Optimal. Leaf size=187 \[ \frac {4}{77 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {186 \sqrt {1-2 x}}{539 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {45040 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 (3+5 x)^{3/2}}+\frac {2976620 \sqrt {1-2 x} \sqrt {2+3 x}}{195657 \sqrt {3+5 x}}-\frac {595324 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5929 \sqrt {33}}-\frac {18016 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5929 \sqrt {33}} \]
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Rubi [A]
time = 0.05, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 {18016 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5929 \sqrt {33}}-\frac {595324 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5929 \sqrt {33}}+\frac {2976620 \sqrt {1-2 x} \sqrt {3 x+2}}{195657 \sqrt {5 x+3}}-\frac {45040 \sqrt {1-2 x} \sqrt {3 x+2}}{17787 (5 x+3)^{3/2}}+\frac {186 \sqrt {1-2 x}}{539 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {4}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}} \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)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac {4}{77 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {2}{77} \int \frac {-\frac {131}{2}-75 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {4}{77 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {186 \sqrt {1-2 x}}{539 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {4}{539} \int \frac {-\frac {1415}{2}+\frac {1395 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {4}{77 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {186 \sqrt {1-2 x}}{539 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {45040 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 (3+5 x)^{3/2}}+\frac {8 \int \frac {-\frac {108295}{4}+16890 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{17787}\\ &=\frac {4}{77 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {186 \sqrt {1-2 x}}{539 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {45040 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 (3+5 x)^{3/2}}+\frac {2976620 \sqrt {1-2 x} \sqrt {2+3 x}}{195657 \sqrt {3+5 x}}-\frac {16 \int \frac {-\frac {1413795}{4}-\frac {2232465 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{195657}\\ &=\frac {4}{77 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {186 \sqrt {1-2 x}}{539 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {45040 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 (3+5 x)^{3/2}}+\frac {2976620 \sqrt {1-2 x} \sqrt {2+3 x}}{195657 \sqrt {3+5 x}}+\frac {9008 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5929}+\frac {595324 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{65219}\\ &=\frac {4}{77 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {186 \sqrt {1-2 x}}{539 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {45040 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 (3+5 x)^{3/2}}+\frac {2976620 \sqrt {1-2 x} \sqrt {2+3 x}}{195657 \sqrt {3+5 x}}-\frac {595324 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5929 \sqrt {33}}-\frac {18016 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5929 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 6.75, size = 104, normalized size = 0.56 \begin {gather*} \frac {2 \left (\frac {8473261+10598372 x-32744810 x^2-44649300 x^3}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+2 \sqrt {2} \left (148831 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-74515 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{195657} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 215, normalized size = 1.15
method | result | size |
default | \(-\frac {2 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \left (743160 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-1488310 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+445896 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-892986 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-44649300 x^{3}-32744810 x^{2}+10598372 x +8473261\right )}{195657 \left (3+5 x \right )^{\frac {3}{2}} \left (6 x^{2}+x -2\right )}\) | \(215\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-18-30 x \right ) \left (\frac {107843}{391314}-\frac {35929 x}{65219}\right )}{\sqrt {\left (x^{2}+\frac {1}{6} x -\frac {1}{3}\right ) \left (-18-30 x \right )}}-\frac {10 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{363 \left (x +\frac {3}{5}\right )^{2}}+\frac {-\frac {77500}{1331} x^{2}-\frac {38750}{3993} x +\frac {77500}{3993}}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}+\frac {1885060 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{1369599 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2976620 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1369599 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(253\) |
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 = 0.24, size = 60, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (44649300 \, x^{3} + 32744810 \, x^{2} - 10598372 \, x - 8473261\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{195657 \, {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\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 = -0.01 \begin {gather*} \int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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