Optimal. Leaf size=191 \[ \frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{5/2}}+\frac {138 \sqrt {1-2 x} \sqrt {3+5 x}}{2695 (2+3 x)^{5/2}}+\frac {10308 \sqrt {1-2 x} \sqrt {3+5 x}}{18865 (2+3 x)^{3/2}}+\frac {733812 \sqrt {1-2 x} \sqrt {3+5 x}}{132055 \sqrt {2+3 x}}-\frac {244604 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005}-\frac {7536 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005} \]
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Rubi [A]
time = 0.05, 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 = {106, 157, 164,
114, 120} \begin {gather*} -\frac {7536 \sqrt {\frac {3}{11}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005}-\frac {244604 \sqrt {\frac {3}{11}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005}+\frac {733812 \sqrt {1-2 x} \sqrt {5 x+3}}{132055 \sqrt {3 x+2}}+\frac {10308 \sqrt {1-2 x} \sqrt {5 x+3}}{18865 (3 x+2)^{3/2}}+\frac {138 \sqrt {1-2 x} \sqrt {5 x+3}}{2695 (3 x+2)^{5/2}}+\frac {4 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{5/2}} \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)^{7/2} \sqrt {3+5 x}} \, dx &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {2}{77} \int \frac {-\frac {123}{2}-75 x}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx\\ &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{5/2}}+\frac {138 \sqrt {1-2 x} \sqrt {3+5 x}}{2695 (2+3 x)^{5/2}}-\frac {4 \int \frac {-\frac {1887}{2}+\frac {1035 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{2695}\\ &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{5/2}}+\frac {138 \sqrt {1-2 x} \sqrt {3+5 x}}{2695 (2+3 x)^{5/2}}+\frac {10308 \sqrt {1-2 x} \sqrt {3+5 x}}{18865 (2+3 x)^{3/2}}-\frac {8 \int \frac {-\frac {131913}{4}+\frac {38655 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{56595}\\ &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{5/2}}+\frac {138 \sqrt {1-2 x} \sqrt {3+5 x}}{2695 (2+3 x)^{5/2}}+\frac {10308 \sqrt {1-2 x} \sqrt {3+5 x}}{18865 (2+3 x)^{3/2}}+\frac {733812 \sqrt {1-2 x} \sqrt {3+5 x}}{132055 \sqrt {2+3 x}}-\frac {16 \int \frac {-\frac {1744335}{4}-\frac {2751795 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{396165}\\ &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{5/2}}+\frac {138 \sqrt {1-2 x} \sqrt {3+5 x}}{2695 (2+3 x)^{5/2}}+\frac {10308 \sqrt {1-2 x} \sqrt {3+5 x}}{18865 (2+3 x)^{3/2}}+\frac {733812 \sqrt {1-2 x} \sqrt {3+5 x}}{132055 \sqrt {2+3 x}}+\frac {11304 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{12005}+\frac {733812 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{132055}\\ &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{5/2}}+\frac {138 \sqrt {1-2 x} \sqrt {3+5 x}}{2695 (2+3 x)^{5/2}}+\frac {10308 \sqrt {1-2 x} \sqrt {3+5 x}}{18865 (2+3 x)^{3/2}}+\frac {733812 \sqrt {1-2 x} \sqrt {3+5 x}}{132055 \sqrt {2+3 x}}-\frac {244604 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005}-\frac {7536 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005}\\ \end {align*}
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Mathematica [A]
time = 6.96, size = 106, normalized size = 0.55 \begin {gather*} \frac {4 \left (\frac {\sqrt {3+5 x} \left (1546591+1424784 x-5720058 x^2-6604308 x^3\right )}{2 \sqrt {1-2 x} (2+3 x)^{5/2}}+\sqrt {2} \left (61151 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-30065 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{132055} \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 | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {16 \left (-30 x^{2}-38 x -12\right )}{26411 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {155052 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{184877 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {244604 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{184877 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{735 \left (\frac {2}{3}+x \right )^{3}}+\frac {316 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{5145 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {133464}{2401} x^{2}-\frac {66732}{12005} x +\frac {200196}{12005}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(277\) |
default | \(\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (1100718 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-559548 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1467624 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-746064 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+489208 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-248688 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+33021540 x^{4}+48413214 x^{3}+10036254 x^{2}-12007307 x -4639773\right )}{132055 \left (2+3 x \right )^{\frac {5}{2}} \left (10 x^{2}+x -3\right )}\) | \(308\) |
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 = 0.22, size = 60, normalized size = 0.31 \begin {gather*} \frac {2 \, {\left (6604308 \, x^{3} + 5720058 \, x^{2} - 1424784 \, x - 1546591\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{132055 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\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 {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{7/2}\,\sqrt {5\,x+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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