185)=370\).
time = 0.10, size = 587, normalized size = 2.36
method | result | size |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {538 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1515591 \left (\frac {2}{3}+x \right )^{5}}-\frac {93382 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{24754653 \left (\frac {2}{3}+x \right )^{4}}+\frac {627806 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{288804285 \left (\frac {2}{3}+x \right )^{3}}+\frac {19417096 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{673876665 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {2610051688}{104825259} x^{2}-\frac {1305025844}{524126295} x +\frac {1305025844}{174708765}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {824710672 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{2201330439 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1305025844 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{2201330439 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{216513 \left (\frac {2}{3}+x \right )^{6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(321\) |
default | \(-\frac {2 \left (75989439306 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-158560640046 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+253298131020 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-528535466820 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+337730841360 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-704713955760 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+225153894240 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-469809303840 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-4756819201380 x^{7}+75051298080 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-156603101280 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-16496888326218 x^{6}+10006839744 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-20880413504 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-21769332100344 x^{5}-11891020005261 x^{4}+140844968748 x^{3}+3218604203112 x^{2}+1399649072964 x +196325700219\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{1572378885 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {11}{2}}}\) | \(587\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.16, size = 80, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (158560640046 \, x^{5} + 534040213536 \, x^{4} + 719808574005 \, x^{3} + 484598540169 \, x^{2} + 162787885893 \, x + 21813966691\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{524126295 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{13/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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