Optimal. Leaf size=156 \[ \frac {2 \sqrt {2+3 x}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {40 \sqrt {1-2 x} \sqrt {2+3 x}}{363 (3+5 x)^{3/2}}-\frac {490 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 \sqrt {3+5 x}}+\frac {98 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{121 \sqrt {33}}-\frac {16 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{121 \sqrt {33}} \]
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Rubi [A]
time = 0.04, antiderivative size = 156, 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 = {101, 157, 164,
114, 120} \begin {gather*} -\frac {16 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{121 \sqrt {33}}+\frac {98 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{121 \sqrt {33}}-\frac {490 \sqrt {1-2 x} \sqrt {3 x+2}}{3993 \sqrt {5 x+3}}-\frac {40 \sqrt {1-2 x} \sqrt {3 x+2}}{363 (5 x+3)^{3/2}}+\frac {2 \sqrt {3 x+2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 101
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {\sqrt {2+3 x}}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac {2 \sqrt {2+3 x}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {2}{11} \int \frac {-\frac {31}{2}-\frac {45 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {2 \sqrt {2+3 x}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {40 \sqrt {1-2 x} \sqrt {2+3 x}}{363 (3+5 x)^{3/2}}+\frac {4}{363} \int \frac {\frac {121}{4}+30 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {2 \sqrt {2+3 x}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {40 \sqrt {1-2 x} \sqrt {2+3 x}}{363 (3+5 x)^{3/2}}-\frac {490 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 \sqrt {3+5 x}}-\frac {8 \int \frac {\frac {309}{4}+\frac {735 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3993}\\ &=\frac {2 \sqrt {2+3 x}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {40 \sqrt {1-2 x} \sqrt {2+3 x}}{363 (3+5 x)^{3/2}}-\frac {490 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 \sqrt {3+5 x}}+\frac {8}{121} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {98 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1331}\\ &=\frac {2 \sqrt {2+3 x}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {40 \sqrt {1-2 x} \sqrt {2+3 x}}{363 (3+5 x)^{3/2}}-\frac {490 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 \sqrt {3+5 x}}+\frac {98 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{121 \sqrt {33}}-\frac {16 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{121 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 6.32, size = 96, normalized size = 0.62 \begin {gather*} \frac {\sqrt {2} \left (\frac {\sqrt {4+6 x} \left (-592+685 x+2450 x^2\right )}{\sqrt {1-2 x} (3+5 x)^{3/2}}-98 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+362 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{3993} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 215, normalized size = 1.38
method | result | size |
default | \(-\frac {2 \sqrt {2+3 x}\, \sqrt {1-2 x}\, \left (660 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+245 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+396 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+147 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+7350 x^{3}+6955 x^{2}-406 x -1184\right )}{3993 \left (3+5 x \right )^{\frac {3}{2}} \left (6 x^{2}+x -2\right )}\) | \(215\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {4 \left (-30 x^{2}-38 x -12\right )}{1331 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}-\frac {206 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{27951 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {70 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{3993 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1815 \left (x +\frac {3}{5}\right )^{2}}-\frac {86 \left (-30 x^{2}-5 x +10\right )}{3993 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(253\) |
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 = 0.23, size = 50, normalized size = 0.32 \begin {gather*} -\frac {2 \, {\left (2450 \, x^{2} + 685 \, x - 592\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{3993 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\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 {\sqrt {3\,x+2}}{{\left (1-2\,x\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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