Optimal. Leaf size=191 \[ \frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {44 \sqrt {1-2 x}}{5 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {6116 \sqrt {1-2 x}}{35 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {36968 \sqrt {1-2 x} \sqrt {2+3 x}}{21 \sqrt {3+5 x}}+\frac {36968}{35} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {1112}{35} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 191, 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 = {100, 157, 164,
114, 120} \begin {gather*} \frac {1112}{35} \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {36968}{35} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {36968 \sqrt {1-2 x} \sqrt {3 x+2}}{21 \sqrt {5 x+3}}+\frac {6116 \sqrt {1-2 x}}{35 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {44 \sqrt {1-2 x}}{5 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {14 \sqrt {1-2 x}}{15 (3 x+2)^{5/2} \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)^{7/2} (3+5 x)^{3/2}} \, dx &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {2}{15} \int \frac {121-165 x}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {44 \sqrt {1-2 x}}{5 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {4}{315} \int \frac {\frac {18249}{2}-10395 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {44 \sqrt {1-2 x}}{5 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {6116 \sqrt {1-2 x}}{35 \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {8 \int \frac {389235-\frac {481635 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{2205}\\ &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {44 \sqrt {1-2 x}}{5 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {6116 \sqrt {1-2 x}}{35 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {36968 \sqrt {1-2 x} \sqrt {2+3 x}}{21 \sqrt {3+5 x}}-\frac {16 \int \frac {\frac {20273715}{4}+\frac {16011765 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{24255}\\ &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {44 \sqrt {1-2 x}}{5 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {6116 \sqrt {1-2 x}}{35 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {36968 \sqrt {1-2 x} \sqrt {2+3 x}}{21 \sqrt {3+5 x}}-\frac {6116}{35} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {36968}{35} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {44 \sqrt {1-2 x}}{5 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {6116 \sqrt {1-2 x}}{35 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {36968 \sqrt {1-2 x} \sqrt {2+3 x}}{21 \sqrt {3+5 x}}+\frac {36968}{35} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {1112}{35} \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 = 7.32, size = 105, normalized size = 0.55 \begin {gather*} \frac {2}{105} \left (-\frac {3 \sqrt {1-2 x} \left (233897+1071882 x+1636038 x^2+831780 x^3\right )}{(2+3 x)^{5/2} \sqrt {3+5 x}}-2 \sqrt {2} \left (9242 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-4655 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right ) \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. \(307\) vs.
\(2(139)=278\).
time = 0.10, size = 308, normalized size = 1.61
method | result | size |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{135 \left (\frac {2}{3}+x \right )^{3}}-\frac {202 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{45 \left (\frac {2}{3}+x \right )^{2}}-\frac {25418 \left (-30 x^{2}-3 x +9\right )}{105 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {23404 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{147 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {36968 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{147 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {110 \left (-30 x^{2}-5 x +10\right )}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(277\) |
default | \(\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (82566 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-166356 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+110088 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-221808 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+36696 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-73936 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-4990680 x^{4}-7320888 x^{3}-1523178 x^{2}+1812264 x +701691\right )}{105 \left (2+3 x \right )^{\frac {5}{2}} \left (10 x^{2}+x -3\right )}\) | \(308\) |
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 = 0.25, size = 60, normalized size = 0.31 \begin {gather*} -\frac {2 \, {\left (831780 \, x^{3} + 1636038 \, x^{2} + 1071882 \, x + 233897\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{35 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 )}^{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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