Optimal. Leaf size=249 \[ \frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {4839325048 \sqrt {1-2 x} \sqrt {2+3 x}}{67110351 \sqrt {3+5 x}}+\frac {4839325048 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{10168235 \sqrt {33}}+\frac {145418632 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{10168235 \sqrt {33}} \]
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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 = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {106, 157, 164,
114, 120} \begin {gather*} \frac {145418632 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{10168235 \sqrt {33}}+\frac {4839325048 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{10168235 \sqrt {33}}-\frac {4839325048 \sqrt {1-2 x} \sqrt {3 x+2}}{67110351 \sqrt {5 x+3}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {499564 \sqrt {1-2 x}}{1452605 (3 x+2)^{3/2} \sqrt {5 x+3}}-\frac {2206 \sqrt {1-2 x}}{207515 (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {1616}{17787 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt {5 x+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 106
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2}{231} \int \frac {-\frac {269}{2}-135 x}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {4 \int \frac {\frac {55457}{4}+21210 x}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx}{17787}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {8 \int \frac {\frac {429823}{4}+\frac {82725 x}{4}}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx}{622545}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {16 \int \frac {\frac {32051607}{8}-\frac {16860285 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{13073445}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {32 \int \frac {\frac {661979505}{4}-\frac {817979805 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{91514115}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {4839325048 \sqrt {1-2 x} \sqrt {2+3 x}}{67110351 \sqrt {3+5 x}}-\frac {64 \int \frac {\frac {34464999645}{16}+\frac {27221203395 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1006655265}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {4839325048 \sqrt {1-2 x} \sqrt {2+3 x}}{67110351 \sqrt {3+5 x}}-\frac {72709316 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{10168235}-\frac {4839325048 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{111850585}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {4839325048 \sqrt {1-2 x} \sqrt {2+3 x}}{67110351 \sqrt {3+5 x}}+\frac {4839325048 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{10168235 \sqrt {33}}+\frac {145418632 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{10168235 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 8.64, size = 115, normalized size = 0.46 \begin {gather*} \frac {2 \left (-\frac {91855922241+53503915182 x-673871013766 x^2-559512908172 x^3+1263428429256 x^4+1306617762960 x^5}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}-2 \sqrt {2} \left (1209831262 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-609979405 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{335551755} \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. \(397\) vs.
\(2(185)=370\).
time = 0.11, size = 398, normalized size = 1.60
method | result | size |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {6250 \left (-30 x^{2}-5 x +10\right )}{1331 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}+\frac {16 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{871563 \left (-\frac {1}{2}+x \right )^{2}}-\frac {17216 \left (-30 x^{2}-38 x -12\right )}{67110351 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}-\frac {6 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1715 \left (\frac {2}{3}+x \right )^{3}}-\frac {817326 \left (-30 x^{2}-3 x +9\right )}{84035 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {6 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{35 \left (\frac {2}{3}+x \right )^{2}}-\frac {3063555524 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{469772457 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {4839325048 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{469772457 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(329\) |
default | \(\frac {2 \sqrt {1-2 x}\, \left (21594666852 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-43553925432 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+17995555710 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-36294937860 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-4798814856 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+9678650096 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-4798814856 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+9678650096 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1306617762960 x^{5}-1263428429256 x^{4}+559512908172 x^{3}+673871013766 x^{2}-53503915182 x -91855922241\right )}{335551755 \left (2+3 x \right )^{\frac {5}{2}} \left (-1+2 x \right )^{2} \sqrt {3+5 x}}\) | \(398\) |
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 = 0.29, size = 80, normalized size = 0.32 \begin {gather*} -\frac {2 \, {\left (1306617762960 \, x^{5} + 1263428429256 \, x^{4} - 559512908172 \, x^{3} - 673871013766 \, x^{2} + 53503915182 \, x + 91855922241\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{335551755 \, {\left (540 \, x^{6} + 864 \, x^{5} + 99 \, x^{4} - 425 \, x^{3} - 154 \, x^{2} + 52 \, x + 24\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 {1}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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