Optimal. Leaf size=160 \[ \frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {97 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)^{3/2}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{1029 \sqrt {2+3 x}}+\frac {458 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1029}-\frac {178 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1029} \]
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Rubi [A]
time = 0.04, antiderivative size = 160, 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, 157, 164,
114, 120} \begin {gather*} -\frac {178 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1029}+\frac {458 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1029}-\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{1029 \sqrt {3 x+2}}-\frac {97 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{3/2}}+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{3/2}} \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 {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^{5/2}} \, dx &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {1}{7} \int \frac {-\frac {181}{2}-160 x}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {97 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)^{3/2}}-\frac {2}{147} \int \frac {-\frac {247}{2}-\frac {485 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {97 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)^{3/2}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{1029 \sqrt {2+3 x}}-\frac {4 \int \frac {\frac {395}{4}+\frac {1145 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1029}\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {97 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)^{3/2}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{1029 \sqrt {2+3 x}}-\frac {458 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1029}+\frac {979 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1029}\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {97 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)^{3/2}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{1029 \sqrt {2+3 x}}+\frac {458 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1029}-\frac {178 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1029}\\ \end {align*}
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Mathematica [A]
time = 7.67, size = 97, normalized size = 0.61 \begin {gather*} \frac {\sqrt {2} \left (\frac {3 \sqrt {6+10 x} \left (11+908 x+1374 x^2\right )}{\sqrt {1-2 x} (2+3 x)^{3/2}}-458 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+3395 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{3087} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 215, normalized size = 1.34
method | result | size |
default | \(-\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (8811 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1374 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+5874 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+916 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+41220 x^{3}+51972 x^{2}+16674 x +198\right )}{3087 \left (2+3 x \right )^{\frac {3}{2}} \left (10 x^{2}+x -3\right )}\) | \(215\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {22 \left (-30 x^{2}-38 x -12\right )}{343 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}-\frac {395 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{21609 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2290 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{21609 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {260 \left (-30 x^{2}-3 x +9\right )}{3087 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1323 \left (\frac {2}{3}+x \right )^{2}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(253\) |
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 = 0.22, size = 50, normalized size = 0.31 \begin {gather*} -\frac {2 \, {\left (1374 \, x^{2} + 908 \, x + 11\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1029 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\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 (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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