Optimal. Leaf size=129 \[ \frac {14 (1-2 x)^{3/2}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {1496 \sqrt {1-2 x} \sqrt {2+3 x}}{15 \sqrt {3+5 x}}+\frac {4636}{75} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {124}{75} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {100, 155, 164,
114, 120} \begin {gather*} \frac {124}{75} \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {4636}{75} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {14 (1-2 x)^{3/2}}{3 \sqrt {3 x+2} \sqrt {5 x+3}}-\frac {1496 \sqrt {3 x+2} \sqrt {1-2 x}}{15 \sqrt {5 x+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 100
Rule 114
Rule 120
Rule 155
Rule 164
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac {14 (1-2 x)^{3/2}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {2}{3} \int \frac {\sqrt {1-2 x} (97+37 x)}{\sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {14 (1-2 x)^{3/2}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {1496 \sqrt {1-2 x} \sqrt {2+3 x}}{15 \sqrt {3+5 x}}+\frac {4}{15} \int \frac {-\frac {1459}{2}-1159 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {14 (1-2 x)^{3/2}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {1496 \sqrt {1-2 x} \sqrt {2+3 x}}{15 \sqrt {3+5 x}}-\frac {682}{75} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {4636}{75} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {14 (1-2 x)^{3/2}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {1496 \sqrt {1-2 x} \sqrt {2+3 x}}{15 \sqrt {3+5 x}}+\frac {4636}{75} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {124}{75} \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 = 6.74, size = 131, normalized size = 1.02 \begin {gather*} \frac {-30 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} (1461+2314 x)-4636 \sqrt {2} \left (6+19 x+15 x^2\right ) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+2590 \sqrt {2} \left (6+19 x+15 x^2\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{225 (2+3 x) (3+5 x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 133, normalized size = 1.03
method | result | size |
default | \(\frac {2 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (1023 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-2318 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-69420 x^{2}-9120 x +21915\right )}{225 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(133\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (15-30 x \right ) \left (\frac {487}{75}+\frac {2314 x}{225}\right )}{\sqrt {\left (x^{2}+\frac {19}{15} x +\frac {2}{5}\right ) \left (15-30 x \right )}}-\frac {2918 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{315 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {4636 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{315 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(201\) |
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 = 0.21, size = 40, normalized size = 0.31 \begin {gather*} -\frac {2 \, {\left (2314 \, x + 1461\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{15 \, {\left (15 \, x^{2} + 19 \, x + 6\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 )}^{5/2}}{{\left (3\,x+2\right )}^{3/2}\,{\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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