Optimal. Leaf size=156 \[ \frac {2129 \sqrt {1-2 x} \sqrt {2+3 x}}{19965 \sqrt {3+5 x}}-\frac {434 (2+3 x)^{3/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {148831 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6050 \sqrt {33}}-\frac {2252 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3025 \sqrt {33}} \]
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Rubi [A]
time = 0.03, 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 = {100, 155, 164,
114, 120} \begin {gather*} -\frac {2252 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3025 \sqrt {33}}-\frac {148831 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6050 \sqrt {33}}+\frac {7 (3 x+2)^{5/2}}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}-\frac {434 (3 x+2)^{3/2}}{363 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {2129 \sqrt {1-2 x} \sqrt {3 x+2}}{19965 \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 {(2+3 x)^{7/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac {7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {1}{33} \int \frac {(2+3 x)^{3/2} \left (\frac {233}{2}+201 x\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=-\frac {434 (2+3 x)^{3/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {1}{363} \int \frac {\left (-3730-\frac {13143 x}{2}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {2129 \sqrt {1-2 x} \sqrt {2+3 x}}{19965 \sqrt {3+5 x}}-\frac {434 (2+3 x)^{3/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {2 \int \frac {-\frac {282759}{4}-\frac {446493 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{19965}\\ &=\frac {2129 \sqrt {1-2 x} \sqrt {2+3 x}}{19965 \sqrt {3+5 x}}-\frac {434 (2+3 x)^{3/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {1126 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3025}+\frac {148831 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{66550}\\ &=\frac {2129 \sqrt {1-2 x} \sqrt {2+3 x}}{19965 \sqrt {3+5 x}}-\frac {434 (2+3 x)^{3/2}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {7 (2+3 x)^{5/2}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {148831 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6050 \sqrt {33}}-\frac {2252 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3025 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 8.46, size = 97, normalized size = 0.62 \begin {gather*} \frac {\frac {5 \sqrt {4+6 x} \left (-28671+66174 x+189851 x^2\right )}{(1-2 x)^{3/2} \sqrt {3+5 x}}+148831 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-74515 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{99825 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 224, normalized size = 1.44
method | result | size |
default | \(-\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (148632 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-297662 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-74316 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+148831 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-5695530 x^{3}-5782240 x^{2}-463350 x +573420\right )}{199650 \left (15 x^{2}+19 x +6\right ) \left (-1+2 x \right )^{2}}\) | \(224\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {-\frac {37975}{2662} x^{2}-\frac {144305}{7986} x -\frac {7595}{1331}}{\sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {94253 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{279510 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {148831 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{279510 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {343 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2904 \left (-\frac {1}{2}+x \right )^{2}}-\frac {2 \left (-30 x^{2}-5 x +10\right )}{33275 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(253\) |
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 = 0.24, size = 50, normalized size = 0.32 \begin {gather*} \frac {{\left (189851 \, x^{2} + 66174 \, x - 28671\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{19965 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\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 (3\,x+2\right )}^{7/2}}{{\left (1-2\,x\right )}^{5/2}\,{\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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