Optimal. Leaf size=129 \[ \frac {7 \sqrt {2+3 x}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {37 \sqrt {1-2 x} \sqrt {2+3 x}}{121 \sqrt {3+5 x}}+\frac {37}{55} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {2}{55} \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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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, 157, 164,
114, 120} \begin {gather*} -\frac {2}{55} \sqrt {\frac {3}{11}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {37}{55} \sqrt {\frac {3}{11}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {37 \sqrt {1-2 x} \sqrt {3 x+2}}{121 \sqrt {5 x+3}}+\frac {7 \sqrt {3 x+2}}{11 \sqrt {1-2 x} \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 {(2+3 x)^{3/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac {7 \sqrt {2+3 x}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1}{11} \int \frac {-\frac {11}{2}-3 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {7 \sqrt {2+3 x}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {37 \sqrt {1-2 x} \sqrt {2+3 x}}{121 \sqrt {3+5 x}}+\frac {2}{121} \int \frac {-30-\frac {111 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {7 \sqrt {2+3 x}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {37 \sqrt {1-2 x} \sqrt {2+3 x}}{121 \sqrt {3+5 x}}+\frac {3}{55} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {111}{605} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {7 \sqrt {2+3 x}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {37 \sqrt {1-2 x} \sqrt {2+3 x}}{121 \sqrt {3+5 x}}+\frac {37}{55} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {2}{55} \sqrt {\frac {3}{11}} 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 = 6.01, size = 122, normalized size = 0.95 \begin {gather*} \frac {10 \sqrt {2+3 x} \sqrt {3+5 x} (20+37 x)-37 \sqrt {2-4 x} (3+5 x) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+70 \sqrt {2-4 x} (3+5 x) F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{605 \sqrt {1-2 x} (3+5 x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 133, normalized size = 1.03
method | result | size |
default | \(-\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (33 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+37 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+1110 x^{2}+1340 x +400\right )}{605 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(133\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-20-30 x \right ) \left (\frac {2}{121}+\frac {37 x}{1210}\right )}{\sqrt {\left (x^{2}+\frac {1}{10} x -\frac {3}{10}\right ) \left (-20-30 x \right )}}-\frac {20 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{847 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {37 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{847 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(201\) |
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 = 0.26, size = 38, normalized size = 0.29 \begin {gather*} -\frac {2 \, {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{121 \, {\left (10 \, x^{2} + x - 3\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 (3\,x+2\right )}^{3/2}}{{\left (1-2\,x\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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