Optimal. Leaf size=160 \[ \frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {88 \sqrt {1-2 x}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {2660 \sqrt {1-2 x} \sqrt {2+3 x}}{9 \sqrt {3+5 x}}+\frac {532}{3} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {16}{3} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {100, 157, 164,
114, 120} \begin {gather*} \frac {16}{3} \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {532}{3} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {2660 \sqrt {1-2 x} \sqrt {3 x+2}}{9 \sqrt {5 x+3}}+\frac {88 \sqrt {1-2 x}}{3 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} \sqrt {5 x+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 100
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {2}{9} \int \frac {88-99 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {88 \sqrt {1-2 x}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {4}{63} \int \frac {\frac {7469}{2}-2310 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {88 \sqrt {1-2 x}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {2660 \sqrt {1-2 x} \sqrt {2+3 x}}{9 \sqrt {3+5 x}}-\frac {8}{693} \int \frac {\frac {97251}{2}+\frac {153615 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {88 \sqrt {1-2 x}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {2660 \sqrt {1-2 x} \sqrt {2+3 x}}{9 \sqrt {3+5 x}}-\frac {88}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {532}{3} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {88 \sqrt {1-2 x}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {2660 \sqrt {1-2 x} \sqrt {2+3 x}}{9 \sqrt {3+5 x}}+\frac {532}{3} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {16}{3} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}
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Mathematica [A]
time = 6.92, size = 100, normalized size = 0.62 \begin {gather*} -\frac {2 \sqrt {1-2 x} \left (1683+5188 x+3990 x^2\right )}{3 (2+3 x)^{3/2} \sqrt {3+5 x}}-\frac {4}{9} \sqrt {2} \left (133 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-67 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 215, normalized size = 1.34
method | result | size |
default | \(\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (396 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-798 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+264 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-532 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-23940 x^{3}-19158 x^{2}+5466 x +5049\right )}{9 \left (2+3 x \right )^{\frac {3}{2}} \left (10 x^{2}+x -3\right )}\) | \(215\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {334 \left (-30 x^{2}-3 x +9\right )}{9 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {1684 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{63 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {380 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{9 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{27 \left (\frac {2}{3}+x \right )^{2}}-\frac {22 \left (-30 x^{2}-5 x +10\right )}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(253\) |
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 = 0.32, size = 50, normalized size = 0.31 \begin {gather*} -\frac {2 \, {\left (3990 \, x^{2} + 5188 \, x + 1683\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{3 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\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 {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^{5/2}\,{\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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