Optimal. Leaf size=158 \[ \frac {14 \sqrt {1-2 x}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {92 \sqrt {1-2 x} \sqrt {2+3 x}}{3 (3+5 x)^{3/2}}+\frac {556 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}-\frac {556}{5} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {184 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5 \sqrt {33}} \]
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Rubi [A]
time = 0.03, antiderivative size = 158, 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 {184 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5 \sqrt {33}}-\frac {556}{5} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {556 \sqrt {1-2 x} \sqrt {3 x+2}}{3 \sqrt {5 x+3}}-\frac {92 \sqrt {1-2 x} \sqrt {3 x+2}}{3 (5 x+3)^{3/2}}+\frac {14 \sqrt {1-2 x}}{3 \sqrt {3 x+2} (5 x+3)^{3/2}} \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)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac {14 \sqrt {1-2 x}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {2}{3} \int \frac {90-103 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {92 \sqrt {1-2 x} \sqrt {2+3 x}}{3 (3+5 x)^{3/2}}-\frac {4}{99} \int \frac {\frac {7359}{2}-2277 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {92 \sqrt {1-2 x} \sqrt {2+3 x}}{3 (3+5 x)^{3/2}}+\frac {556 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}+\frac {8 \int \frac {47916+\frac {151371 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1089}\\ &=\frac {14 \sqrt {1-2 x}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {92 \sqrt {1-2 x} \sqrt {2+3 x}}{3 (3+5 x)^{3/2}}+\frac {556 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}+\frac {92}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {556}{5} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {92 \sqrt {1-2 x} \sqrt {2+3 x}}{3 (3+5 x)^{3/2}}+\frac {556 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}-\frac {556}{5} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {184 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 6.18, size = 100, normalized size = 0.63 \begin {gather*} \frac {2 \sqrt {1-2 x} \left (1583+5144 x+4170 x^2\right )}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {4}{15} \sqrt {2} \left (139 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-70 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 = 0.11, size = 215, normalized size = 1.36
method | result | size |
default | \(-\frac {2 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \left (690 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-1390 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+414 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-834 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-41700 x^{3}-30590 x^{2}+9890 x +7915\right )}{15 \left (3+5 x \right )^{\frac {3}{2}} \left (6 x^{2}+x -2\right )}\) | \(215\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {-420 x^{2}-42 x +126}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {352 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{21 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {556 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{21 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {22 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{75 \left (x +\frac {3}{5}\right )^{2}}+\frac {-692 x^{2}-\frac {346}{3} x +\frac {692}{3}}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(253\) |
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 = 0.15, size = 50, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (4170 \, x^{2} + 5144 \, x + 1583\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{3 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, 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 {{\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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